Chapter 8.4, Problem 59E

Suppose that X is a random variable on a sample space S such that X(s) is a nonnegative integer for all $s\in S$ . The probability generating function for X is

   $G_X\left(x\right)={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}p\left(X\left(s\right)=k\right)x^k}$

(Requires calculus) Show that if G_Xis the probability generating function for a random variable X such that X(s) is a nonnegative integer for all $s\in S$, then

a) $G_X\left(1\right)=1$ .

b) $E\left(X\right)=G_X^{\text{'}}\left(1\right)$

c) $V\left(X\right)=G_x"\left(1\right)+G_x\text{'}\left(1\right)-G_x\text{'}{\left(1\right)}^2$