I am trying to find $E\left[XY|X+Y=m\right]$ where both X, Y are distributed as independent standard normals. I have tried two different approaches but I am not sure if anyone of them is correct. I am having more troubles than expected so it would be great if you could give me a hand.
First, I have tried solving:
$E\left[XY|X+Y=m\right] = \int_{-\infty}^{\infty} x(m-x) \; \frac{f_{x,y}(x, m-x)}{\int_{-\infty}^{\infty} f_{x,y}(x, m-x) dx }dx$
where $f_{x,y}(x, m-x)=\frac{1}{2\pi} exp \{ -\frac{x^2+(m-x)^2}{2} \}$
Second, I have thought that I could use the fact that:
$L\left[XY|X+Y=m\right]= E[XY] + \frac{Cov(XY,X+Y)}{Var(X+Y)}(m-0)$
However, I do not see why $E\left[XY|X+Y=m\right]$ should be linear.
Thanks.