# Thread: conditional expected value of bivariate normal

1. ## conditional expected value of bivariate normal

Let X~N(1,1), Y~N(2,1) be bivariate normal with cov(X,Y)=1/2. Obtain E(Y|X).

I see that you can get E(XY) using the covariance, but I don't see where that leads. The only way I know to calculate the conditional expectation is by integrating over the density function of Y given X, which is so ugly in this case that there's certainly an easier approach.

2. Originally Posted by Beaky
Let X~N(1,1), Y~N(2,1) be bivariate normal with cov(X,Y)=1/2. Obtain E(Y|X).

I see that you can get E(XY) using the covariance, but I don't see where that leads. The only way I know to calculate the conditional expectation is by integrating over the density function of Y given X, which is so ugly in this case that there's certainly an easier approach.
Do you already know that $E(Y|X)=aX+b$ for some $a,b$? (this is a general fact with Gaussian vectors) If so, then you can find $a,b$ from the values of $E[Y]$ and $E[XY]$ (they lead to a system of equations).