# Question

For k random variables X1, X2, . . . , Xk, the values of their joint moment- generating function are given by E(et1X1+ t2X2+ · · · + tkXk)

(a) Show for either the discrete case or the continuous case that the partial derivative of the joint moment-generating function with respect to ti at t1 = t2 = … = tk = 0 is E( Xi).

(b) Show for either the discrete case or the continuous case that the second partial derivative of the joint moment- generating function with respect to ti and tj, i ≠ j, at t1 = t2 = … = tk = 0 is E(XiXj).

(c) If two random variables have the joint density given by

Find their joint moment- generating function and use it to determine the values of E(XY), E(X), E(Y), and cov(X, Y).

(a) Show for either the discrete case or the continuous case that the partial derivative of the joint moment-generating function with respect to ti at t1 = t2 = … = tk = 0 is E( Xi).

(b) Show for either the discrete case or the continuous case that the second partial derivative of the joint moment- generating function with respect to ti and tj, i ≠ j, at t1 = t2 = … = tk = 0 is E(XiXj).

(c) If two random variables have the joint density given by

Find their joint moment- generating function and use it to determine the values of E(XY), E(X), E(Y), and cov(X, Y).

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