Power series distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) A power series random variable X has probability mass function

f(x) = \frac{a_{x}c^{x}}{A(c)} x = 0, 1, 2, ...

for a_{x} > 0, c > 0, and series function A(c) = ∑ a_{x}c^{x}. The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = ∑_{w=0}^{x} \frac{a_{w}c^{w}}{A(c)} x = 0, 1, 2, ... .

The survivor function on the support of X is

S(x) = P(X ≥ x) = ∑_{w=x}^{∞} \frac{a_{w}c^{w}}{A(c)} x = 0, 1, 2, ... .

The hazard function on the support of X is

h(x) = \frac{f(x)}{S(x)} = \frac{a_{x}c^{x}}{∑ _{w=x}^{∞} a_{w}c^{w}} x = 0, 1, 2, ... .

The moment generating function of X is

M(t) = \frac{A(ce^{t})}{A(c)} t > 0.

The population mean of X is

E[X] = c\frac{d}{dc}[lnA(c)].

The variance, skewness, and kurtosis of X are mathematically intractable.