User:BeyondNormality/Abakuks distribution

The probability mass function of the Abakuks distribution is

\mathrm {Abakuks} (x;a,r,n)\equiv \Pr(X=x)={\frac {1}{x(an)^{x}(n-x)!T}}

Recurrence relation

\left\{an(x+1)\Pr(x+1)+\left(x^{2}-nx\right)\Pr(x)=0\right\}

Expected Value

\operatorname {E} [X]={\frac {an\left({\frac {1}{an}}\right)^{r}\,_{2}F_{0}\left(1,-n+r-1;;-{\frac {1}{an}}\right)}{T\Gamma (n-r+2)}}

Moment Generating Function

M_{X}(t)={\frac {\Gamma (r-1)\left({\frac {e^{t}}{an}}\right)^{r-1}\,_{3}{\tilde {F}}_{1}\left(1,r-1,-n+r-1;r;-{\frac {e^{t}}{an}}\right)}{T\Gamma (n-r+2)}}

Characteristic Function

\varphi _{X}(t)={\frac {\Gamma (r-1)\left({\frac {e^{it}}{an}}\right)^{r-1}\,_{3}{\tilde {F}}_{1}\left(1,r-1,-n+r-1;r;-{\frac {e^{it}}{an}}\right)}{T\Gamma (n-r+2)}}

Probability Generating Function

G(t)={\frac {\left({\frac {t}{an}}\right)^{r-1}\,_{3}F_{1}\left(1,r-1,-n+r-1;r;-{\frac {t}{an}}\right)}{(r-1)T(n-r+1)!}}

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
$\Leftarrow$	generalized power series family $\left(a_{x},\theta ,f(.),T'\right)$	$a_{x}={\frac {1}{x(n)^{x}(n-x)!}}\qquad \theta ={\frac {1}{a}}\qquad f(\theta )=\sum _{j\mathop {=} r-1}^{n}{\frac {1}{j(an)^{j}(n-j)!T}}\qquad T'=\{r-1,r,...,n\}$
$\Leftarrow$	$(r-1)$ displaced Kapur-hypergeometric family $\left(a_{i},b_{j},\theta ,k,r\right)$	$k=2\qquad r=1\qquad \theta ={\frac {-1}{an}}$
$\Leftarrow$	Kemp-Dacey-hypergeometric family (II/20)	$(II/20)$

Ababuks, A. (1976). An invariance property of the Poisson distributions for a limited immigrant population. J. of the Royal Statistical Society B 38, 99-101
Wimmer, G., Altmann. (1999). Thesaurus of univariate discrete probability distributions. Stamm; 1. ed (1999), pg 1