# Formula for expectation conditional on two random independent variables

#### GJe

##### New Member
Suppose X, Y, Z are standard normal random variables. I am interested in the expectation E(X | Y, Z), which is obviously messy because it involves a bivariate normal. Is there a simple expression for this conditional expectation when Y and Z are independent random variables?

#### fed2

##### Active Member
thats a good question. best answer i can give is that 'yes it probably does'. I base that on page 269 of http://www.math.chalmers.se/~rootzen/highdimensional/SSP4SE-appA.pdf

It says the formuals are 'easy to remember', so i guess that qualifies as simple. Just staring at them, it looks like they will simplify a bit in the Y indep of Z case.

I believe for example, if y and z are unit normal, then cond dist has variance
var(X) - COV(X,Y)^2 + COV(X,Z)^2, don't quote me on that.

similar for mean, its mean(X) + COV(X,Y)*Y + COV(X,Z)*Z
dont quote me on tht either, but it works out pretty simple.

Good luck and god speeds.