Given joint pdf $f(x,y,z) = \frac{2}{3}(x+y+z)$, how to find $E(X+Y|Z)$?

Question

As the title says, how do you find $E(X+Y|Z)$ given $f(x,y,z) = \frac{2}{3}(x+y+z)$?

Not sure how to go about approaching this problem. The furthest I got was:

$$ E(X+Y|Z) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}(x+y)f(x+y|z)dxdy $$

And now I'm stuck because I'm not sure how to get $f(x+y|z)$.

Answer 1

I think in general you can do as follows:

• $\mathbf{E}[X+Y|Z] = \mathbf{E}[X|Z] + \mathbf{E}[Y|Z]$

• $\mathbf{E}[X|Z]$ can be found from $f_{X,Z}(x,z)$ and $f_Z(z)$. Same for $\mathbf{E}[Y|Z]$.

• Here you are missing the bound of the domain.

Answer 2

Are you sure $f(x,y,z)$ does not have some bound on its domain? If I integrate $f(x,y,z)$ from $-\infty$ to $\infty$ I definitely don't get 1.