A coin is tossed a random number of times $N$, and head comes up with probability $p$. Define the number of times head comes up by $E$. Given that $N$ is Poisson distributed with parameter $\lambda$, show that $E$ is Poisson distributed with parameter $\lambda p$.
My idea was to show this by using the pgf of $E$, since $G_E(s) = G_N(ps + 1 - p)$, where $G$ denotes the pgf. By showing that the pgf of $E$ is equal to that of a $Pois(\lambda p)$ distributed random variable, I would be done. However, when writing out $G_N(ps + 1 - p)$ I don't see how I can get to the expression I want to get to, so any tips or hints regarding this problem would be very welcome!
Let $G_{E}$ be the probability generating function of $E$. Note that $$ G_{E}(t)=G_{N}(1-p+pt)=\exp(\lambda(1-p+pt-1))=\exp(\lambda p(t-1)) $$ whence $E$ is poisson distributed with mean $\lambda p$ where $G$ denotes probability generating function.