(a) Define the probability generating function of a random variable. Calculate the probability generating function of a binomial random variable with parameters $n$ and $p$, and use it to find the mean and variance of the random variable.

(b) $X$ is a binomial random variable with parameters $n$ and $p, Y$ is a binomial random variable with parameters $m$ and $p$, and $X$ and $Y$ are independent. Find the distribution of $X+Y$; that is, determine $P\{X+Y=k\}$ for all possible values of $k$.