User:BeyondNormality/Beall-Rescia distribution


 * $$ a, b, n \ge 0 $$

Moment Generating Function
 * $$ M_X(t)=\exp \left(a \left(n e^{b \left(e^t-1\right)} \left(b \left(e^t-1\right)\right)^{-n} \left(\Gamma (n)-\Gamma \left(n,b

\left(-1+e^t\right)\right)\right)-1\right)\right) $$

Characteristic Function
 * $$ \varphi_X(t)=\exp \left(a \left(n e^{b \left(e^{i t}-1\right)} \left(b \left(e^{i t}-1\right)\right)^{-n} \left(\Gamma (n)-\Gamma

\left(n,b \left(-1+e^{i t}\right)\right)\right)-1\right)\right) $$

Probability Generating Function

\begin{align} G(t) & = \exp (-a) \exp \left(a \Gamma (n+1) \sum _{j=0}^{\infty } \frac{b^j (-1+t)^j}{\Gamma (1+j+n)}\right) \\ & = exp(-a) \, _0F_0\left(a \, _1F_1(1; n+1; b(t-1))\right) \\ & = \frac{\, _0F_0\left(a \, _1F_1(1; n+1; b(t-1))\right)}{\, _0F_0(a)} \\ & = \exp \left(a n e^{b (t-1)} (b (t-1))^{-n} (\Gamma (n)-\Gamma (n,b (t-1)))-a\right) \end{align} $$