Conditional expectation of product of two random variables

Question

I'm quite new to statistics and I'm struggling with this problem.

Two random variables $X$ ~ Normal$(1, 2)$ and conditioning on $X = x$, $Y$ ~ Normal$(x + 2, 3)$. I want to find $\mathbb{E}[XY]$. This is what I've tried:

$$\begin{align}\mathbb{E}[XY] &= \mathbb{E}[\mathbb{E}[XY|X]]\\&=\mathbb{E}[X\mathbb{E}[Y|X]]\end{align}$$

I have two questions:

1. Is moving $X$ from inside the inner expectation out to the outer expectation considered valid? And why is that?
2. Suppose the answer to 1 is yes, how would I proceed from here? I'm thinking of replacing $\mathbb{E}[Y|X]$ with $X + 2$. Am I on the right track?

Answer 1

Yes, moving $X$ out of the conditional expectation is right. This is a general property of conditional expectations: $E(XY|\mathbb G)=XE(Y|\mathbb G)$ if $X$ is measurable w.r.t. $\mathbb G$.

By hypothesis the mean of $Y$ given $X$ is $X+2$. Hence $EXY=E(X(X+2))=EX^{2}+2EX =5$.

Answer 2

1) Yes, it is valid.

Let us work on probability space $(\Omega,\mathcal A,P)$ and let $\mathcal V$ denote a sub $\sigma$-algebra of $\mathcal A$.

By definition we have for suitable random variable $Y$ defined on that space: $$\mathbb E[\mathbf1_AY]=\mathbb E[\mathbf1_A\mathbb E[Y\mid \mathcal V]]\text{ for every }A\in\mathcal V\tag1$$

Now in the special case $\mathcal V=\sigma(X)$ where $X$ denotes a random variable on the space we have: $$\mathcal V=\{\{X\in B\}\mid B\in\mathcal B\}$$ where $\mathcal B$ denotes the Borel $\sigma$-algebra on $\mathbb R$.

That means that $(1)$ can be rewritten as:$$\mathbb E[\mathbf1_B(X)Y]=\mathbb E[\mathbf1_B(X)\mathbb E[Y\mid X]]\text{ for every }B\in\mathcal B\tag2$$

We recognize this as a statement about indicator functions $1_B:\mathbb R\to\mathbb R$ but fortunately it can be proved that this goes further. It can be proved (give it a try) that it works for any suitable Borel-measurable function $f:\mathbb R\to\mathbb R$ leading to:$$\mathbb E[f(X)Y]=\mathbb E[f(X)\mathbb E[Y\mid X]]\tag3$$

So put $(3)$ in your "probability-backpack".

A special case is then the application of this on the identity function, leading to:$$\mathbb E[XY]=\mathbb E[X\mathbb E[Y\mid X]]$$and this is applied in your question.

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2) Yes, you are on the right track.