Equality of Expectations of Mixture Distributions

Question

Let N be a Poisson r.v. with parameter $\lambda$. Let $Y=\sum_{i=1}^NX_i$ and $X_0,X_1,...$ be independent, identically distributed, nonnegative integer valued r.v. with finite mean. Show that for any function g (such that the expectations exist) we have

$E[Yg(Y)]=\lambda E[X_0g(Y+X_0)]$.

Assuming $X_i$'s and N are independent, I obtained

$E[Yg(Y)|N=n]=\sum_{i=1}^nE[X_ig(\sum_{i=1}^nX_i)]$

and then

$E[Yg(Y)]=\sum_{k=0}^{\infty}e^{-\lambda}\frac{\lambda^k}{k!}\left(\sum_{i=1}^kE\left[X_ig\left(\sum_{j=1}^kX_j\right)\right]\right)$

I tried to interchange two summations but couldn't find anything helpful.

Thanks for any help!

Answer 1

I'm so sorry that I don't have time to type my answer here, now. Here's the image of my solution!