User:BeyondNormality/Bartlett distribution

The probability mass function of the Barlett distribution is given by

$$ \begin{align} \mathrm{Barlett}(x;a,q) \equiv \Pr(X = x) & = \sum _{j=0}^x \frac{a^{-j+x} e^{-a} q^j p}{(-j+x)!} \\ & = \frac{p e^{\frac{a p}{q}} q^x \Gamma \left(x+1,\frac{a}{q}\right)}{x!} \end{align}$$


 * $$ x = 0, 1, 2, \dots  $$
 * $$ a \ge 0 $$
 * $$ 0 < p \le 1 $$
 * $$ q = 1 - p $$

Expected Value
 * $$   \operatorname{E}[X] = a+\frac{1}{p}-1 $$

Variance
 * $$ \operatorname{Var}(X) = a+\frac{q}{p^2} $$

Moment Generating Function
 * $$ M_X(t)=\frac{p e^{a \left(e^t-1\right)}}{1-q e^t} $$

Characteristic Function
 * $$ \varphi_X(t)=\frac{p e^{a \left(e^{i t}-1\right)}}{1-q e^{i t}} $$

Probability Generating Function
 * $$ G(t) = \frac{p e^{a (t-1)}}{1-q t} $$