# Expectation and Conditional Independence

This question is an aside from another question here on CV.

We know that the expectation of the product of two independent random variables is the product of expectations, i.e., $$\mathbb{E}[XY]=\mathbb{E}[X]\mathbb{E}[Y]$$

However, is there a conditionally equivalent version of this statement? For example, could I say if $X$ and $Y$ are dependent random variables then by conditional independence we have

$$\mathbb{E}[XY]=\mathbb{E}[X]\mathbb{E}[Y|X]$$

I know that we can factor joint distributions that way using conditional properties, for example, $$f(x,y)=f(x)f(y|x)$$ and so is there an equivalence for expectations?

Your second equality is certainly untrue unless $X$ and $Y$ are independent.
$$E \left[ X Y | Z \right] = E \left[X|Z \right] E\left[Y|Z \right]$$
• @RustyStatistician You can get $E\left[XY \right] = E \left[ X E\left( Y | X \right) \right]$ simply by applying the Law of Iterated Expectations but I have not seen anywhere what you have written. – JohnK Dec 4 '15 at 0:55