The random variable \(X\) has the geometric distribution \(\operatorname{Geo}(p)\).
(a) Show that the probability generating function of \(X\) is \(\frac{p t}{1-q t}\), where \(q=1-p\).
(b) Use the probability generating function of \(X\) to show that \(\operatorname{Var}(X)=\frac{q}{p^{2}}\).

Kenny throws an ordinary fair 6-sided dice repeatedly. The random variable \(X\) is the number of throws that Kenny takes in order to obtain a 6 . The random variable \(Z\) denotes the sum of two independent values of \(X\).
(c) Find the probability generating function of \(Z\).