$ X $ and $ Y $ are both independent and identically distributed random variables with normal distributions $ \mathcal N ( 0 , 1 ) $. What is $ E ( X Y | X + Y = w ) $?
I know this means that $ W=X+Y $ must be normally distributed as well with mean $ 0 $ and variance $ 2 $. I also know that $ E ( X Y ) = E ( X ) E ( Y ) $ because of independence. However, I am confused as to how we calculate the conditional expectation in this situation. Do we just take the integral of the normal pdf? What would the boundaries be?