User:BeyondNormality/Ancker-Gafarian distribution (type 2)

Ancker-Gafarian (type 2)

Notation	$\mathrm {AnckerGafarian2} (b,\rho )$
Support	$x\in \{0,1,2,\dots \}$
PMF	${\displaystyle }$
CDF	${\displaystyle }$
Mean	${\displaystyle }$
Median	${\displaystyle }$
Mode	${\displaystyle }$
Variance	${\displaystyle }$
Skewness	${\displaystyle }$
Excess kurtosis	${\displaystyle }$
Entropy	${\displaystyle }$
MGF	${\displaystyle }$
CF	${\displaystyle }$
PGF	${\displaystyle }$

The probability mass function of the Ancker-Gafarian distribution (type 2) is given by

$\mathrm {AnckerGafarian2} (b,\rho )\equiv \Pr(X=x)={\begin{cases}{\frac {b}{b+e^{\rho }\rho }}&x=0\\{\frac {P_{0}\rho ^{x}}{b(x-1)!}}&x=1,2,\dots \\\end{cases}}$

• $x=0,1,2,\dots$
• $b>0$
• $\rho \geq 0$

Cumulative Distribution Function

$F_{X}(x)={\begin{cases}P_{0}&x=0\\P_{0}\left(1+{\frac {e^{\rho }\rho \Gamma (x,\rho )}{b\Gamma (x)}}\right)&x=1,2,\dots \\\end{cases}}$

Expected Value

$\mathbb {E} (X)={\frac {e^{\rho }\rho (\rho +1)}{b+e^{\rho }\rho }}$

Variance

$\operatorname {Var} (X)={\frac {e^{\rho }\rho \left(b(\rho (\rho +3)+1)+e^{\rho }\rho ^{2}\right)}{\left(b+e^{\rho }\rho \right)^{2}}}$

Recurrence relation

$x\Pr(x+1)-\rho \Pr(x)=0$

Moment Generating Function

$M_{X}(t)={\frac {b+\rho e^{\rho e^{t}+t}}{b+e^{\rho }\rho }}$

Characteristic Function

$\varphi _{X}(t)={\frac {b+\rho e^{\rho e^{it}+it}}{b+e^{\rho }\rho }}$

Probability Generating Function

${\begin{aligned}G(t)&=P_{0}\left(1+{\frac {\rho te^{\rho t}}{b}}\right)\\&={\frac {b+\rho te^{\rho t}}{b+e^{\rho }\rho }}\end{aligned}}$

Interrelations

Symbol	Meaning
$\sim$	$X\sim Y$: the random variable X is distributed as the random variable Y
$\equiv$	the distribution in the title is identical with this distribution
$\Leftarrow$	the distribution in title is a special case of this distribution
$\Rightarrow$	this distribution is a special case of the distribution in the title
$\leftarrow$	this distribution converges to the distribution in the title
$\rightarrow$	the distribution in the title converges to this distribution

Relationship	Distribution	When
$\equiv$	zero-one $(p)$ $\vee$ 1-displaced Poisson $(a)$	$p={\frac {e^{\rho }\rho }{b+e^{\rho }\rho }}\qquad a=\rho$
$\Leftarrow$	generalized zero-one family $\left(a',H(.)\right)$	$a'={\frac {e^{\rho }\rho }{b+e^{\rho }\rho }}$
$\Leftarrow$	power series family $\left(a_{x},\theta ,f(.)\right)$	$\theta =\rho \qquad a_{0}=b\qquad a_{x}={\frac {1}{(x-1)!}}\qquad f(\theta )=b^{2}\left(b+e^{\theta }\theta \right)\qquad x=1,2,\dots$
$\leftarrow$	Ancker-Gafarian (type 1) $\left(a,b,c\right)$	$a\rightarrow \infty \qquad c\rightarrow \infty \qquad {\frac {a}{c}}\rightarrow \rho$
$\Rightarrow$	deterministic $\left(0\right)$	$\rho =0$
$\rightarrow$	1-displaced Poisson $\left(\rho \right)$	$b\rightarrow 0$

References

• Ancker, C.J. Jr., Gafarian, A.V. (1963b) Some queueing problems with balking and reneging II. Operations Research II, 928-937.
• Srivastava, H.M., Kasyap, B.R. (1982). Special Functions in Queuing. New York: Academic Press.
• Wimmer, G., Altmann. (1999). Thesaurus of univariate discrete probability distributions. Stamm; 1. ed (1999), pg 9