Mean value $E(2^X)$

Question

Let $X$ be a random variable and the distribution of $X$ is a Poisson distribution with $\lambda$ parameter. We know that mean value $E(X) = \lambda$, but what we can say about $E(2^X)$?

Answer 1

I guess you want to know what we can say about $E[2^X]$.

For each measurable function g for a given rv X with density $f_X$ we can calculate $E[g(X)]$ by $$\int_{-\infty}^\infty g(x) f_X(x) dx$$ In the discrete case this leads to $$\sum_{k=0}^\infty g(k) P(X=k)$$

In your case is $g(x) = 2^x$

What's $P(X=k)$ if $X$ is a poisson distribution?