Question

Find the mean and variance of a random variable X whose distribution has probability generating moment function is

pr / 1 − (1 − p)r 0 < p < 1

Answer 1

By definition, probability generating function is

$\psi(r) = Er^X = \dfrac{pr}{1-(1-p)r}.$

Moments of $X$ can be found in a following way

$EX = \psi'(1), \quad EX^2 = \psi''(1) + \psi'(1).$

The derivatives of $\psi$ are $\psi'(r) = \dfrac{p}{(1-(1-p)r)^2}, \ \psi''(r) = \dfrac{2p(1-p)}{(1-(1-p)r)^3}$ ,

so $EX = \dfrac{1}{p}, \ EX^2 = \dfrac{2(1-p)}{p^2} + \dfrac{1}{p} = \dfrac{2-p}{p^2}, Var X = \dfrac{1-p}{p^2}.$