Introduction to Probability · Problem 10.1.11

Generating Functions — Grinstead & Snell

Mathematics — Probability · generating-functions-discrete

Problem

Show that if

X

is a random variable with mean

\mu

and variance

\sigma^2

, and if

X^* = (X - \mu)/\sigma

is the standardized version of

X

, then

g_{X^*}(t) = e^{-\mu t/\sigma} g_X\left(\frac{t}{\sigma}\right).