Example expectation of an exponential function

Question

Given a geometric random variable $Y$ with $p = 1/3$, I know that $E[Y] = 1/p = 3$.

However, what is $E[e^{aY}]$ ? for a small value $a$.

Thanks

Answer 1

If $k$ is a positive integer, then the random variable $X$ takes on the value $k$ with probability $p(1-p)^{k-1}$. It follows by the Law of the Unconscious Statistician that $$E(e^{aY})=\sum_{k=1}^\infty e^{ak} p(1-p)^{k-1}.$$ This is an infinite geometric series with first term equal to $pe^a$ and common ratio $r=(1-p)e^a$. If $r\lt 1$, the series converges. For a simplified version of the value, use the ordinary formula for the sum of an infinite geometric series.