Probability theory MOC

Probability-generating function

The probability generating function is a generating function for the probability mass function of a -valued discrete random variable defined by prob

by the Law of the unconscious statistician. This is well-defined as a convergent function .

Properties

1. If the Moment-generating function exists, for

\begin{align*} \mathbb{P}[X=x]= \frac{g_{X}^{(x)}(0)}{x!} \end{align*}

\begin{align*} g_{\lambda X+\mu Y}(t) = g_{X}(t^\lambda) + g_{Y}(t^\mu) \end{align*}