User:BeyondNormality/Barton distribution

Barton

Notation	$\mathrm {Barton} (K,N)$
Parameters	${\displaystyle }$
Support	${\displaystyle }$
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 Barton distribution is

$\mathrm {Barton} (x;K,N)\equiv \Pr(X=x)={\frac {1}{x!}}\sum _{j=x}^{N}{\frac {(-1)^{j-x}(K!)^{j}\left(N_{(j)}\right)^{2}}{(j-x)!(KN)_{(jK)}}}$

• $x=0,1,\dots ,N$
• $K\in \mathbb {N}$
• $N\in \mathbb {N} _{0}$
• $(n)_{(k)}=n(n-1)(n-2)\cdots (n-k+1)$ (Falling Factorial)

Probability Generating Function

$G(t)=\,_{1}F_{K-1}\left(-N;{\frac {1}{K}}-N,{\frac {2}{K}}-N,\dots ,{\frac {K-1}{K}}-N;{\frac {K!(t-1)}{{(-K)}^{K}}}\right)$

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 $\left(N\right)$	$N(K-1)=0$

References

• Barton, D.E. (1958). The matching distributions: Poisson limiting forms and derived method of approximation. J. of the Royal Statistical Society B 20, 73-92.
• 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.
• Wimmer, G., Altmann. (1999). Thesaurus of univariate discrete probability distributions. Stamm; 1. ed (1999), pg 14