What is $E[X|Y]$ if the random variables $X$ and $Y$ are independent?

Question

Just looking for an explanation of how the conditional expectation "$E[X|Y]$" of any two random variables $X$ and $Y$ would change if we included the condition that $X$ and $Y$ be independent.

Normally:

• $E[X|Y] = \sum_xx(P_{X|A}(x|y))$ where $P$ stands for the PMF of $X|A$.

If $X$ and $Y$ are independent, could we just apply the property of independent events

• $P(A|B) = P(A)$

to get that $E[X|Y] = \sum_x x\,P_X(x) = E[X]$?

Thank you!

Answer 1

If $X$ and $Y$ are independent then $E[X|Y]=E[X]$. In the case where $X$ and $Y$ are both discrete, you can relate this back to events through the identity $P(A|B)=P(A)$ for independent events $A,B$, where $P(B)>0$.