08 - Properties of the Probability Generating Function

Properties of the Probability Generating Function

Let $X$ be a random variable taking non-negative integer values then

$$G^{(k)}(0)=k!p_X(k)\text{ for }k=0,1,2,\cdots$$

$$G(1)=1\text{ and }{G}^{(k)}(1)=\mathbb{E}[X(X-1)\cdots (X-k+1)]\text{ for }k=1,2,\cdots$$

Note that a distribution is determined by its generating function