Suppose $f$ is an uniformly distributed random variable with parameters $-1,1$ and $g$ is a Poisson-distributed random variable with parameter $\lambda >0$. We assume that $f$ and $g$ are independent and for a constant $C \in \mathbb{R}$ we define the random variable $h:=f+C^g$. Now I want to calculate $\text{E}[h]$ and $\text{Var}[h]$.
We have $\text{E}[f]=0$, $\text{Var}[f]=\frac{1}{3}$ and $\text{E}[g]=\text{Var}[g]=\lambda$.
$$\text{E}[h]=\text{E}[C^g]$$
$$\text{Var}[h]=\text{E}[h^2]-\text{E}[h]^2=\text{E}[f^2]+2\text{E}[fC^g]+\text{E}[C^{2g}]-\text{E}[h]^2$$ How can I calculate $\text{E}[C^g]$?
To get you started: $$E(C^g)=\sum_{n=0}^\infty C^nP(g=n)=\sum_{n=0}^\infty C^n\mathrm e^{-\lambda}\frac{\lambda^n}{n!}=\mathrm e^{-\lambda}\sum_{n=0}^\infty \frac{(C\lambda)^n}{n!}=\ldots$$