User:BeyondNormality/sandbox

Definitions

Probability Density Function

Relations Between Distribution Properties

Derive Using Formula PDF PDF ${\text{identical properties}}$ PDF CDF ${\text{PDF}}[{\mathcal {D}},x]={\frac {\partial {\text{CDF}}[{\mathcal {D}},x]}{\partial x}}$ PDF Characteristic Function ${\text{PDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},t]}{e^{(it)x}}}\,dt}{2\pi }}$ PDF Characteristic Function ${\text{PDF}}[{\mathcal {D}},x]={\frac {{\mathcal {F}}_{t}[{\text{CharacteristicFunction}}[{\mathcal {D}},t]](-x)}{\sqrt {2\pi }}}$ PDF Survival Function ${\text{PDF}}[{\mathcal {D}},x]=-{\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},x]}{\partial x}}$ PDF Cumulative Hazard Function ${\text{PDF}}[{\mathcal {D}},x]={\frac {\frac {\partial {\text{CumulativeHazardFunction}}({\mathcal {D}},x)}{\partial x}}{e^{{\text{CumulativeHazardFunction}}({\mathcal {D}},x)}}}$ PDF Hazard Function ${\text{PDF}}[{\mathcal {D}},x]={\text{HazardFunction}}[{\mathcal {D}},x]\exp \left(-\int _{-\infty }^{x}{\text{HazardFunction}}[{\mathcal {D}},t]\,dt\right)$ PDF Inverse CDF ${\text{PDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}e^{(i({\text{InverseCDF}}[{\mathcal {D}},\tau ]-x))t}d\tau dt}{2\pi }}$ PDF Inverse CDF ${\text{PDF}}[{\mathcal {D}},x]={\frac {\int _{0}^{1}{\mathcal {F}}_{t}[1]({\text{InverseCDF}}[{\mathcal {D}},\tau ]-x)\,d\tau }{\sqrt {2\pi }}}$ PDF Inverse CDF ${\text{PDF}}[{\mathcal {D}},x]=\int _{0}^{1}\delta ({\text{InverseCDF}}[{\mathcal {D}},\tau ]-x)\,d\tau$ PDF Quantile ${\text{PDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}e^{(i({\text{Quantile}}[{\mathcal {D}},\tau ]-x))t}d\tau dt}{2\pi }}$ PDF Quantile ${\text{PDF}}[{\mathcal {D}},x]={\frac {\int _{0}^{1}{\mathcal {F}}_{t}[1]({\text{Quantile}}[{\mathcal {D}},\tau ]-x)\,d\tau }{\sqrt {2\pi }}}$ PDF Quantile ${\text{PDF}}[{\mathcal {D}},x]=\int _{0}^{1}\delta ({\text{Quantile}}[{\mathcal {D}},\tau ]-x)\,d\tau$ PDF Inverse Survival Function ${\text{PDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}e^{(i({\text{InverseSurvivalFunction}}[{\mathcal {D}},\tau ]-x))t}d\tau dt}{2\pi }}$ PDF Inverse Survival Function ${\text{PDF}}[{\mathcal {D}},x]={\frac {\int _{0}^{1}{\mathcal {F}}_{t}[1]({\text{InverseSurvivalFunction}}[{\mathcal {D}},\tau ]-x)\,d\tau }{\sqrt {2\pi }}}$ PDF Inverse Survival Function ${\text{PDF}}[{\mathcal {D}},x]=\int _{0}^{1}\delta ({\text{InverseSurvivalFunction}}[{\mathcal {D}},\tau ]-x)\,d\tau$ PDF Sparsity ${\text{PDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}\exp \left(\left(i\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{\tau }{\text{Sparsity}}({\mathcal {D}},\xi )\,d\xi -x\right)\right)t\right)d\tau dt}{2\pi }}$ PDF Sparsity ${\text{PDF}}[{\mathcal {D}},x]={\frac {\int _{0}^{1}{\mathcal {F}}_{t}[1]\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{\tau }{\text{Sparsity}}({\mathcal {D}},\xi )\,d\xi -x\right)\,d\tau }{\sqrt {2\pi }}}$ PDF Sparsity ${\text{PDF}}[{\mathcal {D}},x]=\int _{0}^{1}\delta \left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{\tau }{\text{Sparsity}}({\mathcal {D}},\xi )\,d\xi -x\right)\,d\tau$ PDF Lorenz Curve 1 ${\text{PDF}}[{\mathcal {D}},x]={\frac {{\text{Mean}}[{\mathcal {D}}]{\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},x)}{\partial x}}}{x}}$ PDF Lorenz Curve 2 ${\text{PDF}}[{\mathcal {D}},x]={\frac {{\text{Mean}}[{\mathcal {D}}]{\frac {\partial {\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},x])}{\partial x}}}{x}}$ PDF Mean Residual Life Function ${\text{PDF}}[{\mathcal {D}},x]={\frac {\partial ^{2}({\text{MeanResidualLifeFunction}}({\mathcal {D}},x){\text{SurvivalFunction}}[{\mathcal {D}},x])}{\partial x^{2}}}$ PDF Moment ${\text{PDF}}[{\mathcal {D}},x]=\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right){\text{Moment}}[{\mathcal {D}},k]}{k!}}$ PDF Central Moment ${\text{PDF}}[{\mathcal {D}},x]=\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right)\sum _{j=0}^{k}{\text{CentralMoment}}[{\mathcal {D}},j]\left({\binom {k}{j}}{\text{Mean}}[{\mathcal {D}}]^{k-j}\right)}{k!}}$ PDF Factorial Moment ${\text{PDF}}[{\mathcal {D}},x]=\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right)\sum _{j=0}^{k}{\mathcal {S}}_{k}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{k!}}$ PDF Ascending Factorial Moment ${\text{PDF}}[{\mathcal {D}},x]=\sum _{j=0}^{\infty }\left((-1)^{j}{\text{AscendingFactorialMoment}}({\mathcal {D}},j)\right)\sum _{k=j}^{\infty }{\frac {{\mathcal {S}}_{k}^{(j)}\delta ^{(k)}(x)}{k!}}$ PDF Cumulant ${\text{PDF}}[{\mathcal {D}},x]=\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right){\text{BellY}}[k,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k\}]]}{k!}}+\delta (x)$ PDF Factorial Cumulant ${\text{PDF}}[{\mathcal {D}},x]=\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right){\text{BellY}}\left[k,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j}{\mathcal {S}}_{j}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k\}\right]\right]}{k!}}+\delta (x)$ PDF Ascending Factorial Cumulant ${\text{PDF}}[{\mathcal {D}},x]=\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right){\text{BellY}}\left[k,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k\}\right]\right]}{k!}}+\delta (x)$ PDF Moment Generating Function ${\text{PDF}}[{\mathcal {D}},x]={\frac {\int _{-i\infty }^{i\infty }{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},t]}{e^{tx}}}\,dt}{(2\pi )i}}$ PDF Moment Generating Function ${\text{PDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},it]}{e^{(it)x}}}\,dt}{2\pi }}$ PDF Moment Generating Function ${\text{PDF}}[{\mathcal {D}},x]={\frac {{\mathcal {F}}_{t}[{\text{MomentGeneratingFunction}}[{\mathcal {D}},it]](-x)}{\sqrt {2\pi }}}$ PDF Central Moment Generating Function ${\text{PDF}}[{\mathcal {D}},x]={\frac {\int _{-i\infty }^{i\infty }e^{t({\text{Mean}}[{\mathcal {D}}]-x)}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},t]\,dt}{(2\pi )i}}$ PDF Central Moment Generating Function ${\text{PDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }e^{(it)({\text{Mean}}[{\mathcal {D}}]-x)}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},it]\,dt}{2\pi }}$ PDF Central Moment Generating Function ${\text{PDF}}[{\mathcal {D}},x]={\frac {{\mathcal {F}}_{t}\left[e^{(it){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},it]\right](-x)}{\sqrt {2\pi }}}$ PDF Cumulant Generating Function ${\text{PDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }e^{{\text{CumulantGeneratingFunction}}[{\mathcal {D}},it]-(it)x}\,dt}{2\pi }}$ PDF Cumulant Generating Function ${\text{PDF}}[{\mathcal {D}},x]={\frac {{\mathcal {F}}_{t}\left[e^{{\text{CumulantGeneratingFunction}}[{\mathcal {D}},it]}\right](-x)}{\sqrt {2\pi }}}$

Cumulative Distribution Function

Relations Between Distribution Properties

Derive Using Formula CDF PDF ${\text{CDF}}[{\mathcal {D}},x]=\int _{-\infty }^{x}{\text{PDF}}[{\mathcal {D}},t]\,dt$ CDF PDF ${\text{CDF}}[{\mathcal {D}},x]=\int _{-\infty }^{\infty }{\text{Boole}}[t\leq x]{\text{PDF}}[{\mathcal {D}},t]\,dt$ CDF PDF ${\text{CDF}}[{\mathcal {D}},x]=\int _{-\infty }^{\infty }\theta (x-t){\text{PDF}}[{\mathcal {D}},t]\,dt$ CDF CDF ${\text{identical properties}}$ CDF Characteristic Function ${\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{e^{(it)\tau }}}d\tau dt}{2\pi }}$ CDF Characteristic Function ${\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}{\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-t)\,dt}{\sqrt {2\pi }}}$ CDF Characteristic Function ${\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }{\text{Boole}}[t\leq x]\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-t)\right)\,dt}{\sqrt {2\pi }}}$ CDF Characteristic Function ${\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }\theta (x-t)\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-t)\right)\,dt}{\sqrt {2\pi }}}$ CDF Characteristic Function ${\text{CDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ CDF Characteristic Function ${\text{CDF}}[{\mathcal {D}},x]={\frac {1}{2}}-{\frac {\int _{0}^{\infty }{\frac {\Im \left({\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},t]}{e^{(it)x}}}\right)}{t}}\,dt}{\pi }}$ CDF Survival Function ${\text{CDF}}[{\mathcal {D}},x]=1-{\text{SurvivalFunction}}[{\mathcal {D}},x]$ CDF Cumulative Hazard Function ${\text{CDF}}[{\mathcal {D}},x]=1-\exp(-{\text{CumulativeHazardFunction}}({\mathcal {D}},x))$ CDF Hazard Function ${\text{CDF}}[{\mathcal {D}},x]=1-\exp \left(-\int _{-\infty }^{x}{\text{HazardFunction}}[{\mathcal {D}},t]\,dt\right)$ CDF Inverse CDF ${\text{CDF}}[{\mathcal {D}},x]={\text{InverseFunction}}[t,{\text{InverseCDF}}[{\mathcal {D}},t]][x]$ CDF Inverse CDF ${\text{CDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{\tau (i{\text{InverseCDF}}[{\mathcal {D}},t])}\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ CDF Quantile ${\text{CDF}}[{\mathcal {D}},x]={\text{InverseFunction}}[t,{\text{Quantile}}[{\mathcal {D}},t]][x]$ CDF Quantile ${\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }{\frac {\int _{0}^{1}e^{(i{\text{Quantile}}[{\mathcal {D}},\xi ])\tau }\,d\xi }{e^{(it)\tau }}}d\tau dt}{2\pi }}$ CDF Quantile ${\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}{\mathcal {F}}_{\tau }\left[\int _{0}^{1}e^{(i{\text{Quantile}}[{\mathcal {D}},\xi ])\tau }\,d\xi \right](-t)\,dt}{\sqrt {2\pi }}}$ CDF Quantile ${\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }{\text{Boole}}[t\leq x]\left({\mathcal {F}}_{\tau }\left[\int _{0}^{1}e^{(i{\text{Quantile}}[{\mathcal {D}},\xi ])\tau }\,d\xi \right](-t)\right)\,dt}{\sqrt {2\pi }}}$ CDF Quantile ${\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }\theta (x-t)\left({\mathcal {F}}_{\tau }\left[\int _{0}^{1}e^{(i{\text{Quantile}}[{\mathcal {D}},\xi ])\tau }\,d\xi \right](-t)\right)\,dt}{\sqrt {2\pi }}}$ CDF Quantile ${\text{CDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{(i{\text{Quantile}}[{\mathcal {D}},t])\tau }\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ CDF Inverse Survival Function ${\text{CDF}}[{\mathcal {D}},x]={\text{InverseFunction}}[t,{\text{InverseSurvivalFunction}}[{\mathcal {D}},1-t]][x]$ CDF Inverse Survival Function ${\text{CDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{\tau (i{\text{InverseSurvivalFunction}}[{\mathcal {D}},t])}\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ CDF Sparsity ${\text{CDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{\tau }{\text{Sparsity}}({\mathcal {D}},\tau )\,d\tau \right][x]$ CDF Sparsity ${\text{CDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{\left(i\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{t}{\text{Sparsity}}({\mathcal {D}},s)\,ds\right)\right)\tau }\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ CDF Lorenz Curve 1 ${\text{CDF}}[{\mathcal {D}},x]={\text{Mean}}[{\mathcal {D}}]\int _{-\infty }^{x}{\frac {\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},t)}{\partial t}}{t}}\,dt$ CDF Lorenz Curve 1 ${\text{CDF}}[{\mathcal {D}},x]={\text{Mean}}[{\mathcal {D}}]\left(\int _{-\infty }^{x}{\frac {{\text{LorenzCurve1}}({\mathcal {D}},t)}{t^{2}}}\,dt+{\frac {{\text{LorenzCurve1}}({\mathcal {D}},x)}{x}}\right)$ CDF Lorenz Curve 2 ${\text{CDF}}[{\mathcal {D}},x]={\text{Mean}}[{\mathcal {D}}]\int _{-\infty }^{x}{\frac {\frac {\partial {\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},t])}{\partial t}}{t}}\,dt$ CDF Lorenz Curve 2 ${\text{CDF}}[{\mathcal {D}},x]={\text{Mean}}[{\mathcal {D}}]\left(\int _{-\infty }^{x}{\frac {{\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},t])}{t^{2}}}\,dt+{\frac {{\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},x])}{x}}\right)$ CDF Mean Residual Life Function $\left({\begin{array}{c}{\text{SurvivalFunction}}[{\mathcal {D}},x]=\exp \left(-\int _{-\infty }^{x}{\frac {1+{\frac {\partial {\text{MeanResidualLifeFunction}}({\mathcal {D}},t)}{\partial t}}}{{\text{MeanResidualLifeFunction}}({\mathcal {D}},t)}}\,dt\right)\\{\text{CDF}}[{\mathcal {D}},x]=1+{\frac {\partial ({\text{SurvivalFunction}}[{\mathcal {D}},x]{\text{MeanResidualLifeFunction}}({\mathcal {D}},x))}{\partial x}}\\\end{array}}\right)$ CDF Moment ${\text{CDF}}[{\mathcal {D}},x]=\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right){\text{Moment}}[{\mathcal {D}},k+1]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(x)+1)$ CDF Central Moment ${\text{CDF}}[{\mathcal {D}},x]=\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\sum _{j=0}^{k+1}{\text{CentralMoment}}[{\mathcal {D}},j]\left({\binom {k+1}{j}}{\text{Mean}}[{\mathcal {D}}]^{-j+k+1}\right)}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(x)+1)$ CDF Factorial Moment ${\text{CDF}}[{\mathcal {D}},x]=\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(x)+1)$ CDF Ascending Factorial Moment ${\text{CDF}}[{\mathcal {D}},x]=\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\sum _{j=0}^{k+1}\left((-1)^{-j+k-1}{\mathcal {S}}_{k+1}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(x)+1)$ CDF Cumulant ${\text{CDF}}[{\mathcal {D}},x]=\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right){\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(x)+1)$ CDF Factorial Cumulant ${\text{CDF}}[{\mathcal {D}},x]=\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j}{\mathcal {S}}_{j}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(x)+1)$ CDF Ascending Factorial Cumulant ${\text{CDF}}[{\mathcal {D}},x]=\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(x)+1)$ CDF Moment Generating Function ${\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-i\infty }^{i\infty }{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},\tau ]}{e^{t\tau }}}d\tau dt}{(2\pi )i}}$ CDF Moment Generating Function ${\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}{\mathcal {F}}_{\tau }[{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]](-t)\,dt}{\sqrt {2\pi }}}$ CDF Moment Generating Function ${\text{CDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ CDF Central Moment Generating Function ${\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }e^{(i({\text{Mean}}[{\mathcal {D}}]-t))\tau }{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]d\tau dt}{2\pi }}$ CDF Central Moment Generating Function ${\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}{\mathcal {F}}_{\tau }\left[e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]\right](-t)\,dt}{\sqrt {2\pi }}}$ CDF Central Moment Generating Function ${\text{CDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ CDF Cumulant Generating Function ${\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }e^{{\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ]-(it)\tau }d\tau dt}{2\pi }}$ CDF Cumulant Generating Function ${\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}{\mathcal {F}}_{\tau }[\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])](-t)\,dt}{\sqrt {2\pi }}}$ CDF Cumulant Generating Function ${\text{CDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$

Characteristic Function

Relations Between Distribution Properties

Derive Using Formula Characteristic Function PDF ${\text{CharacteristicFunction}}[{\mathcal {D}},x]=\int _{-\infty }^{\infty }e^{(it)x}{\text{PDF}}[{\mathcal {D}},t]\,dt$ Characteristic Function PDF ${\text{CharacteristicFunction}}[{\mathcal {D}},x]={\sqrt {2\pi }}\left({\mathcal {F}}_{t}[{\text{PDF}}[{\mathcal {D}},t]](x)\right)$ Characteristic Function CDF ${\text{CharacteristicFunction}}[{\mathcal {D}},x]=\int _{-\infty }^{\infty }e^{(it)x}{\frac {\partial {\text{CDF}}[{\mathcal {D}},t]}{\partial t}}\,dt$ Characteristic Function CDF ${\text{CharacteristicFunction}}[{\mathcal {D}},x]={\sqrt {2\pi }}\left({\mathcal {F}}_{t}\left[{\frac {\partial {\text{CDF}}[{\mathcal {D}},t]}{\partial t}}\right](x)\right)$ Characteristic Function Characteristic Function ${\text{identical properties}}$ Characteristic Function Survival Function ${\text{CharacteristicFunction}}[{\mathcal {D}},x]=-\int _{-\infty }^{\infty }e^{(it)x}{\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},t]}{\partial t}}\,dt$ Characteristic Function Survival Function ${\text{CharacteristicFunction}}[{\mathcal {D}},x]=-{\sqrt {2\pi }}\left({\mathcal {F}}_{t}\left[{\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},t]}{\partial t}}\right](x)\right)$ Characteristic Function Cumulative Hazard Function ${\text{CharacteristicFunction}}[{\mathcal {D}},x]=\int _{-\infty }^{\infty }{\frac {\partial {\text{CumulativeHazardFunction}}({\mathcal {D}},t)}{\partial t}}e^{-{\text{CumulativeHazardFunction}}({\mathcal {D}},t)+(it)x}\,dt$ Characteristic Function Cumulative Hazard Function ${\text{CharacteristicFunction}}[{\mathcal {D}},x]={\sqrt {2\pi }}\left({\mathcal {F}}_{t}\left[{\frac {\frac {\partial {\text{CumulativeHazardFunction}}({\mathcal {D}},t)}{\partial t}}{e^{{\text{CumulativeHazardFunction}}({\mathcal {D}},t)}}}\right](x)\right)$ Characteristic Function Hazard Function ${\text{CharacteristicFunction}}[{\mathcal {D}},x]=\int _{-\infty }^{\infty }\exp \left((it)x-\int _{-\infty }^{t}{\text{HazardFunction}}[{\mathcal {D}},\tau ]\,d\tau \right){\text{HazardFunction}}[{\mathcal {D}},t]\,dt$ Characteristic Function Hazard Function ${\text{CharacteristicFunction}}[{\mathcal {D}},x]={\sqrt {2\pi }}\left({\mathcal {F}}_{t}\left[{\text{HazardFunction}}[{\mathcal {D}},t]\exp \left(-\int _{-\infty }^{t}{\text{HazardFunction}}[{\mathcal {D}},\tau ]\,d\tau \right)\right](x)\right)$ Characteristic Function Inverse CDF ${\text{CharacteristicFunction}}[{\mathcal {D}},x]=\int _{0}^{1}e^{x(i{\text{InverseCDF}}[{\mathcal {D}},t])}\,dt$ Characteristic Function Quantile ${\text{CharacteristicFunction}}[{\mathcal {D}},x]=\int _{0}^{1}e^{x(i{\text{Quantile}}[{\mathcal {D}},t])}\,dt$ Characteristic Function Inverse Survival Function ${\text{CharacteristicFunction}}[{\mathcal {D}},x]=\int _{0}^{1}e^{x(i{\text{InverseSurvivalFunction}}[{\mathcal {D}},t])}\,dt$ Characteristic Function Sparsity ${\text{CharacteristicFunction}}[{\mathcal {D}},x]=\int _{0}^{1}e^{(ix)\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{t}{\text{Sparsity}}({\mathcal {D}},s)\,ds\right)}\,dt$ Characteristic Function Lorenz Curve 1 ${\text{CharacteristicFunction}}[{\mathcal {D}},x]={\text{Mean}}[{\mathcal {D}}]\int _{-\infty }^{\infty }{\frac {e^{(it)x}{\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},t)}{\partial t}}}{t}}\,dt$ Characteristic Function Lorenz Curve 1 ${\text{CharacteristicFunction}}[{\mathcal {D}},x]=\left({\sqrt {2\pi }}{\text{Mean}}[{\mathcal {D}}]\right)\left({\mathcal {F}}_{t}\left[{\frac {\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},t)}{\partial t}}{t}}\right](x)\right)$ Characteristic Function Lorenz Curve 2 ${\text{CharacteristicFunction}}[{\mathcal {D}},x]={\text{Mean}}[{\mathcal {D}}]\int _{-\infty }^{\infty }{\frac {e^{(it)x}{\frac {\partial {\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},t])}{\partial t}}}{t}}\,dt$ Characteristic Function Lorenz Curve 2 ${\text{CharacteristicFunction}}[{\mathcal {D}},x]=\left({\sqrt {2\pi }}{\text{Mean}}[{\mathcal {D}}]\right)\left({\mathcal {F}}_{t}\left[{\frac {\frac {\partial {\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},t])}{\partial t}}{t}}\right](x)\right)$ Characteristic Function Mean Residual Life Function ${\text{CharacteristicFunction}}[{\mathcal {D}},x]=\int _{-\infty }^{\infty }e^{(it)x}{\frac {\partial ^{2}({\text{SurvivalFunction}}[{\mathcal {D}},t]{\text{MeanResidualLifeFunction}}({\mathcal {D}},t))}{\partial t^{2}}}\,dt$ Characteristic Function Mean Residual Life Function ${\text{CharacteristicFunction}}[{\mathcal {D}},x]={\sqrt {2\pi }}\left({\mathcal {F}}_{t}\left[{\frac {\partial ^{2}({\text{MeanResidualLifeFunction}}({\mathcal {D}},t){\text{SurvivalFunction}}[{\mathcal {D}},t])}{\partial t^{2}}}\right](x)\right)$ Characteristic Function Mean Residual Life Function ${\text{CharacteristicFunction}}[{\mathcal {D}},x]=\left(-{\sqrt {2\pi }}x^{2}\right)\left({\mathcal {F}}_{t}[{\text{SurvivalFunction}}[{\mathcal {D}},t]{\text{MeanResidualLifeFunction}}({\mathcal {D}},t)](x)\right)$ Characteristic Function Moment ${\text{CharacteristicFunction}}[{\mathcal {D}},x]=\sum _{k=0}^{\infty }{\frac {(ix)^{k}{\text{Moment}}[{\mathcal {D}},k]}{k!}}$ Characteristic Function Central Moment ${\text{CharacteristicFunction}}[{\mathcal {D}},x]=e^{(ix){\text{Mean}}[{\mathcal {D}}]}\sum _{k=0}^{\infty }{\frac {(ix)^{k}{\text{CentralMoment}}[{\mathcal {D}},k]}{k!}}$ Characteristic Function Factorial Moment ${\text{CharacteristicFunction}}[{\mathcal {D}},x]=\sum _{k=0}^{\infty }{\frac {\left(-1+e^{ix}\right)^{k}{\text{FactorialMoment}}[{\mathcal {D}},k]}{k!}}$ Characteristic Function Ascending Factorial Moment ${\text{CharacteristicFunction}}[{\mathcal {D}},x]=\sum _{k=0}^{\infty }{\frac {\left(1-e^{-ix}\right)^{k}{\text{AscendingFactorialMoment}}({\mathcal {D}},k)}{k!}}$ Characteristic Function Cumulant ${\text{CharacteristicFunction}}[{\mathcal {D}},x]=\exp \left(\sum _{k=0}^{\infty }{\frac {(ix)^{k}{\text{Cumulant}}[{\mathcal {D}},k]}{k!}}\right)$ Characteristic Function Factorial Cumulant ${\text{CharacteristicFunction}}[{\mathcal {D}},x]=\exp \left(\sum _{k=1}^{\infty }{\frac {\left(-1+e^{ix}\right)^{k}{\text{FactorialCumulant}}({\mathcal {D}},k)}{k!}}\right)$ Characteristic Function Ascending Factorial Cumulant ${\text{CharacteristicFunction}}[{\mathcal {D}},x]=\exp \left(\sum _{k=0}^{\infty }{\frac {\left(1-e^{-ix}\right)^{k}{\text{AscendingFactorialCumulant}}({\mathcal {D}},k)}{k!}}\right)$ Characteristic Function Moment Generating Function ${\text{CharacteristicFunction}}[{\mathcal {D}},x]={\text{MomentGeneratingFunction}}[{\mathcal {D}},ix]$ Characteristic Function Central Moment Generating Function ${\text{CharacteristicFunction}}[{\mathcal {D}},x]=e^{(ix){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},ix]$ Characteristic Function Factorial Moment Generating Function ${\text{CharacteristicFunction}}[{\mathcal {D}},x]={\text{FactorialMomentGeneratingFunction}}\left[{\mathcal {D}},e^{ix}\right]{\text{/;}}-\pi <\Re (x)\leq \pi$ Characteristic Function Descending Factorial Moment Generating Function ${\text{CharacteristicFunction}}[{\mathcal {D}},x]={\text{DescendingFactorialMomentGeneratingFunction}}\left({\mathcal {D}},-1+e^{ix}\right){\text{/;}}-\pi <\Re (x)\leq \pi$ Characteristic Function Ascending Factorial Moment Generating Function ${\text{CharacteristicFunction}}[{\mathcal {D}},x]={\text{AscendingFactorialMomentGeneratingFunction}}\left({\mathcal {D}},1-e^{-ix}\right){\text{/;}}-\pi <\Re (x)\leq \pi$ Characteristic Function Cumulant Generating Function ${\text{CharacteristicFunction}}[{\mathcal {D}},x]=\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},ix])$ Characteristic Function Factorial Cumulant Generating Function ${\text{CharacteristicFunction}}[{\mathcal {D}},x]=\exp \left({\text{FactorialCumulantGeneratingFunction}}\left({\mathcal {D}},e^{ix}\right)\right){\text{/;}}-\pi <\Re (x)\leq \pi$ Characteristic Function Descending Factorial Cumulant Generating Function ${\text{CharacteristicFunction}}[{\mathcal {D}},x]=\exp \left({\text{DescendingFactorialCumulantGeneratingFunction}}\left({\mathcal {D}},-1+e^{ix}\right)\right){\text{/;}}-\pi <\Re (x)\leq \pi$ Characteristic Function Ascending Factorial Cumulant Generating Function ${\text{CharacteristicFunction}}[{\mathcal {D}},x]=\exp \left({\text{AscendingFactorialCumulantGeneratingFunction}}\left({\mathcal {D}},1-e^{-ix}\right)\right){\text{/;}}-\pi <\Re (x)\leq \pi$

Survival Function

Relations Between Distribution Properties

Derive Using Formula Survival Function PDF ${\text{SurvivalFunction}}[{\mathcal {D}},x]=\int _{x}^{\infty }{\text{PDF}}[{\mathcal {D}},t]\,dt$ Survival Function PDF ${\text{SurvivalFunction}}[{\mathcal {D}},x]=\int _{0}^{\infty }{\text{PDF}}[{\mathcal {D}},t+x]\,dt$ Survival Function PDF ${\text{SurvivalFunction}}[{\mathcal {D}},x]=\int _{-\infty }^{\infty }\theta (t-x){\text{PDF}}[{\mathcal {D}},t]\,dt$ Survival Function PDF ${\text{SurvivalFunction}}[{\mathcal {D}},x]=\int _{-\infty }^{\infty }{\text{Boole}}[t\geq x]{\text{PDF}}[{\mathcal {D}},t]\,dt$ Survival Function PDF ${\text{SurvivalFunction}}[{\mathcal {D}},x]=1-\int _{-\infty }^{x}{\text{PDF}}[{\mathcal {D}},t]\,dt$ Survival Function PDF ${\text{SurvivalFunction}}[{\mathcal {D}},x]=1-\int _{-\infty }^{\infty }{\text{Boole}}[t\leq x]{\text{PDF}}[{\mathcal {D}},t]\,dt$ Survival Function PDF ${\text{SurvivalFunction}}[{\mathcal {D}},x]=1-\int _{-\infty }^{\infty }\theta (x-t){\text{PDF}}[{\mathcal {D}},t]\,dt$ Survival Function CDF ${\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\text{CDF}}[{\mathcal {D}},x]$ Survival Function Characteristic Function ${\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{e^{(it)\tau }}}d\tau dt}{2\pi }}$ Survival Function Characteristic Function ${\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\frac {\int _{-\infty }^{x}{\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-t)\,dt}{\sqrt {2\pi }}}$ Survival Function Characteristic Function ${\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }{\text{Boole}}[t\leq x]\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-t)\right)\,dt}{\sqrt {2\pi }}}$ Survival Function Characteristic Function ${\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\frac {\int _{-\infty }^{\infty }\theta (x-t)\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-t)\right)\,dt}{\sqrt {2\pi }}}$ Survival Function Characteristic Function ${\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ Survival Function Survival Function ${\text{identical properties}}$ Survival Function Cumulative Hazard Function ${\text{SurvivalFunction}}[{\mathcal {D}},x]=\exp(-{\text{CumulativeHazardFunction}}({\mathcal {D}},x))$ Survival Function Hazard Function ${\text{SurvivalFunction}}[{\mathcal {D}},x]=\exp \left(-\int _{-\infty }^{x}{\text{HazardFunction}}[{\mathcal {D}},t]\,dt\right)$ Survival Function Inverse CDF ${\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\text{InverseFunction}}[t,{\text{InverseCDF}}[{\mathcal {D}},t]][x]$ Survival Function Inverse CDF ${\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{\tau (i{\text{InverseCDF}}[{\mathcal {D}},t])}\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ Survival Function Quantile ${\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\text{InverseFunction}}[t,{\text{Quantile}}[{\mathcal {D}},t]][x]$ Survival Function Quantile ${\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{\tau (i{\text{Quantile}}[{\mathcal {D}},t])}\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ Survival Function Inverse Survival Function ${\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\text{InverseFunction}}[t,{\text{InverseSurvivalFunction}}[{\mathcal {D}},1-t]][x]$ Survival Function Inverse Survival Function ${\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{\tau (i{\text{InverseSurvivalFunction}}[{\mathcal {D}},t])}\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ Survival Function Sparsity ${\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\text{InverseFunction}}\left[t,{\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{\tau }{\text{Sparsity}}({\mathcal {D}},\tau )\,d\tau \right][x]$ Survival Function Sparsity ${\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{\tau \left(i\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{t}{\text{Sparsity}}({\mathcal {D}},s)\,ds\right)\right)}\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ Survival Function Lorenz Curve 1 ${\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\text{Mean}}[{\mathcal {D}}]\int _{-\infty }^{x}{\frac {\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},t)}{\partial t}}{t}}\,dt$ Survival Function Lorenz Curve 1 ${\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\text{Mean}}[{\mathcal {D}}]\left(\int _{-\infty }^{x}{\frac {{\text{LorenzCurve1}}({\mathcal {D}},t)}{t^{2}}}\,dt+{\frac {{\text{LorenzCurve1}}({\mathcal {D}},x)}{x}}\right)$ Survival Function Lorenz Curve 2 ${\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\text{Mean}}[{\mathcal {D}}]\int _{-\infty }^{x}{\frac {\frac {\partial {\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},t])}{\partial t}}{t}}\,dt$ Survival Function Lorenz Curve 2 ${\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\text{Mean}}[{\mathcal {D}}]\left(\int _{-\infty }^{x}{\frac {{\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},t])}{t^{2}}}\,dt+{\frac {{\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},x])}{x}}\right)$ Survival Function Mean Residual Life ${\text{SurvivalFunction}}[{\mathcal {D}},x]=\exp \left(-\int _{-\infty }^{x}{\frac {1+{\frac {\partial {\text{MeanResidualLifeFunction}}({\mathcal {D}},t)}{\partial t}}}{{\text{MeanResidualLifeFunction}}({\mathcal {D}},t)}}\,dt\right)$ Survival Function Mean Residual Life ${\text{SurvivalFunction}}[{\mathcal {D}},x]=-{\frac {\partial ({\text{MeanResidualLifeFunction}}({\mathcal {D}},x)(1-{\text{CDF}}[{\mathcal {D}},x]))}{\partial x}}$ Survival Function Moment ${\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}(1-{\text{sgn}}(x))-\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right){\text{Moment}}[{\mathcal {D}},k+1]}{(k+1)!}}$ Survival Function Central Moment ${\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}(1-{\text{sgn}}(x))-\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\sum _{j=0}^{k+1}{\text{CentralMoment}}[{\mathcal {D}},j]\left({\binom {k+1}{j}}{\text{Mean}}[{\mathcal {D}}]^{-j+k+1}\right)}{(k+1)!}}$ Survival Function Factorial Moment ${\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}(1-{\text{sgn}}(x))-\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{(k+1)!}}$ Survival Function Ascending Factorial Moment ${\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}(1-{\text{sgn}}(x))-\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\sum _{j=0}^{k+1}\left((-1)^{-j+k-1}{\mathcal {S}}_{k+1}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)}{(k+1)!}}$ Survival Function Cumulant ${\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}(1-{\text{sgn}}(x))-\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right){\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]]}{(k+1)!}}$ Survival Function Factorial Cumulant ${\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}(1-{\text{sgn}}(x))-\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j}{\mathcal {S}}_{j}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}$ Survival Function Ascending Factorial Cumulant ${\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}(1-{\text{sgn}}(x))-\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}$ Survival Function Moment Generating Function ${\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\frac {\int _{-\infty }^{x}\int _{-i\infty }^{i\infty }{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},\tau ]}{e^{t\tau }}}d\tau dt}{(2\pi )i}}$ Survival Function Moment Generating Function ${\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\frac {\int _{-\infty }^{x}{\mathcal {F}}_{\tau }[{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]](-t)\,dt}{\sqrt {2\pi }}}$ Survival Function Moment Generating Function ${\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ Survival Function Central Moment Generating Function ${\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }e^{(i({\text{Mean}}[{\mathcal {D}}]-t))\tau }{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]d\tau dt}{2\pi }}$ Survival Function Central Moment Generating Function ${\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\frac {\int _{-\infty }^{x}{\mathcal {F}}_{\tau }\left[e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]\right](-t)\,dt}{\sqrt {2\pi }}}$ Survival Function Central Moment Generating Function ${\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ Survival Function Cumulant Generating Function ${\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }e^{{\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ]-(it)\tau }d\tau dt}{2\pi }}$ Survival Function Cumulant Generating Function ${\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\frac {\int _{-\infty }^{x}{\mathcal {F}}_{\tau }[\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])](-t)\,dt}{\sqrt {2\pi }}}$ Survival Function Cumulant Generating Function ${\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$

Cumulative Hazard Function

Relations Between Distribution Properties

Derive Using Formula Cumulative Hazard Function PDF ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(\int _{x}^{\infty }{\text{PDF}}[{\mathcal {D}},t]\,dt\right)$ Cumulative Hazard Function PDF ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-\int _{-\infty }^{x}{\text{PDF}}[{\mathcal {D}},t]\,dt\right)$ Cumulative Hazard Function PDF ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-\int _{-\infty }^{\infty }{\text{Boole}}[t\leq x]{\text{PDF}}[{\mathcal {D}},t]\,dt\right)$ Cumulative Hazard Function PDF ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-\int _{-\infty }^{\infty }\theta (x-t){\text{PDF}}[{\mathcal {D}},t]\,dt\right)$ Cumulative Hazard Function CDF ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log(1-{\text{CDF}}[{\mathcal {D}},x])$ Cumulative Hazard Function Characteristic Function ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-{\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{e^{(it)\tau }}}d\tau dt}{2\pi }}\right)$ Cumulative Hazard Function Characteristic Function ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-{\frac {\int _{-\infty }^{x}{\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-t)\,dt}{\sqrt {2\pi }}}\right)$ Cumulative Hazard Function Characteristic Function ${\text{CDF}}[{\mathcal {D}},x]=-\log \left({\frac {\int _{-\infty }^{\infty }{\text{Boole}}[t\leq x]\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-t)\right)\,dt}{\sqrt {2\pi }}}\right)$ Cumulative Hazard Function Characteristic Function ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-{\frac {\int _{-\infty }^{\infty }\theta (x-t)\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-t)\right)\,dt}{\sqrt {2\pi }}}\right)$ Cumulative Hazard Function Characteristic Function ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}\right)$ Cumulative Hazard Function Survival Function ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log({\text{SurvivalFunction}}[{\mathcal {D}},x])$ Cumulative Hazard Function Cumulative Hazard Function ${\text{identical properties}}$ Cumulative Hazard Function Hazard Function ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=\int _{-\infty }^{x}{\text{HazardFunction}}[{\mathcal {D}},t]\,dt$ Cumulative Hazard Function Inverse CDF ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log(1-{\text{InverseFunction}}[t,{\text{InverseCDF}}[{\mathcal {D}},t]][x])$ Cumulative Hazard Function Inverse CDF ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{(i{\text{InverseCDF}}[{\mathcal {D}},t])\tau }\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}\right)$ Cumulative Hazard Function Quantile ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log(1-{\text{InverseFunction}}[t,{\text{Quantile}}[{\mathcal {D}},t]][x])$ Cumulative Hazard Function Quantile ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{(i{\text{Quantile}}[{\mathcal {D}},t])\tau }\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}\right)$ Cumulative Hazard Function Inverse Survival Function ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log(1-{\text{InverseFunction}}[t,{\text{InverseSurvivalFunction}}[{\mathcal {D}},1-t]][x])$ Cumulative Hazard Function Inverse Survival Function ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{\tau (i{\text{InverseSurvivalFunction}}[{\mathcal {D}},t])}\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}\right)$ Cumulative Hazard Function Sparsity ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-{\text{InverseFunction}}\left[t,{\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{\tau }{\text{Sparsity}}({\mathcal {D}},\tau )\,d\tau \right][x]\right)$ Cumulative Hazard Function Sparsity ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{\left(i\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{t}{\text{Sparsity}}({\mathcal {D}},s)\,ds\right)\right)\tau }\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}\right)$ Cumulative Hazard Function Lorenz Curve 1 ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-{\text{Mean}}[{\mathcal {D}}]\int _{-\infty }^{x}{\frac {\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},t)}{\partial t}}{t}}\,dt\right)$ Cumulative Hazard Function Lorenz Curve 1 ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-{\text{Mean}}[{\mathcal {D}}]\left(\int _{-\infty }^{x}{\frac {{\text{LorenzCurve1}}({\mathcal {D}},t)}{t^{2}}}\,dt+{\frac {{\text{LorenzCurve1}}({\mathcal {D}},x)}{x}}\right)\right)$ Cumulative Hazard Function Lorenz Curve 2 ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-{\text{Mean}}[{\mathcal {D}}]\int _{-\infty }^{x}{\frac {\frac {\partial {\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},t])}{\partial t}}{t}}\,dt\right)$ Cumulative Hazard Function Lorenz Curve 2 ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-{\text{Mean}}[{\mathcal {D}}]\left(\int _{-\infty }^{x}{\frac {{\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},t])}{t^{2}}}\,dt+{\frac {{\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},x])}{x}}\right)\right)$ Cumulative Hazard Function Mean Residual Life Function ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(-{\frac {\partial ({\text{MeanResidualLifeFunction}}({\mathcal {D}},x){\text{SurvivalFunction}}[{\mathcal {D}},x])}{\partial x}}\right)$ Cumulative Hazard Function Moment ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}(1-{\text{sgn}}(x))-\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right){\text{Moment}}[{\mathcal {D}},k+1]}{(k+1)!}}\right)$ Cumulative Hazard Function Central Moment ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}(1-{\text{sgn}}(x))-\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\sum _{j=0}^{k+1}{\text{CentralMoment}}[{\mathcal {D}},j]\left({\binom {k+1}{j}}{\text{Mean}}[{\mathcal {D}}]^{-j+k+1}\right)}{(k+1)!}}\right)$ Cumulative Hazard Function Factorial Moment ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}(1-{\text{sgn}}(x))-\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{(k+1)!}}\right)$ Cumulative Hazard Function Ascending Factorial Moment ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}(1-{\text{sgn}}(x))-\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\sum _{j=0}^{k+1}\left((-1)^{-j+k-1}{\mathcal {S}}_{k+1}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)}{(k+1)!}}\right)$ Cumulative Hazard Function Cumulant ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}(1-{\text{sgn}}(x))-\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right){\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]]}{(k+1)!}}\right)$ Cumulative Hazard Function Factorial Cumulant ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}(1-{\text{sgn}}(x))-\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j}{\mathcal {S}}_{j}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}\right)$ Cumulative Hazard Function Ascending Factorial Cumulant ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}(1-{\text{sgn}}(x))-\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}\right)$ Cumulative Hazard Function Moment Generating Function ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-{\frac {\int _{-\infty }^{x}\int _{-i\infty }^{i\infty }{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},\tau ]}{e^{t\tau }}}d\tau dt}{(2\pi )i}}\right)$ Cumulative Hazard Function Moment Generating Function ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-{\frac {\int _{-\infty }^{x}{\mathcal {F}}_{\tau }[{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]](-t)\,dt}{\sqrt {2\pi }}}\right)$ Cumulative Hazard Function Moment Generating Function ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}\right)$ Cumulative Hazard Function Central Moment Generating Function ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-{\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }e^{(i({\text{Mean}}[{\mathcal {D}}]-t))\tau }{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]d\tau dt}{2\pi }}\right)$ Cumulative Hazard Function Central Moment Generating Function ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-{\frac {\int _{-\infty }^{x}{\mathcal {F}}_{\tau }\left[e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]\right](-t)\,dt}{\sqrt {2\pi }}}\right)$ Cumulative Hazard Function Central Moment Generating Function ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}\right)$ Cumulative Hazard Function Cumulant Generating Function ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-{\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }e^{{\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ]-(it)\tau }d\tau dt}{2\pi }}\right)$ Cumulative Hazard Function Cumulant Generating Function ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-{\frac {\int _{-\infty }^{x}{\mathcal {F}}_{\tau }[\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])](-t)\,dt}{\sqrt {2\pi }}}\right)$ Cumulative Hazard Function Cumulant Generating Function ${\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}\right)$

Hazard Function

Relations Between Distribution Properties

Derive Using Formula Hazard Function PDF ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {{\text{PDF}}[{\mathcal {D}},x]}{1-\int _{-\infty }^{x}{\text{PDF}}[{\mathcal {D}},t]\,dt}}$ Hazard Function PDF ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {{\text{PDF}}[{\mathcal {D}},x]}{\int _{x}^{\infty }{\text{PDF}}[{\mathcal {D}},t]\,dt}}$ Hazard Function CDF ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\frac {\partial {\text{CDF}}[{\mathcal {D}},x]}{\partial x}}{1-{\text{CDF}}[{\mathcal {D}},x]}}$ Hazard Function CDF ${\text{HazardFunction}}[{\mathcal {D}},x]=-{\frac {\partial \log(1-{\text{CDF}}[{\mathcal {D}},x])}{\partial x}}$ Hazard Function Characteristic Function ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},t]}{e^{(it)x}}}\,dt}{(2\pi ){\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Hazard Function Characteristic Function ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {2\left({\mathcal {F}}_{t}[{\text{CharacteristicFunction}}[{\mathcal {D}},t]](-x)\right)}{{\sqrt {2\pi }}-(2i)\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{\tau }}\right](-x)\right)}}$ Hazard Function Survival Function ${\text{HazardFunction}}[{\mathcal {D}},x]=-{\frac {\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},x]}{\partial x}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Hazard Function Survival Function ${\text{HazardFunction}}[{\mathcal {D}},x]=-{\frac {\partial \log({\text{SurvivalFunction}}[{\mathcal {D}},x])}{\partial x}}$ Hazard Function Cumulative Hazard Function ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\partial {\text{CumulativeHazardFunction}}({\mathcal {D}},x)}{\partial x}}$ Hazard Function Hazard Function ${\text{identical properties}}$ Hazard Function Inverse CDF ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}e^{(i({\text{InverseCDF}}[{\mathcal {D}},\tau ]-x))t}d\tau dt}{(2\pi ){\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Hazard Function Inverse CDF ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {2\int _{0}^{1}{\mathcal {F}}_{t}[1]({\text{InverseCDF}}[{\mathcal {D}},\tau ]-x)\,d\tau }{{\sqrt {2\pi }}-(2i)\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{(i{\text{InverseCDF}}[{\mathcal {D}},t])\tau }\,dt}{\tau }}\right](-x)\right)}}$ Hazard Function Inverse CDF ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\int _{0}^{1}\delta ({\text{InverseCDF}}[{\mathcal {D}},\tau ]-x)\,d\tau }{{\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{(i{\text{InverseCDF}}[{\mathcal {D}},t])\tau }\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}}$ Hazard Function Quantile ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}e^{(i({\text{Quantile}}[{\mathcal {D}},\tau ]-x))t}d\tau dt}{(2\pi ){\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Hazard Function Quantile ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {2\int _{0}^{1}{\mathcal {F}}_{t}[1]({\text{Quantile}}[{\mathcal {D}},\tau ]-x)\,d\tau }{{\sqrt {2\pi }}-(2i)\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{(i{\text{Quantile}}[{\mathcal {D}},t])\tau }\,dt}{\tau }}\right](-x)\right)}}$ Hazard Function Quantile ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\int _{0}^{1}\delta ({\text{Quantile}}[{\mathcal {D}},\tau ]-x)\,d\tau }{{\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{(i{\text{Quantile}}[{\mathcal {D}},t])\tau }\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}}$ Hazard Function Inverse Survival Function ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}e^{(i({\text{InverseSurvivalFunction}}[{\mathcal {D}},\tau ]-x))t}d\tau dt}{(2\pi ){\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Hazard Function Inverse Survival Function ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {2\int _{0}^{1}{\mathcal {F}}_{t}[1]({\text{InverseSurvivalFunction}}[{\mathcal {D}},\tau ]-x)\,d\tau }{{\sqrt {2\pi }}-(2i)\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{(i{\text{InverseSurvivalFunction}}[{\mathcal {D}},t])\tau }\,dt}{\tau }}\right](-x)\right)}}$ Hazard Function Inverse Survival Function ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\int _{0}^{1}\delta ({\text{InverseSurvivalFunction}}[{\mathcal {D}},\tau ]-x)\,d\tau }{{\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{(i{\text{InverseSurvivalFunction}}[{\mathcal {D}},t])\tau }\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}}$ Hazard Function Sparsity ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}\exp \left(\left(i\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{\tau }{\text{Sparsity}}({\mathcal {D}},\xi )\,d\xi -x\right)\right)t\right)d\tau dt}{(2\pi ){\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Hazard Function Sparsity ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {2\int _{0}^{1}{\mathcal {F}}_{t}[1]\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{\tau }{\text{Sparsity}}({\mathcal {D}},\xi )\,d\xi -x\right)\,d\tau }{{\sqrt {2\pi }}-(2i)\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{\left(i\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{t}{\text{Sparsity}}({\mathcal {D}},s)\,ds\right)\right)\tau }\,dt}{\tau }}\right](-x)\right)}}$ Hazard Function Sparsity ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\int _{0}^{1}\delta \left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{\tau }{\text{Sparsity}}({\mathcal {D}},\xi )\,d\xi -x\right)\,d\tau }{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Hazard Function Lorenz Curve 1 ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {{\text{Mean}}[{\mathcal {D}}]{\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},x)}{\partial x}}}{x\left(1-{\text{Mean}}[{\mathcal {D}}]\int _{-\infty }^{x}{\frac {\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},t)}{\partial t}}{t}}\,dt\right)}}$ Hazard Function Lorenz Curve 1 ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {{\text{Mean}}[{\mathcal {D}}]{\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},x)}{\partial x}}}{x\left(1-{\text{Mean}}[{\mathcal {D}}]\left({\frac {{\text{LorenzCurve1}}({\mathcal {D}},x)}{x}}+\int _{-\infty }^{x}{\frac {{\text{LorenzCurve1}}({\mathcal {D}},t)}{t^{2}}}\,dt\right)\right)}}$ Hazard Function Lorenz Curve 2 ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {{\text{Mean}}[{\mathcal {D}}]{\frac {\partial {\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},x])}{\partial x}}}{x\left(1-{\text{Mean}}[{\mathcal {D}}]\int _{-\infty }^{x}{\frac {\frac {\partial {\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},t])}{\partial t}}{t}}\,dt\right)}}$ Hazard Function Lorenz Curve 2 ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {{\text{Mean}}[{\mathcal {D}}]{\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},x)}{\partial x}}}{x\left(1-{\text{Mean}}[{\mathcal {D}}]\left({\frac {{\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},x])}{x}}+\int _{-\infty }^{x}{\frac {{\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},t])}{t^{2}}}\,dt\right)\right)}}$ Hazard Function Mean Residual Life Function ${\text{HazardFunction}}[{\mathcal {D}},x]=-{\frac {\frac {\partial ^{2}({\text{SurvivalFunction}}[{\mathcal {D}},x]{\text{MeanResidualLifeFunction}}({\mathcal {D}},x))}{\partial x^{2}}}{\frac {\partial ({\text{SurvivalFunction}}[{\mathcal {D}},x]{\text{MeanResidualLifeFunction}}({\mathcal {D}},x))}{\partial x}}}$ Hazard Function Moment ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right){\text{Moment}}[{\mathcal {D}},k]}{k!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Hazard Function Central Moment ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right)\sum _{j=0}^{k}{\text{CentralMoment}}[{\mathcal {D}},j]\left({\binom {k}{j}}{\text{Mean}}[{\mathcal {D}}]^{k-j}\right)}{k!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Hazard Function Factorial Moment ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right)\sum _{j=0}^{k}{\mathcal {S}}_{k}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{k!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Hazard Function Ascending Factorial Moment ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\sum _{j=0}^{\infty }\left((-1)^{j}{\text{AscendingFactorialMoment}}({\mathcal {D}},j)\right)\sum _{k=j}^{\infty }{\frac {{\mathcal {S}}_{k}^{(j)}\delta ^{(k)}(x)}{k!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Hazard Function Cumulant ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right){\text{BellY}}[k,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k\}]]}{k!}}+\delta (x)}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Hazard Function Factorial Cumulant ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right){\text{BellY}}\left[k,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j}{\mathcal {S}}_{j}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k\}\right]\right]}{k!}}+\delta (x)}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Hazard Function Ascending Factorial Cumulant ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right){\text{BellY}}\left[k,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k\}\right]\right]}{k!}}+\delta (x)}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Hazard Function Moment Generating Function ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\int _{-i\infty }^{i\infty }{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},t]}{e^{tx}}}\,dt}{((2\pi )i){\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Hazard Function Moment Generating Function ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},it]}{e^{(it)x}}}\,dt}{(2\pi ){\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Hazard Function Moment Generating Function ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {2\left({\mathcal {F}}_{t}[{\text{MomentGeneratingFunction}}[{\mathcal {D}},it]](-x)\right)}{{\sqrt {2\pi }}-(2i)\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-x)\right)}}$ Hazard Function Central Moment Generating Function ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\int _{-i\infty }^{i\infty }e^{t({\text{Mean}}[{\mathcal {D}}]-x)}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},t]\,dt}{((2\pi )i){\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Hazard Function Central Moment Generating Function ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }e^{(it)({\text{Mean}}[{\mathcal {D}}]-x)}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},it]\,dt}{(2\pi ){\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Hazard Function Central Moment Generating Function ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {2\left({\mathcal {F}}_{t}\left[e^{(it){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},it]\right](-x)\right)}{{\sqrt {2\pi }}-(2i)\left({\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-x)\right)}}$ Hazard Function Cumulant Generating Function ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }e^{{\text{CumulantGeneratingFunction}}[{\mathcal {D}},it]-(it)x}\,dt}{(2\pi ){\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Hazard Function Cumulant Generating Function ${\text{HazardFunction}}[{\mathcal {D}},x]={\frac {2\left({\mathcal {F}}_{t}\left[e^{{\text{CumulantGeneratingFunction}}[{\mathcal {D}},it]}\right](-x)\right)}{{\sqrt {2\pi }}-(2i)\left({\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau }}\right](-x)\right)}}$

Inverse CDF

Relations Between Distribution Properties

Derive Using Formula Inverse CDF PDF ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\int _{-\infty }^{t}{\text{PDF}}[{\mathcal {D}},\tau ]\,d\tau \right][x]$ Inverse CDF PDF ${\text{InverseCDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}e^{\left(i\left(\int _{-\infty }^{\tau }{\text{PDF}}[{\mathcal {D}},s]\,ds-\xi \right)\right)t}d\tau dtd\xi }{2\pi }}$ Inverse CDF CDF ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}[t,{\text{CDF}}[{\mathcal {D}},t]][x]$ Inverse CDF CDF ${\text{InverseCDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}e^{(i({\text{CDF}}[{\mathcal {D}},\tau ]-\xi ))t}d\tau dtd\xi }{2\pi }}$ Inverse CDF CDF ${\text{InverseCDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{(i{\text{CDF}}[{\mathcal {D}},t])\tau }\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ Inverse CDF Characteristic Function ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}\int _{-\infty }^{\infty }{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{e^{(i\xi )\tau }}}d\tau d\xi }{2\pi }}\right][x]$ Inverse CDF Characteristic Function ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}{\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][x]$ Inverse CDF Characteristic Function ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{\infty }{\text{Boole}}[\xi \leq t]\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-\xi )\right)\,d\xi }{\sqrt {2\pi }}}\right][x]$ Inverse CDF Characteristic Function ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{\infty }\theta (t-\xi )\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-\xi )\right)\,d\xi }{\sqrt {2\pi }}}\right][x]$ Inverse CDF Characteristic Function ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][x]$ Inverse CDF Characteristic Function ${\text{InverseCDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left({\frac {i\tau }{2}}-{\frac {\tau \left({\mathcal {F}}_{\xi }\left[{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\xi ]}{\xi }}\right](-t)\right)}{\sqrt {2\pi }}}\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ Inverse CDF Survival Function ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}[t,1-{\text{SurvivalFunction}}[{\mathcal {D}},t]][x]$ Inverse CDF Survival Function ${\text{InverseCDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}e^{(i(1-{\text{SurvivalFunction}}[{\mathcal {D}},\tau ]-\xi ))t}d\tau dtd\xi }{2\pi }}$ Inverse CDF Cumulative Hazard Function ${\text{InverseFunction}}[t,1-\exp(-{\text{CumulativeHazardFunction}}({\mathcal {D}},t))][x]$ Inverse CDF Cumulative Hazard Function ${\text{InverseCDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}e^{t(i(-\exp(-{\text{CumulativeHazardFunction}}({\mathcal {D}},\tau ))-\xi +1))}d\tau dtd\xi }{2\pi }}$ Inverse CDF Hazard Function ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,1-\exp \left(-\int _{-\infty }^{t}{\text{HazardFunction}}[{\mathcal {D}},\tau ]\,d\tau \right)\right][x]$ Inverse CDF Hazard Function ${\text{InverseCDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}\exp \left(t\left(i\left(-\exp \left(-\int _{-\infty }^{\tau }{\text{HazardFunction}}[{\mathcal {D}},s]\,ds\right)-\xi +1\right)\right)\right)d\tau dtd\xi }{2\pi }}$ Inverse CDF Inverse CDF ${\text{identical properties}}$ Inverse CDF Quantile ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{Quantile}}[{\mathcal {D}},x]$ Inverse CDF Inverse Survival Function ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseSurvivalFunction}}[{\mathcal {D}},1-x]$ Inverse CDF Sparsity ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{x}{\text{Sparsity}}({\mathcal {D}},t)\,dt$ Inverse CDF Lorenz Curve 1 ${\text{InverseCDF}}[{\mathcal {D}},x]={\frac {{\text{Mean}}[{\mathcal {D}}]{\text{LorenzCurve1}}^{(0,1)}({\mathcal {D}},{\text{Quantile}}[{\mathcal {D}},x])}{{\text{PDF}}[{\mathcal {D}},{\text{Quantile}}[{\mathcal {D}},x]]}}$ Inverse CDF Lorenz Curve 1 ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{Mean}}[{\mathcal {D}}]{\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},{\text{Quantile}}[{\mathcal {D}},x])}{\partial x}}$ Inverse CDF Lorenz Curve 2 ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{Mean}}[{\mathcal {D}}]{\frac {\partial {\text{LorenzCurve2}}({\mathcal {D}},x)}{\partial x}}$ Inverse CDF Mean Residual Life Function ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\partial ({\text{MeanResidualLifeFunction}}({\mathcal {D}},t){\text{SurvivalFunction}}[{\mathcal {D}},t])}{\partial t}}+1\right][x]$ Inverse CDF Mean Residual Life Function ${\text{InverseCDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}\exp \left(t\left(i\left({\frac {\partial ({\text{MeanResidualLifeFunction}}({\mathcal {D}},\tau ){\text{SurvivalFunction}}[{\mathcal {D}},\tau ])}{\partial \tau }}-\xi +1\right)\right)\right)d\tau dtd\xi }{2\pi }}$ Inverse CDF Moment ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{Moment}}[{\mathcal {D}},k+1]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][x]$ Inverse CDF Moment ${\text{InverseCDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\left(\left({\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{Moment}}[{\mathcal {D}},k+1]}{(k+1)!}}\right)i\right)\tau \right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ Inverse CDF Central Moment ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}\left({\binom {k+1}{j}}{\text{Mean}}[{\mathcal {D}}]^{k-j-1}\right){\text{CentralMoment}}[{\mathcal {D}},j]}{(k+1)!}}\right][x]$ Inverse CDF Central Moment ${\text{InverseCDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\left(\left({\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}\left({\binom {k+1}{j}}{\text{Mean}}[{\mathcal {D}}]^{k+1-j}\right){\text{CentralMoment}}[{\mathcal {D}},j]}{(k+1)!}}\right)i\right)\tau \right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ Inverse CDF Factorial Moment ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{(k+1)!}}\right][x]$ Inverse CDF Factorial Moment ${\text{InverseCDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\tau \left(i\left(\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right)\right)\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ Inverse CDF Ascending Factorial Moment ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}\left((-1)^{-j+k-1}{\mathcal {S}}_{k+1}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][x]$ Inverse CDF Ascending Factorial Moment ${\text{InverseCDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\tau \left(i\left(\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}\left((-1)^{-j+k-1}{\mathcal {S}}_{k+1}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right)\right)\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ Inverse CDF Cumulant ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][x]$ Inverse CDF Cumulant ${\text{InverseCDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\tau \left(i\left(\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right)\right)\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ Inverse CDF Factorial Cumulant ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j+1}{\mathcal {S}}_{j+1}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][x]$ Inverse CDF Factorial Cumulant ${\text{InverseCDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\tau \left(i\left(\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j+1}{\mathcal {S}}_{j+1}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right)\right)\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ Inverse CDF Ascending Factorial Cumulant ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][x]$ Inverse CDF Ascending Factorial Cumulant ${\text{InverseCDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\left(\left({\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}\right)i\right)\tau \right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ Inverse CDF Moment Generating Function ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}\int _{-\infty }^{\infty }{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{e^{(i\xi )\tau }}}d\tau d\xi }{2\pi }}\right][x]$ Inverse CDF Moment Generating Function ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}{\mathcal {F}}_{\tau }[{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][x]$ Inverse CDF Moment Generating Function ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][x]$ Inverse CDF Central Moment Generating Function ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}\int _{-\infty }^{\infty }{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{e^{(i\xi )\tau }}}d\tau d\xi }{2\pi }}\right][x]$ Inverse CDF Central Moment Generating Function ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}{\mathcal {F}}_{\tau }\left[e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]\right](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][x]$ Inverse CDF Central Moment Generating Function ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][x]$ Inverse CDF Central Moment Generating Function ${\text{InverseCDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left({\frac {i\tau }{2}}-{\frac {\tau \left({\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ Inverse CDF Cumulant Generating Function ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}\int _{-\infty }^{\infty }{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{e^{(i\xi )\tau }}}d\tau d\xi }{2\pi }}\right][x]$ Inverse CDF Cumulant Generating Function ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}{\mathcal {F}}_{\tau }[\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][x]$ Inverse CDF Cumulant Generating Function ${\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][x]$ Inverse CDF Cumulant Generating Function ${\text{InverseCDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left({\frac {i\tau }{2}}-{\frac {\tau \left({\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$

Quantile

Relations Between Distribution Properties

Derive Using Formula Quantile PDF ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\int _{-\infty }^{t}{\text{PDF}}[{\mathcal {D}},\tau ]\,d\tau \right][x]$ Quantile PDF ${\text{Quantile}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}e^{t\left(i\left(\int _{-\infty }^{\tau }{\text{PDF}}[{\mathcal {D}},s]\,ds-\xi \right)\right)}d\tau dtd\xi }{2\pi }}$ Quantile CDF ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}[t,{\text{CDF}}[{\mathcal {D}},t]][x]$ Quantile CDF ${\text{Quantile}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}e^{(i({\text{CDF}}[{\mathcal {D}},\tau ]-\xi ))t}d\tau dtd\xi }{2\pi }}$ Quantile CDF ${\text{Quantile}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{(i{\text{CDF}}[{\mathcal {D}},t])\tau }\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ Quantile Characteristic Function ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}\int _{-\infty }^{\infty }{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{e^{(i\xi )\tau }}}d\tau d\xi }{2\pi }}\right][x]$ Quantile Characteristic Function ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}{\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][x]$ Quantile Characteristic Function ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{\infty }{\text{Boole}}[\xi \leq t]\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-\xi )\right)\,d\xi }{\sqrt {2\pi }}}\right][x]$ Quantile Characteristic Function ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{\infty }\theta (t-\xi )\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-\xi )\right)\,d\xi }{\sqrt {2\pi }}}\right][x]$ Quantile Characteristic Function ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][x]$ Quantile Characteristic Function ${\text{Quantile}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left({\frac {i\tau }{2}}-{\frac {\tau \left({\mathcal {F}}_{\xi }\left[{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\xi ]}{\xi }}\right](-t)\right)}{\sqrt {2\pi }}}\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ Quantile Survival Function ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}[t,1-{\text{SurvivalFunction}}[{\mathcal {D}},t]][x]$ Quantile Survival Function ${\text{Quantile}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}e^{(i(1-{\text{SurvivalFunction}}[{\mathcal {D}},\tau ]-\xi ))t}d\tau dtd\xi }{2\pi }}$ Quantile Cumulative Hazard Function ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}[t,1-\exp(-{\text{CumulativeHazardFunction}}({\mathcal {D}},t))][x]$ Quantile Cumulative Hazard Function ${\text{Quantile}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}e^{t(i(-\exp(-{\text{CumulativeHazardFunction}}({\mathcal {D}},\tau ))-\xi +1))}d\tau dtd\xi }{2\pi }}$ Quantile Hazard Function ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,1-\exp \left(-\int _{-\infty }^{t}{\text{HazardFunction}}[{\mathcal {D}},\tau ]\,d\tau \right)\right][x]$ Quantile Hazard Function ${\text{Quantile}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}\exp \left(t\left(i\left(-\exp \left(-\int _{-\infty }^{\tau }{\text{HazardFunction}}[{\mathcal {D}},s]\,ds\right)-\xi +1\right)\right)\right)d\tau dtd\xi }{2\pi }}$ Quantile Inverse CDF ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseCDF}}[{\mathcal {D}},x]$ Quantile Quantile ${\text{identical properties}}$ Quantile Inverse Survival Function ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseSurvivalFunction}}[{\mathcal {D}},1-x]$ Quantile Sparsity ${\text{Quantile}}[{\mathcal {D}},x]={\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{x}{\text{Sparsity}}({\mathcal {D}},t)\,dt$ Quantile Lorenz Curve 1 ${\text{Quantile}}[{\mathcal {D}},x]={\text{Mean}}[{\mathcal {D}}]{\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},{\text{InverseCDF}}[{\mathcal {D}},x])}{\partial x}}$ Quantile Lorenz Curve 1 ${\text{Quantile}}[{\mathcal {D}},x]={\frac {{\text{Mean}}[{\mathcal {D}}]{\text{LorenzCurve1}}^{(0,1)}({\mathcal {D}},{\text{Quantile}}[{\mathcal {D}},x])}{{\text{PDF}}[{\mathcal {D}},{\text{InverseCDF}}[{\mathcal {D}},x]]}}$ Quantile Lorenz Curve 2 ${\text{Quantile}}[{\mathcal {D}},x]={\text{Mean}}[{\mathcal {D}}]{\frac {\partial {\text{LorenzCurve2}}({\mathcal {D}},x)}{\partial x}}$ Quantile Mean Residual Life Function ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\partial ({\text{MeanResidualLifeFunction}}({\mathcal {D}},t){\text{SurvivalFunction}}[{\mathcal {D}},t])}{\partial t}}+1\right][x]$ Quantile Mean Residual Life Function ${\text{Quantile}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}\exp \left(t\left(i\left({\frac {\partial ({\text{MeanResidualLifeFunction}}({\mathcal {D}},\tau ){\text{SurvivalFunction}}[{\mathcal {D}},\tau ])}{\partial \tau }}-\xi +1\right)\right)\right)d\tau dtd\xi }{2\pi }}$ Quantile Moment ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{Moment}}[{\mathcal {D}},k+1]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][x]$ Quantile Moment ${\text{Quantile}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\left(\left({\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{Moment}}[{\mathcal {D}},k+1]}{(k+1)!}}\right)i\right)\tau \right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ Quantile Central Moment ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}\left({\binom {k+1}{j}}{\text{Mean}}[{\mathcal {D}}]^{k-j-1}\right){\text{CentralMoment}}[{\mathcal {D}},j]}{(k+1)!}}\right][x]$ Quantile Central Moment ${\text{Quantile}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\left(\left({\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}\left({\binom {k+1}{j}}{\text{Mean}}[{\mathcal {D}}]^{k+1-j}\right){\text{CentralMoment}}[{\mathcal {D}},j]}{(k+1)!}}\right)i\right)\tau \right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ Quantile Factorial Moment ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{(k+1)!}}\right][x]$ Quantile Factorial Moment ${\text{Quantile}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\tau \left(i\left(\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right)\right)\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ Quantile Ascending Factorial Moment ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}\left((-1)^{-j+k-1}{\mathcal {S}}_{k+1}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][x]$ Quantile Ascending Factorial Moment ${\text{Quantile}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\tau \left(i\left(\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}\left((-1)^{-j+k-1}{\mathcal {S}}_{k+1}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right)\right)\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ Quantile Cumulant ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][x]$ Quantile Cumulant ${\text{Quantile}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\left(\left({\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]]}{(k+1)!}}\right)i\right)\tau \right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ Quantile Factorial Cumulant ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j+1}{\mathcal {S}}_{j+1}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}\right][x]$ Quantile Factorial Cumulant ${\text{Quantile}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\tau \left(i\left(\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j+1}{\mathcal {S}}_{j+1}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right)\right)\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ Quantile Ascending Factorial Cumulant ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][x]$ Quantile Ascending Factorial Cumulant ${\text{Quantile}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\tau \left(i\left(\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right)\right)\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ Quantile Moment Generating Function ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\in t_{-\infty }^{t}\int _{-\infty }^{\infty }{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{e^{(i\xi )\tau }}}d\tau d\xi }{2\pi }}\right][x]$ Quantile Moment Generating Function ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\in t_{-\infty }^{t}{\mathcal {F}}_{\tau }[{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][x]$ Quantile Moment Generating Function ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][x]$ Quantile Moment Generating Function ${\text{Quantile}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left({\frac {i\tau }{2}}-{\frac {\tau \left({\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ Quantile Central Moment Generating Function ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\in t_{-\infty }^{t}\int _{-\infty }^{\infty }{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{e^{(i\xi )\tau }}}d\tau d\xi }{2\pi }}\right][x]$ Quantile Central Moment Generating Function ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\in t_{-\infty }^{t}{\mathcal {F}}_{\tau }\left[e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]\right](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][x]$ Quantile Central Moment Generating Function ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][x]$ Quantile Central Moment Generating Function ${\text{Quantile}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left({\frac {i\tau }{2}}-{\frac {\tau \left({\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$ Quantile Cumulant Generating Function ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}\int _{-\infty }^{\infty }{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{e^{(i\xi )\tau }}}d\tau d\xi }{2\pi }}\right][x]$ Quantile Cumulant Generating Function ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}{\mathcal {F}}_{\tau }[\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][x]$ Quantile Cumulant Generating Function ${\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][x]$ Quantile Cumulant Generating Function ${\text{Quantile}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left({\frac {i\tau }{2}}-{\frac {\tau \left({\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}$

Inverse Survival Function

Relations Between Distribution Properties

Derive Using Formula Inverse Survival Function PDF ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\int _{-\infty }^{t}{\text{PDF}}[{\mathcal {D}},\tau ]\,d\tau \right][1-x]$ Inverse Survival Function PDF ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{1-x}\int _{-\infty }^{\infty }\int _{0}^{1}e^{\left(i\left(\int _{-\infty }^{\tau }{\text{PDF}}[{\mathcal {D}},s]\,ds-\xi \right)\right)t}d\tau dtd\xi }{2\pi }}$ Inverse Survival Function CDF ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}[t,{\text{CDF}}[{\mathcal {D}},t]][1-x]$ Inverse Survival Function CDF ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{1-x}\int _{-\infty }^{\infty }\int _{0}^{1}e^{(i({\text{CDF}}[{\mathcal {D}},\tau ]-\xi ))t}d\tau dtd\xi }{2\pi }}$ Inverse Survival Function CDF ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{(i{\text{CDF}}[{\mathcal {D}},t])\tau }\,dt}{\tau }}\right](x-1)\right)}{\sqrt {2\pi }}}$ Inverse Survival Function Characteristic Function ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}\int _{-\infty }^{\infty }{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{e^{(i\xi )\tau }}}d\tau d\xi }{2\pi }}\right][1-x]$ Inverse Survival Function Characteristic Function ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}{\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][1-x]$ Inverse Survival Function Characteristic Function ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{\infty }{\text{Boole}}[\xi \leq t]\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-\xi )\right)\,d\xi }{\sqrt {2\pi }}}\right][1-x]$ Inverse Survival Function Characteristic Function ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{\infty }\theta (t-\xi )\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-\xi )\right)\,d\xi }{\sqrt {2\pi }}}\right][1-x]$ Inverse Survival Function Characteristic Function ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][1-x]$ Inverse Survival Function Characteristic Function ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left({\frac {i\tau }{2}}-{\frac {\tau \left({\mathcal {F}}_{\xi }\left[{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\xi ]}{\xi }}\right](-t)\right)}{\sqrt {2\pi }}}\right)\,dt}{\tau }}\right](x-1)\right)}{\sqrt {2\pi }}}$ Inverse Survival Function Survival Function ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}[t,{\text{SurvivalFunction}}[{\mathcal {D}},t]][x]$ Inverse Survival Function Survival Function ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{1-x}\int _{-\infty }^{\infty }\int _{0}^{1}e^{(i(1-{\text{SurvivalFunction}}[{\mathcal {D}},\tau ]-\xi ))t}d\tau dtd\xi }{2\pi }}$ Inverse Survival Function Cumulative Hazard Function ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}[t,1-\exp(-{\text{CumulativeHazardFunction}}({\mathcal {D}},t))][1-x]$ Inverse Survival Function Cumulative Hazard Function ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{1-x}\int _{-\infty }^{\infty }\int _{0}^{1}e^{(i(1-\exp(-{\text{CumulativeHazardFunction}}({\mathcal {D}},\tau ))-\xi ))t}d\tau dtd\xi }{2\pi }}$ Inverse Survival Function Hazard Function ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,1-\exp \left(-\int _{-\infty }^{t}{\text{HazardFunction}}[{\mathcal {D}},\tau ]\,d\tau \right)\right][1-x]$ Inverse Survival Function Hazard Function ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{1-x}\int _{-\infty }^{\infty }\int _{0}^{1}\exp \left(\left(i\left(1-\exp \left(-\int _{-\infty }^{\tau }{\text{HazardFunction}}[{\mathcal {D}},s]\,ds\right)-\xi \right)\right)t\right)d\tau dtd\xi }{2\pi }}$ Inverse Survival Function Inverse CDF ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseCDF}}[{\mathcal {D}},1-x]$ Inverse Survival Function Quantile ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{Quantile}}[{\mathcal {D}},1-x]$ Inverse Survival Function Inverse Survival Function ${\text{identical properties}}$ Inverse Survival Function Sparsity ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{1-x}{\text{Sparsity}}({\mathcal {D}},t)\,dt$ Inverse Survival Function Lorenz Curve 1 ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]=-{\text{Mean}}[{\mathcal {D}}]{\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},{\text{InverseCDF}}[{\mathcal {D}},1-x])}{\partial x}}$ Inverse Survival Function Lorenz Curve 1 ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {{\text{Mean}}[{\mathcal {D}}]{\text{LorenzCurve1}}^{(0,1)}({\mathcal {D}},{\text{Quantile}}[{\mathcal {D}},1-x])}{{\text{PDF}}[{\mathcal {D}},{\text{InverseCDF}}[{\mathcal {D}},1-x]]}}$ Inverse Survival Function Lorenz Curve 2 ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]=-{\text{Mean}}[{\mathcal {D}}]{\frac {\partial {\text{LorenzCurve2}}({\mathcal {D}},1-x)}{\partial x}}$ Inverse Survival Function Mean Residual Life Function ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\partial ({\text{MeanResidualLifeFunction}}({\mathcal {D}},t){\text{SurvivalFunction}}[{\mathcal {D}},t])}{\partial t}}+1\right][1-x]$ Inverse Survival Function Mean Residual Life Function ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{1-x}\int _{-\infty }^{\infty }\int _{0}^{1}\exp \left(t\left(i\left({\frac {\partial ({\text{MeanResidualLifeFunction}}({\mathcal {D}},\tau ){\text{SurvivalFunction}}[{\mathcal {D}},\tau ])}{\partial \tau }}-\xi +1\right)\right)\right)d\tau dtd\xi }{2\pi }}$ Inverse Survival Function Moment ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{Moment}}[{\mathcal {D}},k+1]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][1-x]$ Inverse Survival Function Moment ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\left(\left({\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{Moment}}[{\mathcal {D}},k+1]}{(k+1)!}}\right)i\right)\tau \right)\,dt}{\tau }}\right](x-1)\right)}{\sqrt {2\pi }}}$ Inverse Survival Function Central Moment ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}\left({\binom {k+1}{j}}{\text{Mean}}[{\mathcal {D}}]^{k-j-1}\right){\text{CentralMoment}}[{\mathcal {D}},j]}{(k+1)!}}\right][1-x]$ Inverse Survival Function Central Moment ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\tau \left(i\left(\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}{\text{CentralMoment}}[{\mathcal {D}},j]\left({\binom {k+1}{j}}{\text{Mean}}[{\mathcal {D}}]^{-j+k+1}\right)}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right)\right)\right)\,dt}{\tau }}\right](x-1)\right)}{\sqrt {2\pi }}}$ Inverse Survival Function Factorial Moment ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][1-x]$ Inverse Survival Function Factorial Moment ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\tau \left(i\left(\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right)\right)\right)\,dt}{\tau }}\right](x-1)\right)}{\sqrt {2\pi }}}$ Inverse Survival Function Ascending Factorial Moment ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}\left((-1)^{-j+k-1}{\mathcal {S}}_{k+1}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][1-x]$ Inverse Survival Function Ascending Factorial Moment ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\tau \left(i\left(\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}\left((-1)^{-j+k-1}{\mathcal {S}}_{k+1}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right)\right)\right)\,dt}{\tau }}\right](x-1)\right)}{\sqrt {2\pi }}}$ Inverse Survival Function Cumulant ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][1-x]$ Inverse Survival Function Cumulant ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\tau \left(i\left(\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right)\right)\right)\,dt}{\tau }}\right](x-1)\right)}{\sqrt {2\pi }}}$ Inverse Survival Function Factorial Cumulant ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j+1}{\mathcal {S}}_{j+1}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][1-x]$ Inverse Survival Function Factorial Cumulant ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\tau \left(i\left(\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j+1}{\mathcal {S}}_{j+1}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right)\right)\right)\,dt}{\tau }}\right](x-1)\right)}{\sqrt {2\pi }}}$ Inverse Survival Function Ascending Factorial Cumulant ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][1-x]$ Inverse Survival Function Ascending Factorial Cumulant ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\tau \left(i\left(\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right)\right)\right)\,dt}{\tau }}\right](x-1)\right)}{\sqrt {2\pi }}}$ Inverse Survival Function Moment Generating Function ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}\int _{-\infty }^{\infty }{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{e^{(i\xi )\tau }}}d\tau d\xi }{2\pi }}\right][1-x]$ Inverse Survival Function Moment Generating Function ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}{\mathcal {F}}_{\tau }[{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][1-x]$ Inverse Survival Function Moment Generating Function ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][1-x]$ Inverse Survival Function Moment Generating Function ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left({\frac {i\tau }{2}}-{\frac {\tau \left({\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right)\,dt}{\tau }}\right](x-1)\right)}{\sqrt {2\pi }}}$ Inverse Survival Function Central Moment Generating Function ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}\int _{-\infty }^{\infty }{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{e^{(i\xi )\tau }}}d\tau d\xi }{2\pi }}\right][1-x]$ Inverse Survival Function Central Moment Generating Function ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}{\mathcal {F}}_{\tau }\left[e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]\right](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][1-x]$ Inverse Survival Function Central Moment Generating Function ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][1-x]$ Inverse Survival Function Central Moment Generating Function ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left({\frac {i\tau }{2}}-{\frac {\tau \left({\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right)\,dt}{\tau }}\right](x-1)\right)}{\sqrt {2\pi }}}$ Inverse Survival Function Cumulant Generating Function ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}\int _{-\infty }^{\infty }{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{e^{(i\xi )\tau }}}d\tau d\xi }{2\pi }}\right][1-x]$ Inverse Survival Function Cumulant Generating Function ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}{\mathcal {F}}_{\tau }[\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][1-x]$ Inverse Survival Function Cumulant Generating Function ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][1-x]$ Inverse Survival Function Cumulant Generating Function ${\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left({\frac {i\tau }{2}}-{\frac {\tau \left({\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right)\,dt}{\tau }}\right](x-1)\right)}{\sqrt {2\pi }}}$

Sparsity

Relations Between Distribution Properties

Derive Using Formula Sparsity PDF ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}\left[t,\int _{-\infty }^{t}{\text{PDF}}[{\mathcal {D}},\tau ]\,d\tau \right][x]}{\partial x}}$ Sparsity PDF ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}e^{\left(i\left(\int _{-\infty }^{\tau }{\text{PDF}}[{\mathcal {D}},s]\,ds-x\right)\right)t}d\tau dt}{2\pi }}$ Sparsity CDF ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}[t,{\text{CDF}}[{\mathcal {D}},t]][x]}{\partial x}}$ Sparsity CDF ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}e^{(i({\text{CDF}}[{\mathcal {D}},\tau ]-x))t}d\tau dt}{2\pi }}$ Sparsity CDF ${\text{Sparsity}}({\mathcal {D}},x)={\frac {{\mathcal {F}}_{t}\left[\int _{0}^{1}e^{t(i{\text{CDF}}[{\mathcal {D}},\tau ])}\,d\tau \right](-x)}{\sqrt {2\pi }}}$ Sparsity Characteristic Function ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}\int _{-\infty }^{\infty }{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{e^{(i\xi )\tau }}}d\tau d\xi }{2\pi }}\right][x]}{\partial x}}$ Sparsity Characteristic Function ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}{\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][x]}{\partial x}}$ Sparsity Characteristic Function ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][x]}{\partial x}}$ Sparsity Characteristic Function ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}\exp \left(\left({\frac {i\int _{-\infty }^{\tau }\int _{-\infty }^{\infty }{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},s]}{e^{(iu)s}}}dsdu}{2\pi }}-ix\right)t\right)d\tau dt}{2\pi }}$ Sparsity Characteristic Function ${\text{Sparsity}}({\mathcal {D}},x)={\frac {{\mathcal {F}}_{\tau }\left[\int _{0}^{1}\exp \left(-{\frac {\tau \left({\mathcal {F}}_{\tau }\left[{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}+{\frac {i\tau }{2}}\right)\,dt\right](-x)}{\sqrt {2\pi }}}$ Sparsity Survival Function ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}[t,1-{\text{SurvivalFunction}}[{\mathcal {D}},t]][x]}{\partial x}}$ Sparsity Survival Function ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}e^{t(i(-{\text{SurvivalFunction}}[{\mathcal {D}},\tau ]-x+1))}d\tau dt}{2\pi }}$ Sparsity Cumulative Hazard Function ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}[t,1-\exp(-{\text{CumulativeHazardFunction}}({\mathcal {D}},t))][x]}{\partial x}}$ Sparsity Cumulative Hazard Function ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}e^{t(i(-\exp(-{\text{CumulativeHazardFunction}}({\mathcal {D}},\tau ))-x+1))}d\tau dt}{2\pi }}$ Sparsity Hazard Function ${\frac {\partial {\text{InverseFunction}}\left[t,1-\exp \left(-\int _{-\infty }^{t}{\text{HazardFunction}}[{\mathcal {D}},\tau ]\,d\tau \right)\right][x]}{\partial x}}$ Sparsity Hazard Function ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}\exp \left(t\left(i\left(-\exp \left(-\int _{-\infty }^{\tau }{\text{HazardFunction}}[{\mathcal {D}},s]\,ds\right)-x+1\right)\right)\right)d\tau dt}{2\pi }}$ Sparsity Inverse CDF ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseCDF}}[{\mathcal {D}},x]}{\partial x}}$ Sparsity Quantile ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{Quantile}}[{\mathcal {D}},x]}{\partial x}}$ Sparsity Inverse Survival Function ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseSurvivalFunction}}[{\mathcal {D}},1-x]}{\partial x}}$ Sparsity Sparsity ${\text{identical properties}}$ Sparsity Lorenz Curve 1 ${\text{Sparsity}}({\mathcal {D}},x)={\text{Mean}}[{\mathcal {D}}]{\frac {\partial ^{2}{\text{LorenzCurve1}}({\mathcal {D}},{\text{InverseCDF}}[{\mathcal {D}},x])}{\partial x^{2}}}$ Sparsity Lorenz Curve 2 ${\text{Sparsity}}({\mathcal {D}},x)={\text{Mean}}[{\mathcal {D}}]{\frac {\partial ^{2}{\text{LorenzCurve2}}({\mathcal {D}},x)}{\partial x^{2}}}$ Sparsity Mean Residual Life Function ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}\left[t,{\frac {\partial ({\text{MeanResidualLifeFunction}}({\mathcal {D}},t){\text{SurvivalFunction}}[{\mathcal {D}},t])}{\partial t}}+1\right][x]}{\partial x}}$ Sparsity Mean Residual Life Function ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}\exp \left(t\left(i\left({\frac {\partial ({\text{MeanResidualLifeFunction}}({\mathcal {D}},\tau ){\text{SurvivalFunction}}[{\mathcal {D}},\tau ])}{\partial \tau }}-x+1\right)\right)\right)d\tau dt}{2\pi }}$ Sparsity Moment ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{Moment}}[{\mathcal {D}},k+1]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][x]}{\partial x}}$ Sparsity Moment ${\text{Sparsity}}({\mathcal {D}},x)={\frac {{\mathcal {F}}_{\tau }\left[\int _{0}^{1}\exp \left(\left(\left({\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{Moment}}[{\mathcal {D}},k+1]}{(k+1)!}}\right)i\right)\tau \right)\,dt\right](-x)}{\sqrt {2\pi }}}$ Sparsity Factorial Moment ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial }{\partial x}}{\text{InverseFunction}}\left[t,{\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{(k+1)!}}\right][x]$ Sparsity Factorial Moment ${\text{Sparsity}}({\mathcal {D}},x)={\frac {{\mathcal {F}}_{\tau }\left[\int _{0}^{1}\exp \left(\left(\left({\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{(k+1)!}}\right)i\right)\tau \right)\,dt\right](-x)}{\sqrt {2\pi }}}$ Sparsity Ascending Factorial Moment ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial }{\partial x}}{\text{InverseFunction}}\left[t,{\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}\left((-1)^{k-j-1}{\mathcal {S}}_{k+1}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)}{(k+1)!}}\right][x]$ Sparsity Ascending Factorial Moment ${\text{Sparsity}}({\mathcal {D}},x)={\frac {{\mathcal {F}}_{\tau }\left[\int _{0}^{1}\exp \left(\left(\left({\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}\left((-1)^{k-j-1}{\mathcal {S}}_{k+1}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)}{(k+1)!}}\right)i\right)\tau \right)\,dt\right](-x)}{\sqrt {2\pi }}}$ Sparsity Cumulant ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial }{\partial x}}{\text{InverseFunction}}\left[t,{\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]]}{(k+1)!}}\right][x]$ Sparsity Cumulant ${\text{Sparsity}}({\mathcal {D}},x)={\frac {{\mathcal {F}}_{\tau }\left[\int _{0}^{1}\exp \left(\left(\left({\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]]}{(k+1)!}}\right)i\right)\tau \right)\,dt\right](-x)}{\sqrt {2\pi }}}$ Sparsity Factorial Cumulant ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial }{\partial x}}{\text{InverseFunction}}\left[t,{\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j}{\mathcal {S}}_{j}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}\right][x]$ Sparsity Factorial Cumulant ${\text{Sparsity}}({\mathcal {D}},x)={\frac {{\mathcal {F}}_{\tau }\left[\int _{0}^{1}\exp \left(\left(\left({\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j}{\mathcal {S}}_{j}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}\right)i\right)\tau \right)\,dt\right](-x)}{\sqrt {2\pi }}}$ Sparsity Ascending Factorial Cumulant ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial }{\partial x}}{\text{InverseFunction}}\left[t,{\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}\right][x]$ Sparsity Ascending Factorial Cumulant ${\text{Sparsity}}({\mathcal {D}},x)={\frac {{\mathcal {F}}_{\tau }\left[\int _{0}^{1}\exp \left(\left(\left({\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}\right)i\right)\tau \right)\,dt\right](-x)}{\sqrt {2\pi }}}$ Sparsity Moment Generating Function ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial }{\partial x}}{\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}\int _{-i\infty }^{i\infty }{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},\tau ]}{e^{\xi \tau }}}d\tau d\xi }{(2\pi )i}}\right][x]$ Sparsity Moment Generating Function ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial }{\partial x}}{\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}{\mathcal {F}}_{\tau }[{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][x]$ Sparsity Moment Generating Function ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial }{\partial x}}{\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][x]$ Sparsity Moment Generating Function ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}\exp \left(\left(i\left({\frac {\int _{-\infty }^{\tau }\int _{-i\infty }^{i\infty }{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},s]}{e^{us}}}dsdu}{(2\pi )i}}-x\right)\right)t\right)d\tau dt}{2\pi }}$ Sparsity Moment Generating Function ${\text{Sparsity}}({\mathcal {D}},x)={\frac {{\mathcal {F}}_{\tau }\left[\int _{0}^{1}\exp \left({\frac {i\tau }{2}}-{\frac {\tau \left({\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right)\,dt\right](-x)}{\sqrt {2\pi }}}$ Sparsity Central Moment Generating Function ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}\int _{-i\infty }^{i\infty }{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},\tau ]e^{\tau {\text{Mean}}[{\mathcal {D}}]-\xi \tau }d\tau d\xi }{(2\pi )i}}\right][x]}{\partial x}}$ Sparsity Central Moment Generating Function ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}{\mathcal {F}}_{\tau }\left[e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]\right](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][x]}{\partial x}}$ Sparsity Central Moment Generating Function ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][x]}{\partial x}}$ Sparsity Central Moment Generating Function ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}\exp \left(\left(i\left({\frac {\int _{-\infty }^{\tau }\int _{-i\infty }^{i\infty }e^{s{\text{Mean}}[{\mathcal {D}}]-us}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},s]dsdu}{(2\pi )i}}-x\right)\right)t\right)d\tau dt}{2\pi }}$ Sparsity Central Moment Generating Function ${\text{Sparsity}}({\mathcal {D}},x)={\frac {{\mathcal {F}}_{\tau }\left[\int _{0}^{1}\exp \left({\frac {i\tau }{2}}-{\frac {\tau \left({\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right)\,dt\right](-x)}{\sqrt {2\pi }}}$ Sparsity Cumulant Generating Function ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}\int _{-i\infty }^{i\infty }{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},\tau ])}{e^{\xi \tau }}}d\tau d\xi }{(2\pi )i}}\right][x]}{\partial x}}$ Sparsity Cumulant Generating Function ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}{\mathcal {F}}_{\tau }[\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},\tau ])](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][x]}{\partial x}}$ Sparsity Cumulant Generating Function ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},\tau ])}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][x]}{\partial x}}$ Sparsity Cumulant Generating Function ${\text{Sparsity}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}\exp \left(\left({\frac {\int _{-\infty }^{\tau }\int _{-i\infty }^{i\infty }{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},s])}{e^{us}}}dsdu}{2\pi }}-ix\right)t\right)d\tau dt}{2\pi }}$ Sparsity Cumulant Generating Function ${\text{Sparsity}}({\mathcal {D}},x)={\frac {{\mathcal {F}}_{\tau }\left[\int _{0}^{1}\exp \left({\frac {i\tau }{2}}-{\frac {\tau \left({\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right)\,dt\right](-x)}{\sqrt {2\pi }}}$

Lorenz Curve 1

Relations Between Distribution Properties

Derive Using Formula Lorenz Curve 1 PDF ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{x}t{\text{PDF}}[{\mathcal {D}},t]\,dt}{\int _{-\infty }^{\infty }t{\text{PDF}}[{\mathcal {D}},t]\,dt}}$ Lorenz Curve 1 PDF ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{x}t{\text{PDF}}[{\mathcal {D}},t]\,dt}{{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 1 PDF ${\text{LorenzCurve1}}({\mathcal {D}},x)=\left({\begin{array}{cc}\{&{\begin{array}{cc}0&x<u\\{\frac {\int _{u}^{x}t{\text{PDF}}[{\mathcal {D}},t]\,dt}{{\text{Mean}}[{\mathcal {D}}]}}&u\leq x<v\\1&{\text{True}}\\\end{array}}\\\end{array}}\right){\text{/;}}u={\text{support}}[{\mathcal {D}}][[1,1]]\land v={\text{support}}[{\mathcal {D}}][[1,2]]$ Lorenz Curve 1 CDF ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{x}t{\frac {\partial {\text{CDF}}[{\mathcal {D}},t]}{\partial t}}\,dt}{{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 1 CDF ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {x{\text{CDF}}[{\mathcal {D}},x]-\int _{-\infty }^{x}{\text{CDF}}[{\mathcal {D}},t]\,dt}{{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 1 Characteristic Function ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{x}{\frac {t{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{e^{(i\tau )t}}}dtd\tau }{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 1 Characteristic Function ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{x}t\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-t)\right)\,dt}{{\sqrt {2\pi }}{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 1 Characteristic Function ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {1}{2}}+{\frac {{\mathcal {F}}_{\tau }\left[{\frac {(1+\tau (ix)){\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{\tau ^{2}}}\right](-x)}{{\sqrt {2\pi }}{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 1 Survival Function ${\text{LorenzCurve1}}({\mathcal {D}},x)=-{\frac {\int _{-\infty }^{x}t{\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},t]}{\partial t}}\,dt}{{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 1 Survival Function ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {x(1-{\text{SurvivalFunction}}[{\mathcal {D}},x])-\int _{-\infty }^{x}(1-{\text{SurvivalFunction}}[{\mathcal {D}},t])\,dt}{{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 1 Cumulative Hazard Function ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{x}{\frac {t{\frac {\partial {\text{CumulativeHazardFunction}}({\mathcal {D}},t)}{\partial t}}}{e^{{\text{CumulativeHazardFunction}}({\mathcal {D}},t)}}}\,dt}{{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 1 Hazard Function ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{x}{\text{HazardFunction}}[{\mathcal {D}},t]\left(t\exp \left(-\int _{-\infty }^{t}{\text{HazardFunction}}[{\mathcal {D}},\tau ]\,d\tau \right)\right)\,dt}{{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 1 Inverse CDF ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}te^{\xi (i({\text{InverseCDF}}[{\mathcal {D}},\tau ]-t))}d\tau d\xi dt}{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 1 Inverse CDF ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{0}^{{\text{CDF}}[{\mathcal {D}},x]}{\text{InverseCDF}}[{\mathcal {D}},t]\,dt}{{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 1 Quantile ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}te^{\xi (i({\text{Quantile}}[{\mathcal {D}},\tau ]-t))}d\tau d\xi dt}{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 1 Quantile ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{0}^{{\text{CDF}}[{\mathcal {D}},x]}{\text{Quantile}}[{\mathcal {D}},t]\,dt}{{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 1 Inverse Survival Function ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}te^{\xi (i({\text{InverseSurvivalFunction}}[{\mathcal {D}},\tau ]-t))}d\tau d\xi dt}{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 1 Inverse Survival Function ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{{\text{SurvivalFunction}}[{\mathcal {D}},x]}^{1}{\text{InverseSurvivalFunction}}[{\mathcal {D}},t]\,dt}{{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 1 Sparsity ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}t\exp \left(\xi \left(i\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{\tau }{\text{Sparsity}}({\mathcal {D}},s)\,ds-t\right)\right)\right)d\tau d\xi dt}{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 1 Sparsity ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{0}^{{\text{CDF}}[{\mathcal {D}},x]}\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{t}{\text{Sparsity}}({\mathcal {D}},s)\,ds\right)\,dt}{{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 1 Lorenz Curve 1 ${\text{identical properties}}$ Lorenz Curve 1 Lorenz Curve 2 ${\text{LorenzCurve1}}({\mathcal {D}},x)={\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},x])$ Lorenz Curve 1 Mean Residual Life Function ${\text{LorenzCurve1}}({\mathcal {D}},x)=1-{\frac {({\text{MeanResidualLifeFunction}}({\mathcal {D}},x)+x){\text{SurvivalFunction}}[{\mathcal {D}},x]}{{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 1 Moment ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k-1)}(x)\right){\text{Moment}}[{\mathcal {D}},k+1]}{k!}}}{{\text{Mean}}[{\mathcal {D}}]}}+\theta (x)$ Lorenz Curve 1 Central Moment ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k-1)}(x)\right)\sum _{j=0}^{k+1}{\text{CentralMoment}}[{\mathcal {D}},j]\left({\binom {k+1}{j}}{\text{Mean}}[{\mathcal {D}}]^{n-j}\right)}{k!}}}{{\text{Mean}}[{\mathcal {D}}]}}+\theta (x)$ Lorenz Curve 1 Factorial Moment ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k-1)}(x)\right)\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{k!}}}{{\text{Mean}}[{\mathcal {D}}]}}+\theta (x)$ Lorenz Curve 1 Ascending Factorial Moment ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k-1)}(x)\right)\sum _{j=0}^{k+1}\left((-1)^{-j+k-1}{\mathcal {S}}_{k+1}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)}{k!}}}{{\text{Mean}}[{\mathcal {D}}]}}+\theta (x)$ Lorenz Curve 1 Cumulant ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k-1)}(x)\right){\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]]}{k!}}}{{\text{Mean}}[{\mathcal {D}}]}}+\theta (x)$ Lorenz Curve 1 Factorial Cumulant ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k-1)}(x)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j}{\mathcal {S}}_{j}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{k!}}}{{\text{Mean}}[{\mathcal {D}}]}}+\theta (x)$ Lorenz Curve 1 Ascending Factorial Cumulant ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k-1)}(x)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{k!}}}{{\text{Mean}}[{\mathcal {D}}]}}+\theta (x)$ Lorenz Curve 1 Moment Generating Function ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{x}{\frac {t{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{e^{(i\tau )t}}}dtd\tau }{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 1 Moment Generating Function ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {1}{2}}+{\frac {{\mathcal {F}}_{\tau }\left[{\frac {(1+\tau (ix)){\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau ^{2}}}\right](-x)}{{\sqrt {2\pi }}{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 1 Central Moment Generating Function ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{x}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]\left(te^{(i\tau )({\text{Mean}}[{\mathcal {D}}]-t)}\right)dtd\tau }{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 1 Central Moment Generating Function ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {1}{2}}+{\frac {{\mathcal {F}}_{\tau }\left[{\frac {{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]\left((1+\tau (ix))e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}\right)}{\tau ^{2}}}\right](-x)}{{\sqrt {2\pi }}{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 1 Cumulant Generating Function ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{x}t\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ]-(i\tau )t)dtd\tau }{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 1 Cumulant Generating Function ${\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {1}{2}}+{\frac {{\mathcal {F}}_{\tau }\left[{\frac {(1+\tau (ix))\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau ^{2}}}\right](-x)}{{\sqrt {2\pi }}{\text{Mean}}[{\mathcal {D}}]}}$

Lorenz Curve 2

Relations Between Distribution Properties

Derive Using Formula Lorenz Curve 2 PDF ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}t{\text{PDF}}[{\mathcal {D}},t]\,dt}{\int _{-\infty }^{\infty }t{\text{PDF}}[{\mathcal {D}},t]\,dt}}$ Lorenz Curve 2 PDF ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}t{\text{PDF}}[{\mathcal {D}},t]\,dt}{{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 2 CDF ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}t{\frac {\partial {\text{CDF}}[{\mathcal {D}},t]}{\partial t}}\,dt}{{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 2 CDF ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {x{\text{InverseCDF}}[{\mathcal {D}},x]-\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}{\text{CDF}}[{\mathcal {D}},t]\,dt}{{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 2 Characteristic Function ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}{\frac {t{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{e^{(i\tau )t}}}dtd\tau }{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 2 Characteristic Function ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}t\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-t)\right)\,dt}{{\sqrt {2\pi }}{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 2 Characteristic Function ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {1}{2}}+{\frac {{\mathcal {F}}_{\tau }\left[{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ](1+\tau (i{\text{InverseCDF}}[{\mathcal {D}},x]))}{\tau ^{2}}}\right](-{\text{InverseCDF}}[{\mathcal {D}},x])}{{\sqrt {2\pi }}{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 2 Survival Function ${\text{LorenzCurve2}}({\mathcal {D}},x)=-{\frac {\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}t{\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},t]}{\partial t}}\,dt}{{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 2 Survival Function ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {x{\text{InverseCDF}}[{\mathcal {D}},x]-\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}(1-{\text{SurvivalFunction}}[{\mathcal {D}},t])\,dt}{{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 2 Cumulative Hazard Function ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}{\frac {t{\frac {\partial {\text{CumulativeHazardFunction}}({\mathcal {D}},t)}{\partial t}}}{e^{{\text{CumulativeHazardFunction}}({\mathcal {D}},t)}}}\,dt}{{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 2 Hazard Function ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}{\text{HazardFunction}}[{\mathcal {D}},t]\left(t\exp \left(-\int _{-\infty }^{t}{\text{HazardFunction}}[{\mathcal {D}},\tau ]\,d\tau \right)\right)\,dt}{{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 2 Inverse CDF ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}\int _{-\infty }^{\infty }\int _{0}^{1}te^{\xi (i({\text{InverseCDF}}[{\mathcal {D}},\tau ]-t))}d\tau d\xi dt}{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 2 Inverse CDF ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{0}^{x}{\text{InverseCDF}}[{\mathcal {D}},t]\,dt}{{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 2 Quantile ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}\int _{-\infty }^{\infty }\int _{0}^{1}te^{\xi (i({\text{Quantile}}[{\mathcal {D}},\tau ]-t))}d\tau d\xi dt}{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 2 Quantile ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{0}^{x}{\text{Quantile}}[{\mathcal {D}},t]\,dt}{{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 2 Inverse Survival Function ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}\int _{-\infty }^{\infty }\int _{0}^{1}te^{\xi (i({\text{InverseSurvivalFunction}}[{\mathcal {D}},\tau ]-t))}d\tau d\xi dt}{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 2 Inverse Survival Function ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{1-x}^{1}{\text{InverseSurvivalFunction}}[{\mathcal {D}},t]\,dt}{{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 2 Sparsity ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}\int _{-\infty }^{\infty }\int _{0}^{1}t\exp \left(\xi \left(i\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{\tau }{\text{Sparsity}}({\mathcal {D}},s)\,ds-t\right)\right)\right)d\tau d\xi dt}{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 2 Sparsity ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{0}^{x}\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{t}{\text{Sparsity}}({\mathcal {D}},s)\,ds\right)\,dt}{{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 2 Lorenz Curve 1 ${\text{LorenzCurve2}}({\mathcal {D}},x)={\text{LorenzCurve1}}({\mathcal {D}},{\text{InverseCDF}}[{\mathcal {D}},x])$ Lorenz Curve 2 Lorenz Curve 2 ${\text{identical properties}}$ Lorenz Curve 2 Mean Residual Life Function ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}t{\frac {\partial ^{2}({\text{MeanResidualLifeFunction}}({\mathcal {D}},t){\text{SurvivalFunction}}[{\mathcal {D}},t])}{\partial t^{2}}}\,dt}{{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 2 Mean Residual Life Function ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {{\frac {{\text{InverseCDF}}[{\mathcal {D}},x]{\frac {\partial ({\text{MeanResidualLifeFunction}}({\mathcal {D}},{\text{InverseCDF}}[{\mathcal {D}},x]){\text{SurvivalFunction}}[{\mathcal {D}},{\text{InverseCDF}}[{\mathcal {D}},x]])}{\partial x^{1}}}}{\frac {\partial {\text{InverseCDF}}[{\mathcal {D}},x]}{\partial x}}}-{\text{MeanResidualLifeFunction}}({\mathcal {D}},{\text{InverseCDF}}[{\mathcal {D}},x]){\text{SurvivalFunction}}[{\mathcal {D}},{\text{InverseCDF}}[{\mathcal {D}},x]]}{{\text{Mean}}[{\mathcal {D}}]}}+1$ Lorenz Curve 2 Moment ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\sum _{k=1}^{\infty }{\frac {{\text{Moment}}[{\mathcal {D}},k+1]\left((-1)^{k}\delta ^{(k-1)}({\text{InverseCDF}}[{\mathcal {D}},x])\right)}{k!}}}{{\text{Mean}}[{\mathcal {D}}]}}+\theta ({\text{InverseCDF}}[{\mathcal {D}},x])$ Lorenz Curve 2 Central Moment ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k-1)}({\text{InverseCDF}}[{\mathcal {D}},x])\right)\sum _{j=0}^{k+1}{\text{CentralMoment}}[{\mathcal {D}},j]\left({\binom {k+1}{j}}{\text{Mean}}[{\mathcal {D}}]^{n-j}\right)}{k!}}}{{\text{Mean}}[{\mathcal {D}}]}}+\theta ({\text{InverseCDF}}[{\mathcal {D}},x])$ Lorenz Curve 2 Factorial Moment ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k-1)}({\text{InverseCDF}}[{\mathcal {D}},x])\right)\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{k!}}}{{\text{Mean}}[{\mathcal {D}}]}}+\theta ({\text{InverseCDF}}[{\mathcal {D}},x])$ Lorenz Curve 2 Ascending Factorial Moment ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\sum _{k=1}^{\infty }{\frac {\sum _{j=0}^{k+1}\left((-1)^{-j+k-1}{\mathcal {S}}_{k+1}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)\left((-1)^{k}\delta ^{(k-1)}({\text{InverseCDF}}[{\mathcal {D}},x])\right)}{k!}}}{{\text{Mean}}[{\mathcal {D}}]}}+\theta ({\text{InverseCDF}}[{\mathcal {D}},x])$ Lorenz Curve 2 Cumulant ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k-1)}({\text{InverseCDF}}[{\mathcal {D}},x])\right){\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]]}{k!}}}{{\text{Mean}}[{\mathcal {D}}]}}+\theta ({\text{InverseCDF}}[{\mathcal {D}},x])$ Lorenz Curve 2 Factorial Cumulant ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k-1)}({\text{InverseCDF}}[{\mathcal {D}},x])\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j}{\mathcal {S}}_{j}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{k!}}}{{\text{Mean}}[{\mathcal {D}}]}}+\theta ({\text{InverseCDF}}[{\mathcal {D}},x])$ Lorenz Curve 2 Ascending Factorial Cumulant ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k-1)}({\text{InverseCDF}}[{\mathcal {D}},x])\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{k!}}}{{\text{Mean}}[{\mathcal {D}}]}}+\theta (x)$ Lorenz Curve 2 Moment Generating Function ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}{\frac {t{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{e^{(i\tau )t}}}dtd\tau }{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 2 Moment Generating Function ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {1}{2}}+{\frac {{\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ](1+\tau (i{\text{InverseCDF}}[{\mathcal {D}},x]))}{\tau ^{2}}}\right](-{\text{InverseCDF}}[{\mathcal {D}},x])}{{\sqrt {2\pi }}{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 2 Central Moment Generating Function ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]\left(te^{(i\tau )({\text{Mean}}[{\mathcal {D}}]-t)}\right)dtd\tau }{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 2 Central Moment Generating Function ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {1}{2}}+{\frac {{\mathcal {F}}_{\tau }\left[{\frac {{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]\left(e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}(1+\tau (i{\text{InverseCDF}}[{\mathcal {D}},x]))\right)}{\tau ^{2}}}\right](-{\text{InverseCDF}}[{\mathcal {D}},x])}{{\sqrt {2\pi }}{\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 2 Cumulant Generating Function ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}t\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ]-(i\tau )t)dtd\tau }{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}$ Lorenz Curve 2 Cumulant Generating Function ${\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {1}{2}}+{\frac {{\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])(1+\tau (i{\text{InverseCDF}}[{\mathcal {D}},x]))}{\tau ^{2}}}\right](-{\text{InverseCDF}}[{\mathcal {D}},x])}{{\sqrt {2\pi }}{\text{Mean}}[{\mathcal {D}}]}}$

Mean Residual Life Function

Relations Between Distribution Properties

Derive Using Formula Mean Residual Life Function PDF ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\int _{x}^{\infty }(t-x){\text{PDF}}[{\mathcal {D}},t]\,dt}{\int _{x}^{\infty }{\text{PDF}}[{\mathcal {D}},t]\,dt}}$ Mean Residual Life Function PDF ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\int _{0}^{\infty }t{\text{PDF}}[{\mathcal {D}},x+t]\,dt}{\int _{0}^{\infty }{\text{PDF}}[{\mathcal {D}},x+t]\,dt}}$ Mean Residual Life Function PDF ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\int _{x}^{\infty }t{\text{PDF}}[{\mathcal {D}},t]\,dt}{\int _{x}^{\infty }{\text{PDF}}[{\mathcal {D}},t]\,dt}}$ Mean Residual Life Function PDF ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}]-x-\int _{-\infty }^{x}(t-x){\text{PDF}}[{\mathcal {D}},t]\,dt}{1-\int _{-\infty }^{x}{\text{PDF}}[{\mathcal {D}},t]\,dt}}$ Mean Residual Life Function PDF ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=\left({\begin{array}{cc}\{&{\begin{array}{cc}{\text{Mean}}[{\mathcal {D}}]-x&x<u\\-x+{\frac {{\text{Mean}}[{\mathcal {D}}]-\int _{u}^{x}t{\text{PDF}}[{\mathcal {D}},t]\,dt}{\int _{x}^{\infty }{\text{PDF}}[{\mathcal {D}},t]\,dt}}&u\leq x<v\\{\text{Indeterminate}}&{\text{True}}\\\end{array}}\\\end{array}}\right){\text{/;}}u={\text{support}}[{\mathcal {D}}][[1,1]]\land v={\text{support}}[{\mathcal {D}}][[1,2]]$ Mean Residual Life Function CDF ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}]-x+\int _{-\infty }^{x}{\text{CDF}}[{\mathcal {D}},t]\,dt}{1-{\text{CDF}}[{\mathcal {D}},x]}}$ Mean Residual Life Function CDF ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\int _{x}^{\infty }(t-x){\frac {\partial {\text{CDF}}[{\mathcal {D}},t]}{\partial t}}\,dt}{1-{\text{CDF}}[{\mathcal {D}},x]}}$ Mean Residual Life Function CDF ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\int _{0}^{\infty }t{\frac {\partial {\text{CDF}}[{\mathcal {D}},t+x]}{\partial t}}\,dt}{1-{\text{CDF}}[{\mathcal {D}},x]}}$ Mean Residual Life Function CDF ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\int _{x}^{\infty }t{\frac {\partial {\text{CDF}}[{\mathcal {D}},t]}{\partial t}}\,dt}{1-{\text{CDF}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Characteristic Function ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\int _{x}^{\infty }t\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-t)\right)\,dt}{{\sqrt {2\pi }}{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Characteristic Function ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {2\int _{x}^{\infty }t\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-t)\right)\,dt}{{\sqrt {2\pi }}-(2i)\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{\tau }}\right](-x)\right)}}$ Mean Residual Life Function Survival Function ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}]-x+\int _{-\infty }^{x}(1-{\text{SurvivalFunction}}[{\mathcal {D}},t])\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Survival Function ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-{\frac {\int _{x}^{\infty }(t-x){\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},t]}{\partial t}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Survival Function ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-{\frac {\int _{0}^{\infty }t{\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},t+x]}{\partial t}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Survival Function ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x-{\frac {\int _{x}^{\infty }t{\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},t]}{\partial t}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Survival Function ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {{\text{Mean}}[{\mathcal {D}}]+\int _{-\infty }^{x}t{\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},t]}{\partial t}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Cumulative Hazard Function ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-{\frac {\int _{0}^{\infty }{\frac {t{\frac {\partial {\text{CumulativeHazardFunction}}({\mathcal {D}},t+x)}{\partial t}}}{e^{{\text{CumulativeHazardFunction}}({\mathcal {D}},t+x)}}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Cumulative Hazard Function ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {-x\exp(-{\text{CumulativeHazardFunction}}({\mathcal {D}},x))+{\text{Mean}}[{\mathcal {D}}]-\int _{-\infty }^{x}{\frac {t{\frac {\partial {\text{CumulativeHazardFunction}}({\mathcal {D}},t)}{\partial t}}}{e^{{\text{CumulativeHazardFunction}}({\mathcal {D}},t)}}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Hazard Function ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\int _{0}^{\infty }\left(t\exp \left(-\int _{-\infty }^{t+x}{\text{HazardFunction}}[{\mathcal {D}},\tau ]\,d\tau \right)\right){\text{HazardFunction}}[{\mathcal {D}},t+x]\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Hazard Function ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {-x\exp \left(-\int _{-\infty }^{x}{\text{HazardFunction}}[{\mathcal {D}},t]\,dt\right)+{\text{Mean}}[{\mathcal {D}}]-\int _{-\infty }^{x}\left(t\exp \left(-\int _{-\infty }^{t}{\text{HazardFunction}}[{\mathcal {D}},\tau ]\,d\tau \right)\right){\text{HazardFunction}}[{\mathcal {D}},t]\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Inverse CDF ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{0}^{1}te^{(i({\text{InverseCDF}}[{\mathcal {D}},\tau ]-t-x))\xi }d\tau d\xi dt}{(2\pi ){\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Inverse CDF ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {{\text{Mean}}[{\mathcal {D}}]-\int _{0}^{{\text{CDF}}[{\mathcal {D}},x]}{\text{InverseCDF}}[{\mathcal {D}},t]\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Quantile ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {{\text{Mean}}[{\mathcal {D}}]-{\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}te^{(i({\text{Quantile}}[{\mathcal {D}},\tau ]-t))\xi }d\tau d\xi dt}{2\pi }}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Quantile ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{0}^{1}te^{(i({\text{Quantile}}[{\mathcal {D}},\tau ]-t-x))\xi }d\tau d\xi dt}{(2\pi ){\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Quantile ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {{\text{Mean}}[{\mathcal {D}}]-\int _{0}^{{\text{CDF}}[{\mathcal {D}},x]}{\text{Quantile}}[{\mathcal {D}},t]\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Inverse Survival Function ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{0}^{1}te^{(i({\text{InverseSurvivalFunction}}[{\mathcal {D}},\tau ]-t-x))\xi }d\tau d\xi dt}{(2\pi ){\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Inverse Survival Function ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {{\text{Mean}}[{\mathcal {D}}]-\int _{{\text{SurvivalFunction}}[{\mathcal {D}},x]}^{1}{\text{InverseSurvivalFunction}}[{\mathcal {D}},t]\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Sparsity ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{0}^{1}t\exp \left(\left(i\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{\tau }{\text{Sparsity}}({\mathcal {D}},s)\,ds-t-x\right)\right)\xi \right)d\tau d\xi dt}{(2\pi ){\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Sparsity ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {-x(1-{\text{CDF}}[{\mathcal {D}},x])+{\text{Mean}}[{\mathcal {D}}]-{\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}t\exp \left(\left(i\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{\tau }{\text{Sparsity}}({\mathcal {D}},s)\,ds-t\right)\right)\xi \right)d\tau d\xi dt}{2\pi }}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Sparsity ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {-x(1-{\text{CDF}}[{\mathcal {D}},x])+{\text{Mean}}[{\mathcal {D}}]-\int _{0}^{{\text{CDF}}[{\mathcal {D}},x]}\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{t}{\text{Sparsity}}({\mathcal {D}},s)\,ds\right)\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Lorenz Curve 1 ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}](1-{\text{LorenzCurve1}}({\mathcal {D}},x))}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}-x$ Mean Residual Life Function Lorenz Curve 1 ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}]\int _{0}^{\infty }{\frac {t{\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},t+x)}{\partial t}}}{t+x}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Lorenz Curve 1 ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}](1-{\text{LorenzCurve1}}({\mathcal {D}},x))-x+(x{\text{Mean}}[{\mathcal {D}}])\int _{-\infty }^{x}{\frac {\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},t)}{\partial t}}{t}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Lorenz Curve 1 ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}]-x+({\text{Mean}}[{\mathcal {D}}]x)\int _{-\infty }^{x}{\frac {{\text{LorenzCurve1}}({\mathcal {D}},t)}{t^{2}}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Lorenz Curve 2 ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}](1-{\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},x]))}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}-x$ Mean Residual Life Function Lorenz Curve 2 ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}]\int _{0}^{\infty }{\frac {t{\frac {\partial {\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},t+x])}{\partial t}}}{t+x}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Mean Residual Life Function ${\text{identical properties}}$ Mean Residual Life Function Moment ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}](1-\theta (x))-x\theta (-x)+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)(x{\text{Moment}}[{\mathcal {D}},k+1]+{\text{Moment}}[{\mathcal {D}},k+2])}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Moment ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\theta (-x){\text{Mean}}[{\mathcal {D}}]+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right){\text{Moment}}[{\mathcal {D}},k+2]}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Central Moment ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}](1-\theta (x))-x\theta (-x)+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\left(x\sum _{j=0}^{k+1}\left({\binom {k+1}{j}}{\text{Mean}}[{\mathcal {D}}]^{k+1-j}\right){\text{CentralMoment}}[{\mathcal {D}},j]+\sum _{j=0}^{k+2}\left({\binom {k+2}{j}}{\text{Mean}}[{\mathcal {D}}]^{k+2-j}\right){\text{CentralMoment}}[{\mathcal {D}},j]\right)}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Central Moment ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\theta (-x){\text{Mean}}[{\mathcal {D}}]+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right)\sum _{j=0}^{k+2}\left({\binom {k+2}{j}}{\text{Mean}}[{\mathcal {D}}]^{k+2-j}\right){\text{CentralMoment}}[{\mathcal {D}},j]}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Factorial Moment ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}](1-\theta (x))-x\theta (-x)+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\left(x\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]+\sum _{j=0}^{k+2}{\mathcal {S}}_{k+2}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]\right)}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Factorial Moment ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\theta (-x){\text{Mean}}[{\mathcal {D}}]+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right)\sum _{j=0}^{k+2}{\mathcal {S}}_{k+2}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Ascending Factorial Moment ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}](1-\theta (x))-x\theta (-x)+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\left(x\sum _{j=0}^{k+1}\left((-1)^{k-j-1}{\mathcal {S}}_{k+1}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)+\sum _{j=0}^{k+2}\left((-1)^{k-j}{\mathcal {S}}_{k+2}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)\right)}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Ascending Factorial Moment ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\theta (-x){\text{Mean}}[{\mathcal {D}}]+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right)\sum _{j=0}^{k+2}\left((-1)^{k-j}{\mathcal {S}}_{k+2}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Cumulant ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)({\text{BellY}}[k+2,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+2\}]]+x{\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]])}{(k+1)!}}+(1-\theta (x)){\text{Mean}}[{\mathcal {D}}]-x\theta (-x)}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Cumulant ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\theta (-x){\text{Mean}}[{\mathcal {D}}]+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right){\text{BellY}}[k+2,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+2\}]]}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Factorial Cumulant ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}](1-\theta (x))-x\theta (-x)+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\left(x{\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j}{\mathcal {S}}_{j}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]+{\text{BellY}}\left[k+2,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j}{\mathcal {S}}_{j}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+2\}\right]\right]\right)}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Factorial Cumulant ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\theta (-x){\text{Mean}}[{\mathcal {D}}]+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right){\text{BellY}}\left[k+2,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j}{\mathcal {S}}_{j}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+2\}\right]\right]}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Ascending Factorial Cumulant ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\left({\text{BellY}}\left[k+2,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+2\}\right]\right]+x{\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]\right)}{(k+1)!}}+(1-\theta (x)){\text{Mean}}[{\mathcal {D}}]-x\theta (-x)}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Ascending Factorial Cumulant ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\theta (-x){\text{Mean}}[{\mathcal {D}}]+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right){\text{BellY}}\left[k+2,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+2\}\right]\right]}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Moment Generating Function ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\frac {1}{2}}({\text{Mean}}[{\mathcal {D}}]-x)-{\frac {{\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau ^{2}}}\right](-x)}{\sqrt {2\pi }}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Moment Generating Function ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\int _{x}^{\infty }t\left({\mathcal {F}}_{\tau }[{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]](-t)\right)\,dt}{{\sqrt {2\pi }}{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Moment Generating Function ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {2\int _{x}^{\infty }t\left({\mathcal {F}}_{\tau }[{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]](-t)\right)\,dt}{{\sqrt {2\pi }}-(2i)\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-x)\right)}}$ Mean Residual Life Function Central Moment Generating Function ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\frac {1}{2}}({\text{Mean}}[{\mathcal {D}}]-x)-{\frac {{\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau ^{2}}}\right](-x)}{\sqrt {2\pi }}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Central Moment Generating Function ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\int _{x}^{\infty }t\left({\mathcal {F}}_{\tau }\left[e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]\right](-t)\right)\,dt}{{\sqrt {2\pi }}{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Central Moment Generating Function ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {2\int _{x}^{\infty }t\left({\mathcal {F}}_{\tau }\left[e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]\right](-t)\right)\,dt}{{\sqrt {2\pi }}-(2i)\left({\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-x)\right)}}$ Mean Residual Life Function Cumulant Generating Function ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\frac {1}{2}}({\text{Mean}}[{\mathcal {D}}]-x)-{\frac {{\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau ^{2}}}\right](-x)}{\sqrt {2\pi }}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Cumulant Generating Function ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\int _{x}^{\infty }t\left({\mathcal {F}}_{\tau }[\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])](-t)\right)\,dt}{{\sqrt {2\pi }}{\text{SurvivalFunction}}[{\mathcal {D}},x]}}$ Mean Residual Life Function Cumulant Generating Function ${\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {2\int _{x}^{\infty }t\left({\mathcal {F}}_{\tau }[\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])](-t)\right)\,dt}{{\sqrt {2\pi }}-(2i)\left({\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau }}\right](-x)\right)}}$

Moments

Mean

Relations Between Distribution Properties

Derive Using Formula Mean PDF ${\text{Mean}}[{\mathcal {D}}]=\int _{-\infty }^{\infty }t{\text{PDF}}[{\mathcal {D}},t]\,dt$ Mean CDF ${\text{Mean}}[{\mathcal {D}}]=\int _{-\infty }^{\infty }t{\frac {\partial {\text{CDF}}[{\mathcal {D}},t]}{\partial t}}\,dt$ Mean CDF ${\text{Mean}}[{\mathcal {D}}]=\int _{0}^{\infty }(1-{\text{CDF}}[{\mathcal {D}},t]-{\text{CDF}}[{\mathcal {D}},-t])\,dt$ Mean Characteristic Function ${\text{Mean}}[{\mathcal {D}}]=-i{\text{CharacteristicFunction}}^{(0,1)}({\mathcal {D}},0)$ Mean Survival Function ${\text{Mean}}[{\mathcal {D}}]=-\int _{-\infty }^{\infty }t{\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},t]}{\partial t}}\,dt$ Mean Survival Function ${\text{Mean}}[{\mathcal {D}}]=\int _{0}^{\infty }({\text{SurvivalFunction}}[{\mathcal {D}},t]+{\text{SurvivalFunction}}[{\mathcal {D}},-t]-1)\,dt$ Mean Cumulative Hazard Function ${\text{Mean}}[{\mathcal {D}}]=\int _{-\infty }^{\infty }{\frac {t{\frac {\partial {\text{CumulativeHazardFunction}}({\mathcal {D}},t)}{\partial t}}}{e^{{\text{CumulativeHazardFunction}}({\mathcal {D}},t)}}}\,dt$ Mean Hazard Function ${\text{Moment}}[{\mathcal {D}},1]={\text{Too complicated}}$ Mean Inverse CDF ${\text{Mean}}[{\mathcal {D}}]=\int _{0}^{1}{\text{InverseCDF}}[{\mathcal {D}},t]\,dt$ Mean Quantile ${\text{Mean}}[{\mathcal {D}}]=\int _{0}^{1}{\text{Quantile}}[{\mathcal {D}},t]\,dt$ Mean Inverse Survival Function ${\text{Mean}}[{\mathcal {D}}]=\int _{0}^{1}{\text{InverseSurvivalFunction}}[{\mathcal {D}},t]\,dt$ Mean Sparsity ${\text{Mean}}[{\mathcal {D}}]={\text{Median}}[{\mathcal {D}}]+\int _{0}^{1}\int _{\frac {1}{2}}^{t}{\text{Sparsity}}({\mathcal {D}},\tau )d\tau dt$ Mean Mean Residual Life Function ${\text{Mean}}[{\mathcal {D}}]=\int _{-\infty }^{\infty }t{\frac {\partial ^{2}({\text{MeanResidualLifeFunction}}({\mathcal {D}},t){\text{SurvivalFunction}}[{\mathcal {D}},t])}{\partial t^{2}}}\,dt$ Mean Mean ${\text{identical properties}}$ Mean Moment ${\text{Mean}}[{\mathcal {D}}]={\text{Moment}}[{\mathcal {D}},1]$ Mean Factorial Moment ${\text{Mean}}[{\mathcal {D}}]={\text{FactorialMoment}}[{\mathcal {D}},1]$ Mean Ascending Factorial Moment ${\text{Mean}}[{\mathcal {D}}]={\text{AscendingFactorialMoment}}({\mathcal {D}},1)$ Mean Cumulant ${\text{Mean}}[{\mathcal {D}}]={\text{Cumulant}}[{\mathcal {D}},1]$ Mean Factorial Cumulant ${\text{Mean}}[{\mathcal {D}}]={\text{FactorialCumulant}}({\mathcal {D}},1)$ Mean Ascending Factorial Cumulant ${\text{Mean}}[{\mathcal {D}}]={\text{AscendingFactorialCumulant}}({\mathcal {D}},1)$ Mean Moment Generating Function ${\text{Mean}}[{\mathcal {D}}]={\text{MomentGeneratingFunction}}^{(0,1)}({\mathcal {D}},0)$ Mean Factorial Moment Generating Function ${\text{Mean}}[{\mathcal {D}}]={\text{FactorialMomentGeneratingFunction}}^{(0,1)}({\mathcal {D}},1)$ Mean Descending Factorial Moment Generating Function ${\text{Mean}}[{\mathcal {D}}]={\text{DescendingFactorialMomentGeneratingFunction}}^{(0,1)}({\mathcal {D}},0)$ Mean Ascending Factorial Moment Generating Function ${\text{Mean}}[{\mathcal {D}}]={\text{AscendingFactorialMomentGeneratingFunction}}^{(0,1)}({\mathcal {D}},0)$ Mean Cumulant Generating Function ${\text{Mean}}[{\mathcal {D}}]={\text{CumulantGeneratingFunction}}^{(0,1)}({\mathcal {D}},1)$ Mean Factorial Cumulant Generating Function ${\text{Mean}}[{\mathcal {D}}]={\text{FactorialCumulantGeneratingFunction}}^{(0,1)}({\mathcal {D}},1)$ Mean Descending Factorial Cumulant Generating Function ${\text{Mean}}[{\mathcal {D}}]={\text{DescendingFactorialCumulantGeneratingFunction}}^{(0,1)}({\mathcal {D}},0)$ Mean Ascending Factorial Cumulant Generating Function ${\text{Mean}}[{\mathcal {D}}]={\text{AscendingFactorialCumulantGeneratingFunction}}^{(0,1)}({\mathcal {D}},0)$

Variance

Relations Between Distribution Properties

Derive Using Formula Variance PDF ${\text{Variance}}[{\mathcal {D}}]=\int _{-\infty }^{\infty }t^{2}{\text{PDF}}[{\mathcal {D}},t]\,dt-\left(\int _{-\infty }^{\infty }t{\text{PDF}}[{\mathcal {D}},t]\,dt\right){}^{2}$ Variance CDF ${\text{Variance}}[{\mathcal {D}}]=\int _{-\infty }^{\infty }t^{2}{\frac {\partial {\text{CDF}}[{\mathcal {D}},t]}{\partial t}}\,dt-\left(\int _{-\infty }^{\infty }t{\frac {\partial {\text{CDF}}[{\mathcal {D}},t]}{\partial t}}\,dt\right){}^{2}$ Variance Characteristic Function ${\text{Variance}}[{\mathcal {D}}]={\text{CharacteristicFunction}}^{(0,1)}({\mathcal {D}},0)^{2}-{\text{CharacteristicFunction}}^{(0,2)}({\mathcal {D}},0)$ Variance Survival Function ${\text{Variance}}[{\mathcal {D}}]=-\int _{-\infty }^{\infty }t^{2}{\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},t]}{\partial t}}\,dt-\left(\int _{-\infty }^{\infty }t{\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},t]}{\partial t}}\,dt\right){}^{2}$ Variance Cumulative Hazard Function ${\text{Variance}}[{\mathcal {D}}]=\int _{-\infty }^{\infty }{\frac {t^{2}{\frac {\partial {\text{CumulativeHazardFunction}}({\mathcal {D}},t)}{\partial t}}}{e^{{\text{CumulativeHazardFunction}}({\mathcal {D}},t)}}}\,dt-\left(\int _{-\infty }^{\infty }{\frac {t{\frac {\partial {\text{CumulativeHazardFunction}}({\mathcal {D}},t)}{\partial t}}}{e^{{\text{CumulativeHazardFunction}}({\mathcal {D}},t)}}}\,dt\right){}^{2}$ Variance Hazard Function ${\text{Moment}}[{\mathcal {D}},1]={\text{Too complicated}}$ Variance Inverse CDF ${\text{Variance}}[{\mathcal {D}}]=\int _{0}^{1}{\text{InverseCDF}}[{\mathcal {D}},t]^{2}\,dt-\left(\int _{0}^{1}{\text{InverseCDF}}[{\mathcal {D}},t]\,dt\right){}^{2}$ Variance Quantile ${\text{Variance}}[{\mathcal {D}}]=\int _{0}^{1}{\text{Quantile}}[{\mathcal {D}},t]^{2}\,dt-\left(\int _{0}^{1}{\text{Quantile}}[{\mathcal {D}},t]\,dt\right){}^{2}$ Variance Inverse Survival Function ${\text{Variance}}[{\mathcal {D}}]=\int _{0}^{1}{\text{InverseSurvivalFunction}}[{\mathcal {D}},t]^{2}\,dt-\left(\int _{0}^{1}{\text{InverseSurvivalFunction}}[{\mathcal {D}},t]\,dt\right){}^{2}$ Variance Sparsity ${\text{Variance}}[{\mathcal {D}}]=\int _{0}^{1}\left(\int _{\frac {1}{2}}^{t}{\text{Sparsity}}({\mathcal {D}},\tau )\,d\tau \right){}^{2}\,dt-\left(\int _{0}^{1}\int _{\frac {1}{2}}^{t}{\text{Sparsity}}({\mathcal {D}},\tau )d\tau dt\right){}^{2}$ Variance Mean Residual Life Function ${\text{Variance}}[{\mathcal {D}}]=\int _{-\infty }^{\infty }t^{2}{\frac {\partial ^{2}({\text{MeanResidualLifeFunction}}({\mathcal {D}},t){\text{SurvivalFunction}}[{\mathcal {D}},t])}{\partial t^{2}}}\,dt-\left(\int _{-\infty }^{\infty }t{\frac {\partial ^{2}({\text{MeanResidualLifeFunction}}({\mathcal {D}},t){\text{SurvivalFunction}}[{\mathcal {D}},t])}{\partial t^{2}}}\,dt\right){}^{2}$ Variance Variance ${\text{identical properties}}$ Variance Standard Deviation ${\text{Variance}}[{\mathcal {D}}]={\text{StandardDeviation}}[{\mathcal {D}}]^{2}$ Variance Moment ${\text{Variance}}[{\mathcal {D}}]={\text{Moment}}[{\mathcal {D}},2]-{\text{Moment}}[{\mathcal {D}},1]^{2}$ Variance Central Moment ${\text{Variance}}[{\mathcal {D}}]={\text{CentralMoment}}[{\mathcal {D}},2]$ Variance Factorial Moment ${\text{Variance}}[{\mathcal {D}}]=-{\text{FactorialMoment}}[{\mathcal {D}},1]^{2}+{\text{FactorialMoment}}[{\mathcal {D}},1]+{\text{FactorialMoment}}[{\mathcal {D}},2]$ Variance Ascending Factorial Moment Failed to parse (unknown function "\tex"): {\displaystyle \text{Variance}[\mathcal{D}]=-\text{AscendingFactorialMoment}(\mathcal{D},1)^2-\text{AscendingFactorialMoment}(\mathcal{D},1)+\tex t{AscendingFactorialMoment}(\mathcal{D},2) } Variance Cumulant ${\text{Variance}}[{\mathcal {D}}]={\text{Cumulant}}[{\mathcal {D}},2]$ Variance Factorial Cumulant ${\text{Variance}}[{\mathcal {D}}]={\text{FactorialCumulant}}({\mathcal {D}},1)+{\text{FactorialCumulant}}({\mathcal {D}},2)$ Variance Ascending Factorial Cumulant ${\text{Variance}}[{\mathcal {D}}]={\text{AscendingFactorialCumulant}}({\mathcal {D}},2)-{\text{AscendingFactorialCumulant}}({\mathcal {D}},1)$ Variance Moment Generating Function Failed to parse (unknown function "\mathc"): {\displaystyle \text{Variance}[\mathcal{D}]=\text{MomentGeneratingFunction}^{(0,2)}(\mathcal{D},0)-\text{MomentGeneratingFunction}^{(0,1)}(\mathc al{D},0)^2 } Variance Central Moment Generating Function ${\text{Variance}}[{\mathcal {D}}]={\text{CentralMomentGeneratingFunction}}^{(0,2)}({\mathcal {D}},0)$ Variance Factorial Moment Generating Function Failed to parse (syntax error): {\displaystyle \text{Variance}[\mathcal{D}]=-\text{FactorialMomentGeneratingFunction}^{(0,1)}(\mathcal{D},1)^2+\text{FactorialMomentGeneratingFun ction}^{(0,1)}(\mathcal{D},1)+\text{FactorialMomentGeneratingFunction}^{(0,2)}(\mathcal{D},1) } Variance Descending Factorial Moment Generating Function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \text{Variance}[\mathcal{D}]=\text{DescendingFactorialMomentGeneratingFunction}^{(0,1)}(\mathcal{D},0)-\text{DescendingFactorialMo mentGeneratingFunction}^{(0,1)}(\mathcal{D},0)^2+\text{DescendingFactorialMomentGeneratingFunction}^{(0,2)}(\mathcal{D},0) } Variance Ascending Factorial Moment Generating Function Failed to parse (syntax error): {\displaystyle \text{Variance}[\mathcal{D}]=-\text{AscendingFactorialMomentGeneratingFunction}^{(0,1)}(\mathcal{D},0)^2-\text{AscendingFactorialM omentGeneratingFunction}^{(0,1)}(\mathcal{D},0)+\text{AscendingFactorialMomentGeneratingFunction}^{(0,2)}(\mathcal{D},0) } Variance Cumulant Generating Function ${\text{Variance}}[{\mathcal {D}}]={\text{CumulantGeneratingFunction}}^{(0,2)}({\mathcal {D}},1)$ Variance Factorial Cumulant Generating Function Failed to parse (syntax error): {\displaystyle \text{Variance}[\mathcal{D}]=\text{FactorialCumulantGeneratingFunction}^{(0,1)}(\mathcal{D},1)+\text{FactorialCumulantGeneratingFu nction}^{(0,2)}(\mathcal{D},1) } Variance Descending Factorial Cumulant Generating Function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \text{Variance}[\mathcal{D}]=\text{DescendingFactorialCumulantGeneratingFunction}^{(0,1)}(\mathcal{D},0)+\text{DescendingFactorial CumulantGeneratingFunction}^{(0,2)}(\mathcal{D},0) } Variance Ascending Factorial Cumulant Generating Function Failed to parse (syntax error): {\displaystyle \text{Variance}[\mathcal{D}]=\text{AscendingFactorialCumulantGeneratingFunction}^{(0,2)}(\mathcal{D},0)-\text{AscendingFactorialCu mulantGeneratingFunction}^{(0,1)}(\mathcal{D},0) }

Standard Deviation

Relations Between Distribution Properties

Derive Using Formula Standard Deviation PDF ${\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {\int _{-\infty }^{\infty }t^{2}{\text{PDF}}[{\mathcal {D}},t]\,dt-\left(\int _{-\infty }^{\infty }t{\text{PDF}}[{\mathcal {D}},t]\,dt\right){}^{2}}}$ Standard Deviation CDF ${\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {\int _{-\infty }^{\infty }t^{2}{\text{CDF}}^{(0,1)}({\mathcal {D}},t)\,dt-\left(\int _{-\infty }^{\infty }t{\text{CDF}}^{(0,1)}({\mathcal {D}},t)\,dt\right){}^{2}}}$ Standard Deviation Characteristic Function ${\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {{\text{CharacteristicFunction}}^{(0,1)}({\mathcal {D}},0)^{2}-{\text{CharacteristicFunction}}^{(0,2)}({\mathcal {D}},0)}}$ Standard Deviation Survival Function ${\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {-\int _{-\infty }^{\infty }t^{2}{\text{SurvivalFunction}}^{(0,1)}({\mathcal {D}},t)\,dt-\left(\int _{-\infty }^{\infty }t{\text{SurvivalFunction}}^{(0,1)}({\mathcal {D}},t)\,dt\right){}^{2}}}$ Standard Deviation Cumulative Hazard Function ${\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {\int _{-\infty }^{\infty }t^{2}e^{-{\text{CumulativeHazardFunction}}({\mathcal {D}},t)}{\text{CumulativeHazardFunction}}^{(0,1)}({\mathcal {D}},t)\,dt-\left(\int _{-\infty }^{\infty }te^{-{\text{CumulativeHazardFunction}}({\mathcal {D}},t)}{\text{CumulativeHazardFunction}}^{(0,1)}({\mathcal {D}},t)\,dt\right){}^{2}}}$ Standard Deviation Hazard Function ${\text{Too complicated}}$ Standard Deviation Inverse CDF ${\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {\int _{0}^{1}{\text{InverseCDF}}[{\mathcal {D}},t]^{2}\,dt-\left(\int _{0}^{1}{\text{InverseCDF}}[{\mathcal {D}},t]\,dt\right){}^{2}}}$ Standard Deviation Quantile ${\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {\int _{0}^{1}{\text{Quantile}}[{\mathcal {D}},t]^{2}\,dt-\left(\int _{0}^{1}{\text{Quantile}}[{\mathcal {D}},t]\,dt\right){}^{2}}}$ Standard Deviation Inverse Survival Function ${\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {\int _{0}^{1}{\text{InverseSurvivalFunction}}[{\mathcal {D}},t]^{2}\,dt-\left(\int _{0}^{1}{\text{InverseSurvivalFunction}}[{\mathcal {D}},t]\,dt\right){}^{2}}}$ Standard Deviation Sparsity ${\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {\int _{0}^{1}\left(\int _{\frac {1}{2}}^{t}{\text{Sparsity}}({\mathcal {D}},\tau )\,d\tau \right){}^{2}\,dt-\left(\int _{0}^{1}\int _{\frac {1}{2}}^{t}{\text{Sparsity}}({\mathcal {D}},\tau )d\tau dt\right){}^{2}}}$ Standard Deviation Mean Residual Life Function ${\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {\int _{-\infty }^{\infty }t^{2}{\frac {\partial ^{2}({\text{SurvivalFunction}}[{\mathcal {D}},t]{\text{MeanResidualLifeFunction}}({\mathcal {D}},t))}{\partial t^{2}}}\,dt-\left(\int _{-\infty }^{\infty }t{\frac {\partial ^{2}({\text{SurvivalFunction}}[{\mathcal {D}},t]{\text{MeanResidualLifeFunction}}({\mathcal {D}},t))}{\partial t^{2}}}\,dt\right){}^{2}}}$ Standard Deviation Variance ${\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {{\text{Variance}}[{\mathcal {D}}]}}$ Standard Deviation Standard Deviation ${\text{identical properties}}$ Standard Deviation Moment ${\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {{\text{Moment}}[{\mathcal {D}},2]-{\text{Moment}}[{\mathcal {D}},1]^{2}}}$ Standard Deviation Central Moment ${\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {{\text{CentralMoment}}[{\mathcal {D}},2]}}$ Standard Deviation Factorial Moment Failed to parse (syntax error): {\displaystyle \text{StandardDeviation}[\mathcal{D}]=\sqrt{\text{FactorialMoment}[\mathcal{D},1]-\text{FactorialMoment}[\mathcal{D},1]^2+\text{Fa ctorialMoment}[\mathcal{D},2]} } Standard Deviation Ascending Factorial Moment Failed to parse (unknown function "\mathc"): {\displaystyle \text{StandardDeviation}[\mathcal{D}]=\sqrt{-\text{AscendingFactorialMoment}(\mathcal{D},1)-\text{AscendingFactorialMoment}(\mathc al{D},1)^2+\text{AscendingFactorialMoment}(\mathcal{D},2)} } Standard Deviation Cumulant ${\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {{\text{Cumulant}}[{\mathcal {D}},2]}}$ Standard Deviation Factorial Cumulant Failed to parse (syntax error): {\displaystyle \text{StandardDeviation}[\mathcal{D}]=\sqrt{\text{FactorialMoment}[\mathcal{D},1]-\text{FactorialMoment}[\mathcal{D},1]^2+\text{Fa ctorialMoment}[\mathcal{D},2]} } Standard Deviation Ascending Factorial Cumulant Failed to parse (unknown function "\mathc"): {\displaystyle \text{StandardDeviation}[\mathcal{D}]=\sqrt{-\text{AscendingFactorialMoment}(\mathcal{D},1)-\text{AscendingFactorialMoment}(\mathc al{D},1)^2+\text{AscendingFactorialMoment}(\mathcal{D},2)} } Standard Deviation Moment Generating Function Failed to parse (syntax error): {\displaystyle \text{StandardDeviation}[\mathcal{D}]=\sqrt{-\text{MomentGeneratingFunction}^{(0,1)}(\mathcal{D},0)^2+\text{MomentGeneratingFuncti on}^{(0,2)}(\mathcal{D},0)} } Standard Deviation Central Moment Generating Function ${\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {{\text{CentralMomentGeneratingFunction}}^{(0,2)}({\mathcal {D}},0)}}$ Standard Deviation Factorial Moment Generating Function Failed to parse (syntax error): {\displaystyle \text{StandardDeviation}[\mathcal{D}]=\sqrt{\text{FactorialMomentGeneratingFunction}^{(0,1)}(\mathcal{D},1)-\text{FactorialMomentG eneratingFunction}^{(0,1)}(\mathcal{D},1)^2+\text{FactorialMomentGeneratingFunction}^{(0,2)}(\mathcal{D},1)} } Standard Deviation Descending Factorial Moment Generating Function Failed to parse (syntax error): {\displaystyle \text{StandardDeviation}[\mathcal{D}]=\sqrt{\text{DescendingFactorialMomentGeneratingFunction}^{(0,1)}(\mathcal{D},0)-\text{Descen dingFactorialMomentGeneratingFunction}^{(0,1)}(\mathcal{D},0)^2+\text{DescendingFactorialMomentGeneratingFunction}^{(0,2)}(\mat hcal{D},0)} } Standard Deviation Ascending Factorial Moment Generating Function Failed to parse (syntax error): {\displaystyle \text{StandardDeviation}[\mathcal{D}]=\sqrt{-\text{AscendingFactorialMomentGeneratingFunction}^{(0,1)}(\mathcal{D},0)-\text{Ascend ingFactorialMomentGeneratingFunction}^{(0,1)}(\mathcal{D},0)^2+\text{AscendingFactorialMomentGeneratingFunction}^{(0,2)}(\mathc al{D},0)} } Standard Deviation Cumulant Generating Function ${\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {{\text{CumulantGeneratingFunction}}^{(0,2)}({\mathcal {D}},1)}}$ Standard Deviation Factorial Cumulant Generating Function Failed to parse (syntax error): {\displaystyle \text{StandardDeviation}[\mathcal{D}]=\sqrt{\text{FactorialCumulantGeneratingFunction}^{(0,1)}(\mathcal{D},1)+\text{FactorialCumul antGeneratingFunction}^{(0,2)}(\mathcal{D},1)} } Standard Deviation Descending Factorial Cumulant Generating Function Failed to parse (syntax error): {\displaystyle \text{StandardDeviation}[\mathcal{D}]=\sqrt{\text{DescendingFactorialCumulantGeneratingFunction}^{(0,1)}(\mathcal{D},0)+\text{Desc endingFactorialCumulantGeneratingFunction}^{(0,2)}(\mathcal{D},0)} } Standard Deviation Ascending Factorial Cumulant Generating Function Failed to parse (syntax error): {\displaystyle \text{StandardDeviation}[\mathcal{D}]=\sqrt{-\text{AscendingFactorialCumulantGeneratingFunction}^{(0,1)}(\mathcal{D},0)+\text{Asce ndingFactorialCumulantGeneratingFunction}^{(0,2)}(\mathcal{D},0)} }

Skewness

Relations Between Distribution Properties

Derive Using Formula Skewness Mean Residual Life Function ${\text{Skewness}}[{\mathcal {D}}]={\frac {\int _{-\infty }^{\infty }(t-{\text{Mean}}[{\mathcal {D}}])^{3}{\frac {\partial ^{2}{\text{MeanResidualLifeFunction}}({\mathcal {D}},t)}{\partial t^{2}}}\,dt}{{\text{StandardDeviation}}[{\mathcal {D}}]^{3}}}$ Skewness Moment Failed to parse (unknown function "\r"): {\displaystyle \text{Skewness}[\mathcal{D}]=\frac{2 \text{Moment}[\mathcal{D},1]^3-(3 \text{Moment}[\mathcal{D},1]) \text{Moment}[\mathcal{D},2]+\text{Moment}[\mathcal{D},3]}{\left(-\text{Moment}[\mathcal{D},1]^2+\text{Moment}[\mathcal{D},2]\r ight)^{3/2}} } Skewness Central Moment ${\text{Skewness}}[{\mathcal {D}}]={\frac {{\text{CentralMoment}}[{\mathcal {D}},3]}{{\text{CentralMoment}}[{\mathcal {D}},2]^{3/2}}}$ Skewness Factorial Moment Failed to parse (syntax error): {\displaystyle \text{Skewness}[\mathcal{D}]=\frac{\text{FactorialMoment}[\mathcal{D},1]-3 \text{FactorialMoment}[\mathcal{D},1]^2+2 \text{FactorialMoment}[\mathcal{D},1]^3+3 \text{FactorialMoment}[\mathcal{D},2]-(3 \text{FactorialMoment}[\mathcal{D},1]) \text{FactorialMoment}[\mathcal{D},2]+\text{FactorialMoment}[\mathcal{D},3]}{\left(\text{FactorialMoment}[\mathcal{D},1]-\text{ FactorialMoment}[\mathcal{D},1]^2+\text{FactorialMoment}[\mathcal{D},2]\right)^{3/2}} } Skewness Ascending Factorial Moment Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \text{Skewness}[\mathcal{D}]=\frac{\text{AscendingFactorialMoment}(\mathcal{D},1)+3 \text{AscendingFactorialMoment}(\mathcal{D},1)^2+2 \text{AscendingFactorialMoment}(\mathcal{D},1)^3-3 \text{AscendingFactorialMoment}(\mathcal{D},2)-(3 \text{AscendingFactorialMoment}(\mathcal{D},1)) \text{AscendingFactorialMoment}(\mathcal{D},2)+\text{AscendingFactorialMoment}(\mathcal{D},3)}{\left(-\text{AscendingFactorialM oment}(\mathcal{D},1)-\text{AscendingFactorialMoment}(\mathcal{D},1)^2+\text{AscendingFactorialMoment}(\mathcal{D},2)\right)^{3 /2}} } Skewness Cumulant ${\text{Skewness}}[{\mathcal {D}}]={\frac {{\text{Cumulant}}[{\mathcal {D}},3]}{{\text{Cumulant}}[{\mathcal {D}},2]^{3/2}}}$ Skewness Factorial Cumulant ${\text{Skewness}}[{\mathcal {D}}]={\frac {{\text{FactorialCumulant}}({\mathcal {D}},1)+3{\text{FactorialCumulant}}({\mathcal {D}},2)+{\text{FactorialCumulant}}({\mathcal {D}},3)}{({\text{FactorialCumulant}}({\mathcal {D}},1)+{\text{FactorialCumulant}}({\mathcal {D}},2))^{3/2}}}$ Skewness Ascending Factorial Cumulant Failed to parse (syntax error): {\displaystyle \text{Skewness}[\mathcal{D}]=\frac{\text{CheckAbort}[\text{AscendingFactorialCumulant}(\mathcal{D},1)-3 \text{AscendingFactorialCumulant}(\mathcal{D},2)+\text{AscendingFactorialCumulant}(\mathcal{D},3),\text{AscendingFactorialCumul ant}(\mathcal{D},1)-3 \text{AscendingFactorialCumulant}(\mathcal{D},2)+\text{AscendingFactorialCumulant}(\mathcal{D},3)]}{(\text{AscendingFactorialCu mulant}(\mathcal{D},2)-\text{AscendingFactorialCumulant}(\mathcal{D},1))^{3/2}} } Skewness Moment Generating Function Failed to parse (syntax error): {\displaystyle \text{Skewness}[\mathcal{D}]=\frac{2 \text{MomentGeneratingFunction}^{(0,1)}(\mathcal{D},0)^3-\left(3 \text{MomentGeneratingFunction}^{(0,1)}(\mathcal{D},0)\right) \text{MomentGeneratingFunction}^{(0,2)}(\mathcal{D},0)+\text{MomentGeneratingFunction}^{(0,3)}(\mathcal{D},0)}{\left(-\text{Mom entGeneratingFunction}^{(0,1)}(\mathcal{D},0)^2+\text{MomentGeneratingFunction}^{(0,2)}(\mathcal{D},0)\right)^{3/2}} } Skewness Central Moment Generating Function Failed to parse (syntax error): {\displaystyle \text{Skewness}[\mathcal{D}]=\frac{\text{CentralMomentGeneratingFunction}^{(0,3)}(\mathcal{D},0)}{\text{CentralMomentGeneratingFun ction}^{(0,2)}(\mathcal{D},0)^{3/2}} } Skewness Factorial Moment Generating Function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \text{Skewness}[\mathcal{D}]=\frac{-3 \text{FactorialMomentGeneratingFunction}^{(0,1)}(\mathcal{D},1)^2+2 \text{FactorialMomentGeneratingFunction}^{(0,1)}(\mathcal{D},1)^3+\text{FactorialMomentGeneratingFunction}^{(0,1)}(\mathcal{D}, 1) \left(1-3 \text{FactorialMomentGeneratingFunction}^{(0,2)}(\mathcal{D},1)\right)+3 \text{FactorialMomentGeneratingFunction}^{(0,2)}(\mathcal{D},1)+\text{FactorialMomentGeneratingFunction}^{(0,3)}(\mathcal{D},1) }{\left(\text{FactorialMomentGeneratingFunction}^{(0,1)}(\mathcal{D},1)-\text{FactorialMomentGeneratingFunction}^{(0,1)}(\mathc al{D},1)^2+\text{FactorialMomentGeneratingFunction}^{(0,2)}(\mathcal{D},1)\right)^{3/2}} } Skewness Descending Factorial Moment Generating Function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \text{Skewness}[\mathcal{D}]=\frac{-3 \text{DescendingFactorialMomentGeneratingFunction}^{(0,1)}(\mathcal{D},0)^2+2 \text{DescendingFactorialMomentGeneratingFunction}^{(0,1)}(\mathcal{D},0)^3+\text{DescendingFactorialMomentGeneratingFunction}^ {(0,1)}(\mathcal{D},0) \left(1-3 \text{DescendingFactorialMomentGeneratingFunction}^{(0,2)}(\mathcal{D},0)\right)+3 \text{DescendingFactorialMomentGeneratingFunction}^{(0,2)}(\mathcal{D},0)+\text{DescendingFactorialMomentGeneratingFunction}^{( 0,3)}(\mathcal{D},0)}{\left(\text{DescendingFactorialMomentGeneratingFunction}^{(0,1)}(\mathcal{D},0)-\text{DescendingFactorial MomentGeneratingFunction}^{(0,1)}(\mathcal{D},0)^2+\text{DescendingFactorialMomentGeneratingFunction}^{(0,2)}(\mathcal{D},0)\ri ght)^{3/2}} } Skewness Ascending Factorial Moment Generating Function Failed to parse (syntax error): {\displaystyle \text{Skewness}[\mathcal{D}]=\frac{3 \text{AscendingFactorialMomentGeneratingFunction}^{(0,1)}(\mathcal{D},0)^2+2 \text{AscendingFactorialMomentGeneratingFunction}^{(0,1)}(\mathcal{D},0)^3+\text{AscendingFactorialMomentGeneratingFunction}^{( 0,1)}(\mathcal{D},0) \left(1-3 \text{AscendingFactorialMomentGeneratingFunction}^{(0,2)}(\mathcal{D},0)\right)-3 \text{AscendingFactorialMomentGeneratingFunction}^{(0,2)}(\mathcal{D},0)+\text{AscendingFactorialMomentGeneratingFunction}^{(0, 3)}(\mathcal{D},0)}{\left(-\text{AscendingFactorialMomentGeneratingFunction}^{(0,1)}(\mathcal{D},0)-\text{AscendingFactorialMom entGeneratingFunction}^{(0,1)}(\mathcal{D},0)^2+\text{AscendingFactorialMomentGeneratingFunction}^{(0,2)}(\mathcal{D},0)\right) ^{3/2}} } Skewness Cumulant Generating Function ${\text{Skewness}}[{\mathcal {D}}]={\frac {{\text{CumulantGeneratingFunction}}^{(0,3)}({\mathcal {D}},1)}{{\text{CumulantGeneratingFunction}}^{(0,2)}({\mathcal {D}},1)^{3/2}}}$ Skewness Factorial Cumulant Generating Function ${\text{Skewness}}[{\mathcal {D}}]={\frac {{\text{FactorialCumulantGeneratingFunction}}^{(0,1)}({\mathcal {D}},1)+3{\text{FactorialCumulantGeneratingFunction}}^{(0,2)}({\mathcal {D}},1)+{\text{FactorialCumulantGeneratingFunction}}^{(0,3)}({\mathcal {D}},1)}{\left({\text{FactorialCumulantGeneratingFunction}}^{(0,1)}({\mathcal {D}},1)+{\text{FactorialCumulantGeneratingFunction}}^{(0,2)}({\mathcal {D}},1)\right)^{3/2}}}$ Skewness Descending Factorial Cumulant Generating Function Failed to parse (syntax error): {\displaystyle \text{Skewness}[\mathcal{D}]=\frac{\text{DescendingFactorialCumulantGeneratingFunction}^{(0,1)}(\mathcal{D},0)+3 \text{DescendingFactorialCumulantGeneratingFunction}^{(0,2)}(\mathcal{D},0)+\text{DescendingFactorialCumulantGeneratingFunction }^{(0,3)}(\mathcal{D},0)}{\left(\text{DescendingFactorialCumulantGeneratingFunction}^{(0,1)}(\mathcal{D},0)+\text{DescendingFac torialCumulantGeneratingFunction}^{(0,2)}(\mathcal{D},0)\right)^{3/2}} } Skewness Ascending Factorial Cumulant Generating Function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \text{Skewness}[\mathcal{D}]=\frac{\text{AscendingFactorialCumulantGeneratingFunction}^{(0,1)}(\mathcal{D},0)-3 \text{AscendingFactorialCumulantGeneratingFunction}^{(0,2)}(\mathcal{D},0)+\text{AscendingFactorialCumulantGeneratingFunction}^ {(0,3)}(\mathcal{D},0)}{\left(-\text{AscendingFactorialCumulantGeneratingFunction}^{(0,1)}(\mathcal{D},0)+\text{AscendingFactor ialCumulantGeneratingFunction}^{(0,2)}(\mathcal{D},0)\right)^{3/2}} }

Kurtosis

Relations Between Distribution Properties

Derive Using Formula Kurtosis Mean Residual Life Function ${\text{Kurtosis}}[{\mathcal {D}}]={\frac {\int _{-\infty }^{\infty }(t-{\text{Mean}}[{\mathcal {D}}])^{4}{\frac {\partial ^{2}{\text{MeanResidualLifeFunction}}({\mathcal {D}},t)}{\partial t^{2}}}\,dt}{{\text{StandardDeviation}}[{\mathcal {D}}]^{4}}}$ Kurtosis Moment Failed to parse (syntax error): {\displaystyle \text{Kurtosis}[\mathcal{D}]=\frac{-3 \text{Moment}[\mathcal{D},1]^4+\left(6 \text{Moment}[\mathcal{D},1]^2\right)\text{Moment}[\mathcal{D},2]-(4 \text{Moment}[\mathcal{D},1]) \text{Moment[\mathcal{D},3]+\text{Moment}[\mathcal{D},4]}{\left(-\text{Moment}[\mathcal{D},1]^2+\text{Moment}[\mathcal{D},2]\right)^2} } Kurtosis Central Moment ${\text{Kurtosis}}[{\mathcal {D}}]={\frac {{\text{CentralMoment}}[{\mathcal {D}},4]}{{\text{CentralMoment}}[{\mathcal {D}},2]^{2}}}$ Kurtosis Factorial Moment Failed to parse (syntax error): {\displaystyle \text{Kurtosis}[\mathcal{D}]=\frac{6 \text{FactorialMoment}[\mathcal{D},1]^3-3 \text{FactorialMoment}[\mathcal{D},1]^4+7 \text{FactorialMoment}[\mathcal{D},2]+\text{FactorialMoment}[\mathcal{D},1]^2 (-4+6 \text{FactorialMoment}[\mathcal{D},2])+\text{FactorialMoment}[\mathcal{D},1] (1-12 \text{FactorialMoment}[\mathcal{D},2]-4 \text{FactorialMoment}[\mathcal{D},3])+6 \text{FactorialMoment}[\mathcal{D},3]+\text{FactorialMoment}[\mathcal{D},4]}{\left(\text{FactorialMoment}[\mathcal{D},1]-\text{ FactorialMoment}[\mathcal{D},1]^2+\text{FactorialMoment}[\mathcal{D},2]\right)^2} } Kurtosis Ascending Factorial Moment Failed to parse (syntax error): {\displaystyle \text{Kurtosis}[\mathcal{D}]=\frac{-\text{AscendingFactorialMoment}(\mathcal{D},1)-3 \text{AscendingFactorialMoment}(\mathcal{D},1)^4+7 \text{AscendingFactorialMoment}(\mathcal{D},2)+\left(6 \text{AscendingFactorialMoment}(\mathcal{D},1)^2\right) (-\text{AscendingFactorialMoment}(\mathcal{D},1)+\text{AscendingFactorialMoment}(\mathcal{D},2))-6 \text{AscendingFactorialMoment}(\mathcal{D},3)-(4 \text{AscendingFactorialMoment}(\mathcal{D},1)) (\text{AscendingFactorialMoment}(\mathcal{D},1)-3 \text{AscendingFactorialMoment}(\mathcal{D},2)+\text{AscendingFactorialMoment}(\mathcal{D},3))+\text{AscendingFactorialMoment}( \mathcal{D},4)}{\left(-\text{AscendingFactorialMoment}(\mathcal{D},1)-\text{AscendingFactorialMoment}(\mathcal{D},1)^2+\text{As cendingFactorialMoment}(\mathcal{D},2)\right)^2} } Kurtosis Cumulant ${\text{Kurtosis}}[{\mathcal {D}}]=3+{\frac {{\text{Cumulant}}[{\mathcal {D}},4]}{{\text{Cumulant}}[{\mathcal {D}},2]^{2}}}$ Kurtosis Factorial Cumulant ${\text{Kurtosis}}[{\mathcal {D}}]=3+{\frac {{\text{FactorialCumulant}}({\mathcal {D}},1)+7{\text{FactorialCumulant}}({\mathcal {D}},2)+6{\text{FactorialCumulant}}({\mathcal {D}},3)+{\text{FactorialCumulant}}({\mathcal {D}},4)}{({\text{FactorialCumulant}}({\mathcal {D}},1)+{\text{FactorialCumulant}}({\mathcal {D}},2))^{2}}}$ Kurtosis Ascending Factorial Cumulant Failed to parse (syntax error): {\displaystyle \text{Kurtosis}[\mathcal{D}]=3+\frac{-\text{AscendingFactorialCumulant}(\mathcal{D},1)+7 \text{AscendingFactorialCumulant}(\mathcal{D},2)-6 \text{AscendingFactorialCumulant}(\mathcal{D},3)+\text{AscendingFactorialCumulant}(\mathcal{D},4)}{(\text{AscendingFactorialCum ulant}(\mathcal{D},2)-\text{AscendingFactorialCumulant}(\mathcal{D},1))^2} } Kurtosis Moment Generating Function Failed to parse (syntax error): {\displaystyle \text{Kurtosis}[\mathcal{D}]=\frac{-3 \text{MomentGeneratingFunction}^{(0,1)}(\mathcal{D},0)^4+\left(6 \text{MomentGeneratingFunction}^{(0,1)}(\mathcal{D},0)^2\right) \text{MomentGeneratingFunction}^{(0,2)}(\mathcal{D},0)-\left(4 \text{MomentGeneratingFunction}^{(0,1)}(\mathcal{D},0)\right) \text{MomentGeneratingFunction}^{(0,3)}(\mathcal{D},0)+\text{MomentGeneratingFunction}^{(0,4)}(\mathcal{D},0)}{\left(-\text{Mom entGeneratingFunction}^{(0,1)}(\mathcal{D},0)^2+\text{MomentGeneratingFunction}^{(0,2)}(\mathcal{D},0)\right)^2} } Kurtosis Central Moment Generating Function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \text{Kurtosis}[\mathcal{D}]=\frac{\text{CentralMomentGeneratingFunction}^{(0,4)}(\mathcal{D},0)}{\text{CentralMomentGeneratingFun ction}^{(0,2)}(\mathcal{D},0)^2} } Kurtosis Factorial Moment Generating Function Failed to parse (unknown function "\mathca"): {\displaystyle \text{Kurtosis}[\mathcal{D}]=\frac{6 \text{FactorialMomentGeneratingFunction}^{(0,1)}(\mathcal{D},1)^3-3 \text{FactorialMomentGeneratingFunction}^{(0,1)}(\mathcal{D},1)^4+7 \text{FactorialMomentGeneratingFunction}^{(0,2)}(\mathcal{D},1)+\text{FactorialMomentGeneratingFunction}^{(0,1)}(\mathcal{D},1) ^2 \left(-4+6 \text{FactorialMomentGeneratingFunction}^{(0,2)}(\mathcal{D},1)\right)+\text{FactorialMomentGeneratingFunction}^{(0,1)}(\mathca l{D},1) \left(1-12 \text{FactorialMomentGeneratingFunction}^{(0,2)}(\mathcal{D},1)-4 \text{FactorialMomentGeneratingFunction}^{(0,3)}(\mathcal{D},1)\right)+6 \text{FactorialMomentGeneratingFunction}^{(0,3)}(\mathcal{D},1)+\text{FactorialMomentGeneratingFunction}^{(0,4)}(\mathcal{D},1) }{\left(\text{FactorialMomentGeneratingFunction}^{(0,1)}(\mathcal{D},1)-\text{FactorialMomentGeneratingFunction}^{(0,1)}(\mathc al{D},1)^2+\text{FactorialMomentGeneratingFunction}^{(0,2)}(\mathcal{D},1)\right)^2} } Kurtosis Descending Factorial Moment Generating Function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \text{Kurtosis}[\mathcal{D}]=\frac{6 \text{DescendingFactorialMomentGeneratingFunction}^{(0,1)}(\mathcal{D},0)^3-3 \text{DescendingFactorialMomentGeneratingFunction}^{(0,1)}(\mathcal{D},0)^4+7 \text{DescendingFactorialMomentGeneratingFunction}^{(0,2)}(\mathcal{D},0)+\text{DescendingFactorialMomentGeneratingFunction}^{( 0,1)}(\mathcal{D},0)^2 \left(-4+6 \text{DescendingFactorialMomentGeneratingFunction}^{(0,2)}(\mathcal{D},0)\right)+\text{DescendingFactorialMomentGeneratingFunct ion}^{(0,1)}(\mathcal{D},0) \left(1-12 \text{DescendingFactorialMomentGeneratingFunction}^{(0,2)}(\mathcal{D},0)-4 \text{DescendingFactorialMomentGeneratingFunction}^{(0,3)}(\mathcal{D},0)\right)+6 \text{DescendingFactorialMomentGeneratingFunction}^{(0,3)}(\mathcal{D},0)+\text{DescendingFactorialMomentGeneratingFunction}^{( 0,4)}(\mathcal{D},0)}{\left(\text{DescendingFactorialMomentGeneratingFunction}^{(0,1)}(\mathcal{D},0)-\text{DescendingFactorial MomentGeneratingFunction}^{(0,1)}(\mathcal{D},0)^2+\text{DescendingFactorialMomentGeneratingFunction}^{(0,2)}(\mathcal{D},0)\ri ght)^2} } Kurtosis Ascending Factorial Moment Generating Function Failed to parse (syntax error): {\displaystyle \text{Kurtosis}[\mathcal{D}]=\frac{-6 \text{AscendingFactorialMomentGeneratingFunction}^{(0,1)}(\mathcal{D},0)^3-3 \text{AscendingFactorialMomentGeneratingFunction}^{(0,1)}(\mathcal{D},0)^4+7 \text{AscendingFactorialMomentGeneratingFunction}^{(0,2)}(\mathcal{D},0)+\text{AscendingFactorialMomentGeneratingFunction}^{(0, 1)}(\mathcal{D},0)^2 \left(-4+6 \text{AscendingFactorialMomentGeneratingFunction}^{(0,2)}(\mathcal{D},0)\right)+\text{AscendingFactorialMomentGeneratingFunctio n}^{(0,1)}(\mathcal{D},0) \left(-1+12 \text{AscendingFactorialMomentGeneratingFunction}^{(0,2)}(\mathcal{D},0)-4 \text{AscendingFactorialMomentGeneratingFunction}^{(0,3)}(\mathcal{D},0)\right)-6 \text{AscendingFactorialMomentGeneratingFunction}^{(0,3)}(\mathcal{D},0)+\text{AscendingFactorialMomentGeneratingFunction}^{(0, 4)}(\mathcal{D},0)}{\left(-\text{AscendingFactorialMomentGeneratingFunction}^{(0,1)}(\mathcal{D},0)-\text{AscendingFactorialMom entGeneratingFunction}^{(0,1)}(\mathcal{D},0)^2+\text{AscendingFactorialMomentGeneratingFunction}^{(0,2)}(\mathcal{D},0)\right) ^2} } Kurtosis Cumulant Generating Function Failed to parse (syntax error): {\displaystyle \text{Kurtosis}[\mathcal{D}]=\frac{3 \text{CumulantGeneratingFunction}^{(0,2)}(\mathcal{D},1)^2+\text{CumulantGeneratingFunction}^{(0,4)}(\mathcal{D},1)}{\text{Cumu lantGeneratingFunction}^{(0,2)}(\mathcal{D},1)^2} } Kurtosis Factorial Cumulant Generating Function ${\text{Kurtosis}}[{\mathcal {D}}]={\frac {3{\text{FactorialCumulantGeneratingFunction}}^{(0,1)}({\mathcal {D}},1)^{2}+7{\text{FactorialCumulantGeneratingFunction}}^{(0,2)}({\mathcal {D}},1)+3{\text{FactorialCumulantGeneratingFunction}}^{(0,2)}({\mathcal {D}},1)^{2}+{\text{FactorialCumulantGeneratingFunction}}^{(0,1)}({\mathcal {D}},1)\left(1+6{\text{FactorialCumulantGeneratingFunction}}^{(0,2)}({\mathcal {D}},1)\right)+6{\text{FactorialCumulantGeneratingFunction}}^{(0,3)}({\mathcal {D}},1)+{\text{FactorialCumulantGeneratingFunction}}^{(0,4)}({\mathcal {D}},1)}{\left({\text{FactorialCumulantGeneratingFunction}}^{(0,1)}({\mathcal {D}},1)+{\text{FactorialCumulantGeneratingFunction}}^{(0,2)}({\mathcal {D}},1)\right)^{2}}}$ Kurtosis Descending Factorial Cumulant Generating Function Failed to parse (syntax error): {\displaystyle \text{Kurtosis}[\mathcal{D}]=\frac{3 \text{DescendingFactorialCumulantGeneratingFunction}^{(0,1)}(\mathcal{D},0)^2+7 \text{DescendingFactorialCumulantGeneratingFunction}^{(0,2)}(\mathcal{D},0)+3 \text{DescendingFactorialCumulantGeneratingFunction}^{(0,2)}(\mathcal{D},0)^2+\text{DescendingFactorialCumulantGeneratingFuncti on}^{(0,1)}(\mathcal{D},0) \left(1+6 \text{DescendingFactorialCumulantGeneratingFunction}^{(0,2)}(\mathcal{D},0)\right)+6 \text{DescendingFactorialCumulantGeneratingFunction}^{(0,3)}(\mathcal{D},0)+\text{DescendingFactorialCumulantGeneratingFunction }^{(0,4)}(\mathcal{D},0)}{\left(\text{DescendingFactorialCumulantGeneratingFunction}^{(0,1)}(\mathcal{D},0)+\text{DescendingFac torialCumulantGeneratingFunction}^{(0,2)}(\mathcal{D},0)\right)^2} } Kurtosis Ascending Factorial Cumulant Generating Function Failed to parse (syntax error): {\displaystyle \!\( TagBox[ RowBox[{ RowBox[{"Kurtosis", "[", "\[ScriptCapitalD]", "]"}], "==", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"3", " ", SuperscriptBox[ RowBox[{ SuperscriptBox["AscendingFactorialCumulantGeneratingFunction", TagBox[ RowBox[{"(", RowBox[{"0", ",", "1"}], ")"}], Derivative], MultilineFunction->None], "[", RowBox[{"\[ScriptCapitalD]", ",", "0"}], "]"}], "2"]}], "+", RowBox[{"7", " ", RowBox[{ SuperscriptBox["AscendingFactorialCumulantGeneratingFunction", TagBox[ RowBox[{"(", RowBox[{"0", ",", "2"}], ")"}], Derivative], MultilineFunction->None], "[", RowBox[{"\[ScriptCapitalD]", ",", "0"}], "]"}]}], "+", RowBox[{"3", " ", SuperscriptBox[ RowBox[{ SuperscriptBox["AscendingFactorialCumulantGeneratingFunction", TagBox[ RowBox[{"(", RowBox[{"0", ",", "2"}], ")"}], Derivative], MultilineFunction->None], "[", RowBox[{"\[ScriptCapitalD]", ",", "0"}], "]"}], "2"]}], "-", RowBox[{ RowBox[{ SuperscriptBox["AscendingFactorialCumulantGeneratingFunction", TagBox[ RowBox[{"(", RowBox[{"0", ",", "1"}], ")"}], Derivative], MultilineFunction->None], "[", RowBox[{"\[ScriptCapitalD]", ",", "0"}], "]"}], " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"6", " ", RowBox[{ SuperscriptBox["AscendingFactorialCumulantGeneratingFunction", TagBox[ RowBox[{"(", RowBox[{"0", ",", "2"}], ")"}], Derivative], MultilineFunction->None], "[", RowBox[{"\[ScriptCapitalD]", ",", "0"}], "]"}]}]}], ")"}]}], "-", RowBox[{"6", " ", RowBox[{ SuperscriptBox["AscendingFactorialCumulantGeneratingFunction", TagBox[ RowBox[{"(", RowBox[{"0", ",", "3"}], ")"}], Derivative], MultilineFunction->None], "[", RowBox[{"\[ScriptCapitalD]", ",", "0"}], "]"}]}], "+", RowBox[{ SuperscriptBox["AscendingFactorialCumulantGeneratingFunction", TagBox[ RowBox[{"(", RowBox[{"0", ",", "4"}], ")"}], Derivative], MultilineFunction->None], "[", RowBox[{"\[ScriptCapitalD]", ",", "0"}], "]"}]}], ")"}], "/", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", RowBox[{ SuperscriptBox["AscendingFactorialCumulantGeneratingFunction", TagBox[ RowBox[{"(", RowBox[{"0", ",", "1"}], ")"}], Derivative], MultilineFunction->None], "[", RowBox[{"\[ScriptCapitalD]", ",", "0"}], "]"}]}], "+", RowBox[{ SuperscriptBox["AscendingFactorialCumulantGeneratingFunction", TagBox[ RowBox[{"(", RowBox[{"0", ",", "2"}], ")"}], Derivative], MultilineFunction->None], "[", RowBox[{"\[ScriptCapitalD]", ",", "0"}], "]"}]}], ")"}], "2"]}]}], HoldForm]\) }