User:BeyondNormality/Anderson distribution

The probability mass function of the Anderson distribution is

A n d e r s o n (x; K, N) \equiv Pr (X = x) = \sum_{j = 0}^{N - x} \frac{(- 1)^{j}}{j! x!} {(\frac{(- j + N - x)!}{N!})}^{K - 2}

Probability Generating Function

\begin{matrix} G (t) & =_{1} F_{K - 1} (- N; \underset{K - 1}{\underset{⏟}{- N, \dots, - N}}; (- 1)^{K} (t - 1)) \\ = (N!)^{2 - K} \sum_{j = 0}^{N} \frac{(t - 1)^{j} ((N - j)!)^{K - 2}}{j!} \end{matrix}

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
$\to$	the distribution in the title converges to this distribution

Relationship	Distribution	When
$\Leftarrow$	Kemp-hypergeometric family Type I
$\Rightarrow$	deterministic $(0)$	$N = 0$

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