User:BeyondNormality/Ahmad-Kudo-Poisson distribution

Ahmad-Kudo-Poisson distribution

Notation	$\mathrm {AhmadKudoPoisson} (a,c,\alpha )$
Support	x ∈ { 0, 1, 2, ... }
PMF	${\displaystyle }$
CDF	${\displaystyle }$
Mean	${\displaystyle }$
Median	${\displaystyle }$
Mode	${\displaystyle }$
Variance	${\displaystyle }$
Skewness	${\displaystyle }$
Excess kurtosis	${\displaystyle }$
Entropy	${\displaystyle }$
MGF	${\displaystyle }$
CF	${\displaystyle }$
PGF	${\displaystyle }$

Also known as the modified Poisson distribution, the probability mass function of the Ahmad-Kudo-Poisson distribution is given by

$\mathrm {AhmadKudoPoisson} (x;a,c,\alpha )\equiv \Pr(X=x)={\begin{cases}{\frac {e^{-a}a^{x}}{x!}}&x=0,1,\dots ,c-1,\\{\frac {e^{-a}(1-\alpha )a^{c}}{c!}}&x=c,\\{\frac {e^{-a}a^{c+1}\left({\frac {\alpha (c+1)}{a}}+1\right)}{(c+1)!}}&x=c+1,\\{\frac {e^{-a}a^{x}}{x!}}&x=c+2,c+3,\dots \\\end{cases}}$

• $x=0,1,2,\dots$
• $a\geq 0$
• $c\in \mathbb {N}$
• $0\leq \alpha \leq 1$

Cumulative Distribution Function

$F_{X}(x)={\begin{cases}{\frac {\Gamma (x+1,a)}{\Gamma (x+1)}}&x=0,1,\dots ,c-1,\\{\frac {e^{-a}(1-\alpha )a^{c}}{c!}}+{\frac {\Gamma (c,a)}{\Gamma (c)}}&x=c,\\{\frac {e^{-a}a^{c+1}\left({\frac {\alpha (c+1)}{a}}+1\right)}{(c+1)!}}+{\frac {e^{-a}(1-\alpha )a^{c}}{c!}}+{\frac {\Gamma (c,a)}{\Gamma (c)}}&x=c+1,\\{\frac {\Gamma (x+1,a)}{\Gamma (x+1)}}&x=c+2,c+3,\dots \\\end{cases}}$

Expected Value

$\mathbb {E} (X)=a+{\frac {e^{-a}\alpha a^{c}}{\Gamma (c+1)}}$

Variance

$\operatorname {Var} (X)=a+{\frac {e^{-2a}\alpha a^{c}\left(e^{a}(-2a+2c+1)\Gamma (c+1)-\alpha a^{c}\right)}{\Gamma (c+1)^{2}}}$

Moment Generating Function

$M_{X}(t)={\frac {e^{-a}\alpha \left(e^{t}-1\right)\left(ae^{t}\right)^{c}}{c!}}+e^{a\left(e^{t}-1\right)}$

Characteristic Function

$\varphi _{X}(t)={\frac {e^{-a}\alpha \left(e^{it}-1\right)\left(ae^{it}\right)^{c}}{c!}}+e^{a\left(e^{it}-1\right)}$

Probability Generating Function

$G(t)={\frac {e^{-a}\alpha (t-1)(at)^{c}}{c!}}+e^{a(t-1)}$

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
$\Rightarrow$	deterministic	$a=0$
$\Rightarrow$	Poisson $\left(a\right)$	$\alpha =0$

References

• Ahmad, M., Kudo, A. (1967). Modified and partially truncated Poisson distribution Bulletin of the Institute of Statistical Research and Training, University of Dacca 1, 82-90
• Wimmer, G., Altmann. (1999). Thesaurus of univariate discrete probability distributions. Stamm; 1. ed (1999), pg 5