$\mathbb{E}(X_{Y+1}X_{2}^{2}X_{2}|x_{1})$ with $X\sim N(0,1)$ and $Y\sim Pois(1)$ both independent

Question

Let $\{X_{i},i\in\mathbb{N}\}$ be a sequence of independent standard normal random variables. Furthermore, $Y$ is a Poisson distributed random variable with parameter $\lambda=1$, i.e., $\mathbb{P}(Y=n)=\frac{e^{-1}}{n!}, n=0,1,2,...$, and Y is independent of the sequence $\{X_{i},i\in\mathbb{N}\}$. Compute the following (conditional) expectation $$ \mathbb{E}(X_{Y+1}X_{1}^{2}X_{2}|X_{1}) $$

Answer 1

I would suggest the following:

$$\mathbb{E}\left[ X_{Y+1} X_1^2 X_2 | X_1 \right] = \mathbb{P}(Y=1) \mathbb{E} \left[ X_2X_1^2 X_2 | X_1 \right] + \mathbb{P} (Y\geq 2) \mathbb{E}\left[X_3X_1^2X_2 | X_1\right]\\ = \mathbb{P}(Y=1) \mathbb{E}[X_2^2] X_1^2 + \mathbb{P}(Y\geq 2) \mathbb{E}[X_2]^2 \cdot X_1^2 $$

And now you simply use the distribution of $Y$ and $X_i$...