Math Help - stupid normal distribution question

1. Computing conditional expectation...

if I have X1,X2 iid normal with N(0,1)

and I want to find E(X1*X2 | X1 + X2 = x)

Can I simply say: X1 = x - X2 and thus

E(X1*X2 | X1 + X2 = x) =

E[ (x - X2)*X2) = E[ (x * X2) - ((X2)^2) ] <=>

x*E[X] - E[X2^2] =
0 - 1 =
-1

???

2. Also,

I think I heard somewhere that if you have X and Y are two random variables, both of which have the standard normal distribution N(0,1), then X and Y are independent...

This sounds strange to me, is it true?

3. no and no
you need the covariance to be zero in the second question.

4. i see, how do i find the first one then?

i see, for the second one, thanks!

5. I'm bet there is a clever way to do the first, but the straight fowrard way will work.

Obtain the conditional density.
let W=XY and Z=X+Y.
From the joint density of X and Y you can obtain the joint density of W and Z.

Then $f(W|Z)={f(W,Z)\over f(Z)}$

So $E(W|Z)=\int wf(w|Z)dw$

The marginal density of Z is obvious.