Past exam of the computer science course of the University of Oxford / 2015 / Probability and computing / 1 / f / Solution

For $X$ with binomial distribution $B(n,p)$, where $n$ is a nonnegative integer and $0\le p\le 1$, use its probability mass function and the binomial theorem:This moment-generating function of a binomial distribution is finite for every real $t$, since the random variable has finite support. The formula also includes the degenerate cases $p=0$, $p=1$ and $n=0$.