# Conditional expectation of two independent RV

The expectation of the product of two independent random variables $$X$$ and $$Y$$ is the product of the expectations: \begin{align} E(XY) = E(X)E(Y) \end{align} Let's add another random variable $$Z$$ in the mix, do we have the following equality: \begin{align} E(XY \vert Z) = E(X \vert Z)E(Y \vert Z) \end{align}

• $E(XY) = E(X)E(Y)$ is, generally speaking, not true. – Viktor Mar 2 at 9:09
• Yes of course, I forgot to write the independent hypothesis, just edited the post – Victor Mar 2 at 9:13
• The short answer is: you need conditional independence of $X$ and $Y$ given $Z$. Independence of $X$ and $Y$ is insufficient. See stats.stackexchange.com/a/184933/80704 – Viktor Mar 2 at 9:24

Definitely not true (in general). For example, let's say, $$X,Y$$ are the two different coin tosses. And, $$Z$$ is the number of heads you get after these tosses. Apparently, $$X$$ and $$Y$$ are independent; but given $$Z$$, they're not. Also, independence means $$E[XY]=E[X]E[Y]$$, but having it holds doesn't mean they're independent.