I have a multipart problem with an explicit PDF given, but really I am just looking for a hint or an idea to go on.
I am either given or have solved for the following:
- Joint PDF: $$\frac{2}{ \pi}e^{x(y+z-x-4)-\frac{1}{2}(y^2+z^2)}, \ x \geq 0;\ y, z \in \mathbb{R}$$
- Joint density for $X, Y$: $f_{X, Y}(x, y)$
- The conditional expectation $E(Y| X= x)$
I am looking to solve for the conditional expectation $E(YZ|X=x)$. How would I proceed? Thank you.
Some thoughts (that could be very wrong): I know $f_Z$ can be obtained from integrating $f_{X, Y, Z}$ w.r.t. to $X$ and $Y$. But is $f_Z$ just $f_{X, Y, Z}/f_{X,Y}$? Then I can integrate $\frac{f_Z \cdot f_{X, Y}}{f_{X,Y}\cdot f_{X}}$?
The reason I am not doing the first part (integrating $x$ and $y$) is because there seems to be some sort of shortcut/logical progression from the way the questions are asked and I was wondering if I can make use of the previous parts that I have already done.