# Conditional Expectation with 2 distributions

you could help me with this problem I would really appreciate it. This is not a task just an exercise to understand the subject. Let the random variable $X \sim \mathrm{Bernoulli}(p)$. Suppose that given an $X = i$, $Y$ is a random variable of type $\mathrm{Poisson} (3 (i + 1))$. Find $\mathbb{E} \left[(X +1) Y^2\right]$.

• Weirdest title I've seen here in a while. – Asaf Karagila Nov 20 '17 at 20:55
• In some languages, the word for "expectation" could be translated to "hope". Of course, this is a bad translation (in this context). – madprob Nov 20 '17 at 20:56
• @AsafKaragila: I guess the OP is Italian or French. We use the term speranza / ésperance (i.e. hope) to denote the expected value. – Jack D'Aurizio Nov 20 '17 at 21:05
• @Jack: Well, I expect that it's not unreasonable to hope that people use the correct English term regardless. Or maybe I hope it's not unreasonable to expect. I'm not which one, and for Italian people, I guess that would be interchangeable anyway. :P – Asaf Karagila Nov 20 '17 at 21:07

HINT Let $V\sim\mathcal{P}(3)$ and $W\sim \mathcal{P}(6)$. Then, by the Law of Iterated Expectation (aka Law of Total Expectation, Tower Property of Expectation):
$$\begin{split} \mathbb{E}\left[(X+1)Y^2\right] & = \mathbb P[X=0]\,\Bbb E\left[V^2\right] +\mathbb P[X=1]~\mathbb E\left[2W^2\right] \\ & = (1-p)\,\mathbb{E}\left[V^2\right] + 2p\, \mathbb{E}\left[W^2\right]\\ \end{split}$$