User:BeyondNormality/Ancker-Gafarian distribution (type 2)

The probability mass function of the Ancker-Gafarian distribution (type 2) is given by

$$    \mathrm{AnckerGafarian2}(b,\rho) \equiv \Pr(X = x) = \begin{cases} \frac{b}{b+e^{\rho } \rho } & x= 0 \\ \frac{P_0 \rho ^x}{b (x-1)!} & x= 1, 2, \dots \\ \end{cases} $$


 * $$ x = 0,1,2, \dots $$
 * $$ b > 0 $$
 * $$ \rho \ge 0 $$

Cumulative Distribution Function

F_X(x) = \begin{cases} P_0 & x= 0 \\ P_0 \left( 1+ \frac{e^{\rho } \rho \Gamma (x,\rho )}{b \Gamma (x)} \right) & x= 1, 2, \dots \\ \end{cases} $$

Expected Value
 * $$ \mathbb E(X) = \frac{e^{\rho } \rho (\rho +1)}{b+e^{\rho } \rho } $$

Variance
 * $$ \operatorname{Var}(X) = \frac{e^{\rho } \rho  \left(b (\rho  (\rho +3)+1)+e^{\rho } \rho ^2\right)}{\left(b+e^{\rho } \rho \right)^2} $$

Recurrence relation
 * $$ x \Pr (x+1)-\rho  \Pr (x)=0 $$

Moment Generating Function
 * $$ M_X(t)=\frac{b+\rho e^{\rho  e^t+t}}{b+e^{\rho } \rho } $$

Characteristic Function
 * $$ \varphi_X(t)=\frac{b+\rho e^{\rho  e^{i t}+i t}}{b+e^{\rho } \rho } $$

Probability Generating Function
 * $$ \begin{align}

G(t) & = P_0 \left( 1+ \frac{\rho t e^{\rho t}}{b} \right) \\ & = \frac{b+\rho t e^{\rho  t}}{b+e^{\rho } \rho } \end{align}$$