A show for either the discrete case or the continuous case

Question

For k random variables X₁, X₂, ... , X_k, the values of their joint moment-generating function are given by

(a) Show for either the discrete case or the continuous case that the partial derivative of the joint momentgenerating function with respect to t_i at t₁ = t₂ = ··· = t_k = 0 is E(X_i).

(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

 

(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).