Let X and Y be jointly continuous random variables with joint density function

f(x, y) = cx²/ y

0

0≤x≤y²,

where c is a constant to be determined.

(a) Determine the value of c.

(b) Find E[XY].

(c) Are X and Y independent? Why or why not?

(d) Evaluate the marginal density function of X, fx(x).

(e) Write down the conditional density function of Y given X = x.

(f) Give an expression for E[Y | X = x] in terms of x.

(g) What is the limit of E[Y | X = x] as x → 0+?

(h) What does this suggest about the "conditional distribution of Y given X = 0"? 2 pts

(i) Let U = X/Y². Evaluate and simplify the joint density function fu,y(u, y).

(j) What is the conditional expectation of Y given that U = u?