Let $\displaystyle X_1, X_2,... $~N(0,1) be an iid sequence and let N ~ Poi$\displaystyle (\lambda) $ independently.

Find the MGF of the random sum

$\displaystyle S = \sum_{k=1}^N X_k $

What I do know is the pmf and pdf of the N and $\displaystyle X_n $ , but im confused with the random sum and how to calculate it. Can anyone please help?

I know that the MGF of the random sum is the product of the MGFs' of the RV's that make up that random sum.