how to use correlation to calculate condition expectation

Question

Suppose (X, Y) is Bivariate Normal, with X,Y ~ N(0, 1), and Corr(X, Y) = $\rho$ , please find E(Y|X) and E(X|Y).

I don't have a clue how to solve this, anyone could give me a pointer? Just for the record, this is not my homework, I just study math in my spare time as a hobby.

Thanks.

Answer 1

One approach: The joint density of $(X,Y)$ is $$f(x,y)=\frac1{2\pi\sqrt{1-\rho^2}}\exp \left[-\frac1{2(1-\rho^2)}(x^2-2\rho xy+y^2)\right]\tag1$$ and the marginal density of $X$ is $$ f(x)=\frac1{\sqrt{2\pi}}\exp\left[-\left(\frac{x^2}2\right)\right].\tag2 $$ Compute the conditional density of $Y$ given $X=x$ by dividing (1) by (2). After some algebraic simplification you'll find that this conditional density given $X=x$ belongs to a certain distribution with a specific mean, and therefore you've discovered $E(Y\mid X=x)$.

In fact the conditional distribution of $Y$ given $X=x$ is normal with mean $\rho x$ and variance $1-\rho^2$. It follows that $E(Y\mid X=x)=\rho x$ and so $E(Y\mid X)=\rho X$. By symmetry, you get $E(X\mid Y)=\rho Y$.