The random variable \(X\) has the binomial distribution \(\mathrm{B}(n, p)\).
(a) Write down an expression for \(\mathrm{P}(X=r)\) and hence show that the probability generating function of \(X\) is \((q+p t)^{n}\), where \(q=1-p\).
(b) Use the probability generating function of \(X\) to prove that \(\mathrm{E}(X)=n p\) and \(\operatorname{Var}(X)=n p(1-p)\).