statistics questions

Problem 1 (a) Suppose that Z ~ N(0, 1). Find E[|ZI], E[Z-'] and E[Z-1/3], if they exist. (b) Suppose that K is a random variable that only takes values in {0, 1, 2, 3. . . .). Show that EK] = > P(K > k). (b) If X - Gamma(or, #), find E[ X'ed ], specifying the real values of s and f for which it exists. (b) Suppose that a random variable X has the property that, for any a > 1, the conditional dis- tribution of X/a given that X > a is the same as the distribution of X. Specify all possible distributions of X (by giving formulas for their c.d.f. or p.d.f.), as well as their expected value. Problem 2 Let (X, Y) be a jointly continuous pair of random variables such that the marginal density of X is fx (x ) = 2 V4-x2, -2