User:BeyondNormality/Anderson distribution

Anderson

Notation	$\mathrm {Anderson} (K,N)$
Parameters	${\displaystyle }$
Support	$x=0,1,2,\dots ,N$
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 Anderson distribution is

$\mathrm {Anderson} (x;K,N)\equiv \Pr(X=x)=\sum _{j=0}^{N-x}{\frac {(-1)^{j}}{j!x!}}\left({\frac {(-j+N-x)!}{N!}}\right)^{K-2}$

• $x=0,1,\dots ,N$
• $K\in \mathbb {N} -\{0,1\}$
• $N\in \mathbb {N} _{0}$

Probability Generating Function

${\begin{aligned}G(t)&=\,_{1}F_{K-1}(-N;\underbrace {-N,\dots ,-N} _{K-1};(-1)^{K}(t-1))\\&=(N!)^{2-K}\sum _{j=0}^{N}{\frac {(t-1)^{j}((N-j)!)^{K-2}}{j!}}\end{aligned}}$

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
$\Leftarrow$	Kemp-hypergeometric family Type I
$\Rightarrow$	deterministic $\left(0\right)$	$N=0$

References

• Anderson, T.W. (1943). On card matching. Annals of Mathematical Statistics 14, 426-435
• Johnson, N.L., Kotz, S., Kemp, A.W. (1992) Univariate Discrete Distributions. New York: Wiley
• Kemp, A.W. (1978) Cluster size probabilities for generalized Poisson distributions. Communications in Statistics - Theory and Methods A 7, 1433-1438.
• Kemp, A.W., Kemp, C.D. (1974a) A family of discrete distributions defined via their factorial moments. Communications in Statistics 3, 1187-1196.
• Wimmer, G., Altmann. (1999). Thesaurus of univariate discrete probability distributions. Stamm; 1. ed (1999), pg 10