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\noindent
{\bf Power series distribution} (from \color{blue}\url{http://www.math.wm.edu/~leemis/chart/UDR/UDR.html}\color{black})

\noindent
A power series random variable $X$ has probability mass function 
$$
f(x) = \frac{a_{x} \kern 0.08 em c ^ {\kern 0.08 em x}} {A (c)} \qquad \qquad x= 0,\,1,\,2, \,\ldots 
$$
for $a_{x} > 0$, $c > 0$, and series function $A(c) = \sum a_{x} \kern 0.08 em  c ^ {\kern 0.08 em x}$.

\vspace{0.05in}

\noindent
The cumulative distribution function on the support of $X$ is
$$
F(x) = P(X \le x) = \sum_{w = 0} ^ {x} \frac{a_{w}\kern 0.08 em c ^ {\kern 0.08 em w}} {A(c)} \qquad \qquad x= 0,\,1,\,2,\, \ldots \, .
$$
The survivor function on the support of $X$ is
$$
S(x) = P(X \ge x) = \sum_{w=x} ^ {\infty} \frac{a_ {w}\kern 0.08 em c ^ {\kern 0.08 em w}} {A(c)} \qquad \qquad x= 0,\,1,\,2,\, \ldots \, .
$$
The hazard function on the support of $X$ is
$$
h(x) = \frac{f(x)}{S(x)} = \frac{a_{x} c ^ {x}}{\sum_{w = x} ^ {\infty} a_{w} \kern 0.04 em c ^ {\kern 0.04 em w}} \qquad \qquad x=
0,\,1,\,2,\, \ldots \, .
$$
The moment generating function of $X$ is
$$
M(t) = \frac{A \left( c e ^ {\kern 0.08 em t} \right)} {A(c)} \qquad \qquad t > 0.
$$
The population mean of $X$ is
$$
E[X] = c \frac{d} {d c} [\ln A(c)].
$$
The variance, skewness, and kurtosis of $X$ are mathematically intractable.\\
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