The random variable Z is defined as a function of the random variables X and Y as follows: Z=₁. Moreover, it is known that E[X|Z] = Z² and that Z is zero mean. Compute E[Y|Z = 2] and E[Y].

Given that the MGF of the sum of independent random variables is the product of individual MGFs. If X and Y are independent random variables, then MX+Y(t) = MX(t)MY(t) Prove this statement.

. Let X and Y be random variables and let AE B. Prove that the function X(w), if we A, Y(w). if w€ AC is a random variable. Z(w) =

Suppose the discrete random variables X and Y are jointly distributed according to the given table.  a. Compute the expected values E(X) and E(Y ), the variances V (X) and V (Y ), and also the covariance Cov(X, Y ) of X and Y. b. Determine whether X and Y are independent. c. Let T = X − Y . Compute E(T) and V (T)

3. Determine the value c so that each of the following functions can serve as probability distributions of the discrete random variable X: a. f(x) = c(x² +4) for x = 0,1, 2, 3; 3 b. f(x)%3Dс for x= 0,1, 2 3-x

Suppose X and S2 are calculated from a random sample X₁,..., Xn drawn from a population with finite variance o2. We know that ES2 = o². Prove that ES ≤ 0, and if o2 > 0, then ES

You are given the following information regarding random variable X: I. E[X] = 2 II. E[X³] = 10 III. Var[X] = 1 Find E[(X – 1) ³].

For random variables X and Y such that E(X) = 2, E(Y) = 1, E(X²) = 10, E(Y²) = 3, and E (XY) = c, Find: а. Var(3X — 5Y). b. The correlation coefficient between X and Y, p. The range of values of c for which the assumptions of the problem are consistent. С.