1. Consider a two-dimensional random vector (X, Y) with joint density
$$f_{X,Y}(x, y) = \frac{4}{x^2y^2}, \quad 1 < x < 2, \quad 1 < y < 2.$$
(a) Verify that $f_{X,Y}$ is a properly defined joint density.
(b) Find the marginal densities of X and Y.
(c) Find the conditional density of X given Y, and use this to determine if X and Y are independent.
(d) Calculate $E[X^2Y]$.
(e) Calculate the covariance between X and XY.
