Let \(X\) have the geometric distribution

\[f(x)=p(1-p)^{x-1} \quad \text { for } x=1,2, \ldots\]

(a) Obtain the moment generating function for

\[t<-\ln (1-p)\]

Recall that \(\sum_{k=0}^{\infty} r^{k}=\frac{1}{1-r}\) for \(|r|<1\).]

(b) Obtain \(E(X)\) and \(E\left(X^{2}\right)\) by differentiating the moment generating function.