Gamma–Poisson distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ gamma–Poisson(α, β) is used to indicate that the random variable X has the gamma–Poisson distribution with positive parameters α andβ. A gamma–Poisson random variable X has probability mass function

f(x) = \frac{Γ(x + β)α^{x}}{Γ(β)(1 + α)^{β+x} x!} x = 0,1,2,...

for any α,β > 0. A gamma–Poisson random variable is a Poisson random variable with a random parameter μ which has the gamma distribution with parameters α and β. The probability mass function for three different parameter settings is illustrated below.

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x f(x) α = 2 β = 1 α = 1 β = 2 β = 5

The cumulative distribution function on the support of X is

F (x) = ∑_{w=0}^{x} \frac{Γ(w + β)α^{w}}{Γ(β)(1 + α)^{β+w} w!} x = 0,1,2,...

The moment generating function of X is

M(t) = E[e^{tx}] = (1 + α − αe^{t})^{−β} − ∞ < t < ∞.

The characteristic function of X is

ϕ(t) = E[e^{itx}] = (1 + α − αe^{it})^{−β} − ∞ < t < ∞.

The population mean, variance, skewness, and kurtosis of X are

E[X] = αβ V [X] = αβ + α^{2}β

E [( \frac{X − μ}{σ} ) ^{3}] = \frac{1 + 2α}{\sqrt{αβ(1 + α)}} E [( \frac{X − μ}{σ} ) ^{4}] = \frac{3α^{2}β + 6α^{2}+ 3αβ + 6α + 1}{αβ(1 + α)} .