# Conditional expectation of two continuous variables

So I have continuous random variables $X$ and $Y$ which have $\mu_x$ and $\mu_y$ as their means and variances $\sigma_x^2$ and $\sigma_y^2$ and correlation $\rho$.

Find $E(Y\mid X)$.

I know that $E(Y\mid X)=\int_{R} y \left[\frac{f(x,y)}{f_X(x)}\right]dy$.

I don't understand how to find the PDFs based on the given information. I realize that this is the regression function, but I don't understand how to approach this.

Regards

• Do you know the distribution of $X$ and $Y$? Such as,are they normally distributed? – Deep North Oct 5 '15 at 0:55
• @DeepNorth Knowing that $X$ and $Y$ are normally distributed does not help in this problem. Knowing that $X$ and $Y$ are jointly normally distributed does; cf. the answer by Vimal. – Dilip Sarwate Oct 5 '15 at 1:36

You haven't specified the probability densities for the two random variables, but if you assume a multivariate normal distribution, you can easily compute the entire conditional distribution $p(Y|X=x)$. Its expectation is simply:
$E[Y|X=x] = \mu_y + \frac{\sigma_y}{\sigma_x} \rho (x - \mu_x)$.
• Does this have to do with $E(Y|X)=\int y(f(x,y)/f_x(x))dy$ – jimduquettesucked Oct 5 '15 at 2:49
• Yes, that is correct. A brute-force way is to substitute the value of $f(x,y)$ with the formula for a bivariate normal and simplify it. There's another clever way, which is easy to argue once you do the brute-force way. Both approaches are described in this post: stats.stackexchange.com/questions/30588/…. – Vimal Oct 5 '15 at 4:34