I am looking to minimizing the following function:
$$\min_{f(\cdot)} E_{XY}[(x - f(\cdot))]$$
where $f(\cdot)$ describes linear functions of the form $Ay + d$. I know the following about distributions:
$$P_X(x) = \frac{1}{a}e^{\frac{-x}{a}}u(x)$$ $$P_{Y|X}(y|x) = xe^{-xy}u(y)$$
How would I go about solving this problem? I think that taking derivatives would be good and I know that I can find the joint distribution to use for the expectation, but the expectation of y is divergent. Any hints?