principle of mathematic statistics

Problem 1 (a) Suppose that Z - (0, 1). Find E [|Z|] , 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 E[X] = > P(K > k). (b) If X - Gamma(o, 8), 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