Given X,Y,Z are 3 N(1,1) random variables,
Find E[ XY | Y + Z = 1]
The Attempt at a Solution
I'm honestly completely lost in statistics... I didn't quite grasp the intuitive aspect of expectation because my professor lives in the numbers side and not the relating it to others side. So these processes are turning into foggy abstract generalities and any light would be superb.
My book's explanation of the multi-variable cases are that they're essentially the same as the joint case which is the same as the singular case.
For two jointly continuous random variables,
E[X| Y = y] = ∫-∞ , ∞ x*fx|y(x|y)dx
*note that X could easily be replaced by g(X) and that fxy(x|y) is defined as fxy(x,y) / fy(y)
My thoughts are to define two variables
W = XY
V = Y + Z
**note: X + Y = N(2,2) = 1
This allows us to write (1) as
E[W | V = 1]
which would be defined as listed above for the X,Y case.
At this point I'm stuck because finding the pdf seems to essentially be the struggle and I don't know how.