The conditional expected value of a two-dimensional variable

Question

I have problem:

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

$$ f(x,y) = \left\{ \begin{array}{ll} 2x^2 & \textrm{when $-1<x<1$ and 0<y<|x|}\\ 0 & \textrm{in other case}\\ \end{array} \right. $$

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.

Thanks in advance.

Answer 1

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 $\int_{-0.5}^0 2x^2(2x+ 0.5) dx$.