Prove $E[XY]=E[YE[X|Y]]$
Solution 1:
Basically $E[XY]=E[E[XY|Y]]=E[YE[X|Y]]$. The first step is the iterated rule of conditional expectation. For the second, use the fact that given Y, Y is like a constant.
However if you are looking for the usage of rigorous definition of conditional expectation, the solution by Davide Giraudo is the one to go for.
Solution 2:
Just use the following:
- For an integrable random variable $Z$, $E[Z]=E[E[Z\mid\mathcal F]]$ for any $\sigma$-algebra $\mathcal F$.
- If $Y$ is $\mathcal F$-measurable, then $E[XY\mid\mathcal F]=YE[X\mid\mathcal F]$.