Conditional Expectation - using Wald's Equation

Question

Let $N\sim\!\mathcal{P}(\lambda)$ and $(X_i)_{i\!\geq{1}}$ iid, $X_i\sim\!Be(p)$. If $N$ and $(X_i)_{i\!\geq{1}}$ are independent for all $i$, calculate $P(\mathbb{E}(X_1+\ldots+X_N|N)=0)$.

So using Wald's equation and the fact that $(X_i)_{i\!\geq{1}}$ are iid, I know that $\mathbb{E}(\sum_{i=1}^{N}X_i|N)=N\mathbb{E}(X_1)=Np$

But, how do I calculate $P(\mathbb{E}(X_1+\ldots+X_N|N)=0)$?

Thanks for the help!

Answer 1

As you found, $\mathsf E\left(\sum\limits_{j=1}^N X_j\;\middle\vert\; N\right)= Np$

You know that $N\sim\mathcal P(\lambda)$ so you can find $\mathsf P(Np{=}0)$ from

$$\mathsf P(N{=}k) \;=\; \dfrac{\lambda^k\, {\sf e}^{-\lambda}}{k!}$$