Conditional Expectation given the correlation

Question

I'm taking an econometrics course and have been stumped by a seemingly easy question regarding conditional expectation. We know that $E(X_1)=E(X_2)=0, Var(X_1)=Var(X_2)=1, Corr(X_1,X_2)=-0.35$

I am trying to find $E(X_2|X_1)$ in terms of $X_1$ which apparently is: $E(X_2|X_1)=\rho_{12}X_1$

But I struggle to see how you can make that conclusion.

So far I know that $cov(X_1,X_2)=E((X_1-0)(X_2-0))=E(X_1X_2)=-0.35$

But I'm stumped from this point.

Thanks

Answer 1

There is a general result concerning bivariate normal random variables:

If $X$ and $Y$ have bivariate normal distribution with means $\mu_1$ and $\mu_2$ respectively, and variances $\sigma^2_1$ and $\sigma^2_2$ respectively, and correlation $\rho$, then:

$$E(X\mid Y) = \mu_1 + \rho\frac{\sigma_1}{\sigma_2}(Y-\mu_2)$$

and

$$\operatorname{Var}(X\mid Y)=\sigma_1^2(1-\rho^2).$$