User:BeyondNormality/Ahmad-Kudo-Poisson distribution

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

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


 * $$ x = 0,1,2, \dots $$
 * $$ a \ge 0 $$
 * $$ c \in \mathbb{N}  $$
 * $$ 0 \le \alpha \le 1 $$

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


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

Moment Generating Function


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

Characteristic Function


 * $$ \varphi_X(t)=\frac{e^{-a} \alpha \left(e^{i t}-1\right) \left(a e^{i t}\right)^c}{c!}+e^{a \left(e^{i t}-1\right)} $$

Probability Generating Function


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