# Probability - Expectation and conditional distribution

I'm finding it hard to solve a specific question... Tried solving it for the past hour without success. I'll state the problem and show my attempt of solving it so maybe someone could help me. The question:

"Let $n$ random variables be given, $X[1],X[2],...,X[n];$ (The number in the brackets is the index and $n$ is natural) such that all of them are independent with the same probability distribution; Poisson with parameter $Lambda=3$. The random variable $Sn$ is defined as follows : $S[n]=X[1]+X[2]+...+X[n]$.

For $m>=1$ find: $E(Exp(S[n+m])|S[n])$; in other words the expectation of $Exp(S[n+m])$ given $S[n]$."

My attempt of solving it: Since I can neither upload any images showing the solution (low rating) nor type in the required mathematics language for this website, I'll summarize what I did==>

1) Using the convolution theorem, the probability distribution of $S[n]$ is Poisson with parameter $3n$.

2) $S[n+m]= S[n]+ X[n+1]+...+X[n+m]$, so if I define a R.V $M=X[n+1]+...+X[n+m]$ and call $U=S[n+m]$ we get that $U=S[n]+M$ such that M has a Poisson probability distribution with parameter $3m$.

3) $P(U=u|S[n]=s)=P(U=u\;AND\;S[n]=s)/P(S[n]=s)=P(M=u-s)/P(S[n]=s)$ and by substituting the relative probabilities one can find the probability distribution of $P(U|S[n])$ for $u>=s+1$.

4) Hence the answer is the sum over $Exp(U)*P(U|S[n])$ for all $u>=s+1$ which unfortunately doesn't give me the correct answer which is: $Exp(S[n]+3m(e-1))$.

Sorry for the long post, and thanks in advance for any help.

$$\frac{P(U=u\land S[n]=s)}{P(S[n]=s)}=\frac{P(M=u-s)}{P(S[n]=s)}\;.$$
$$P(U=u\mid S[n]=s)=P(M=u-s\mid S[n]=s)=P(M=u-s)\;.$$