Conditional Expectation And Covariance with Random Variables

Question

For any two random variables X and Y, the covariance is defined as Cov(X, Y ) = E [X − E[X]] [Y − E[Y ]]. • If E [Y |X = x] = x, show that Cov(X, Y ) = E [X − E[X]]2 My approach is that Cov(X,Y)=E(XY)-E(X)*E(Y) If X=Y then maybe the proof becomes true. But I am unable to prove it. The meaning of E [Y |X = x] = x Expectation of Y given X=x is x but how can I use this to prove the solution?

Answer 1

For any two random variables $X$ and $Y$, the covariance is defined as $$Cov(X, Y ) = E\big[(X − E[X]) (Y − E[Y ])\big]$$

Use this definition.

If $E [Y \mid X = x] = x$, show that $Cov(X, Y ) = E\big[(X − E[X])^2\big]$

You shall also need the Tower Rule (or Law of Iterated Expectation). Here's the easy one to get you started.

$$\begin{align}Cov(X, Y ) & = E\big[(X − E[X]) (Y − E[Y ])\big]\\&=E\big[(X − E[X]) (Y − E[E[Y\mid X] ])\big]\\&=E\big[(X − E[X]) (Y − E[X])\big]\\&~~\vdots\end{align}$$

The next is a little more complicated, but not too much. Give it a try...