Expected value of product of dependent Poisson random variables

Question

I would like to compute

$$\mathbb{E}\bigg[Y\sum_{j=1}^{Y}X_j\bigg]$$

where $Y\sim\text{Poisson}(\lambda)$ for some $\lambda>0$ and the $\big\{X_j\big\}$ are iid with $X_j\sim\text{Bernoulli}(p)$ for some $p\in(0,1)$. The variable $Y$ and the $X_j$ are assumed to be independent.

I'm not sure how to go about this, because it's the product of two dependent Poisson random variables. Tips appreciated.

Answer 1

$\newcommand{\e}{\operatorname{E}}$According to the law of total expectation, \begin{align} & \e\left( Y \sum_{j=1}^Y X_j \right) = \e\left( \e\left( Y \sum_{j=1}^Y X_j \,\Big\vert\, Y \right) \right) = \e\left( Y \e\left( \sum_{j=1}^Y X_j \right) \right) \\[10pt] = {} & \e\left( Y (Yp) \right) = p\e\left( Y^2\right) = p \left( \lambda+\lambda^2 \right) \end{align}