Problem

The random variable $X$ has the geometric distribution $\operatorname{Geo}(p)$.

(a) Show that the probability generating function of $X$ is $\dfrac{pt}{1-qt}$, where $q=1-p$.

(b) Use the probability generating function of $X$ to show that $\operatorname{Var}(X)=\dfrac{q}{p^2}$.

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 $Z$.