# The conditional expected value of a two-dimensional variable

I have problem:

We have two-dimensional variable: ($$X,$$Y), which has density:

$$f(x,y) = \left\{ \begin{array}{ll} 2x^2 & \textrm{when -1

Compute: $$E(X-Y|X+Y=-0.5)$$

I tried calculate common distribution for $$X-Y$$ and $$X+Y$$, but I do not know how.

If X+ Y= -0.5 then Y= -0.5- X so that X- Y= X- (-0.5- X)= 2X+ 0.5. Further, the graph of X+ Y= -0.5 is the line from (-0.5, 0) to (0, -0.5). The expected value is $$\int_{-0.5}^0 2x^2(2x+ 0.5) dx$$.
• Thank you. But I think the line will be from $(-1,-0.5)$ to $(0.-0.5)$. The limit should be from $-1$ to $-0.5$. This integral is equal $-31/48$. I have to select answer from:$-8/7$, $-5/4$, $-7/8$, $-31/28$, $-15/14$. This solution is for sure good? – John1357 Jan 19 at 13:42