Given a sequence of independent variables {$X_i, i \in \mathbb{N}$} with $X_i$ ~ $N(0,1)$ for all $i$ and a Poisson distributed random variable $Y$ with parameter $1$ independent of the aforementioned sequence. We want to compute:
$\mathbb{E} [X_{Y+1}X^2_{1}X_{2} | X_{1}]$
My steps were the following:
1) First we can take out the $X^2_{1}$ due to the fact that we are conditioning on $X_1$. This leaves us with:
$X^2_{1}*\mathbb{E} [X_{Y+1}X_{2} | X_{1}]$.
2) This is where I get stuck because our variable $Y$ can also have value $1$ which would give us $X^2_{2}$ within the expectation. However, there is also a probability that it is not equal to $1$. This gave me the following idea; we can turn the expectation above into a sum of the probabilities as follows:
$X^2_{1}*\mathbb{E} [X_{Y+1}X_{2} | X_{1}]$ = $X^2_{1}(\mathbb{P}(Y=1)*\mathbb{E}[X^2_{2}]+(1-\mathbb{P}(Y=1))*\mathbb{E}[X_{Y+1}]*\mathbb{E}[X_{2}])$.
This could then be further solved using rules for the moments of a standard normal distribution. My question is, is this step correct? Can i write the expectation as the sum of a probability times expectation?
I must add that this question is for a homework exercise so it would be preferred to not give the final answer immediately but rather a tip on how to get there.