[math]X[/math] and [math]Y[/math] are independent random variables with common moment generating function [math]M(t) = \exp(t^2/ 2).[/math] Let [math]W = X + Y[/math] and [math]Z = Y - X[/math]. Determine the joint moment generating function, [math]M(t_1,t_2)[/math] of [math]W[/math] and [math]Z[/math].

• [math]\exp(2t_1^2 + 2t_2^2 )[/math]
• [math]\exp[(t_1 − t_2 )^2 ][/math]
• [math]\exp[(t_1 + t_2 )^2 ][/math]
• [math]\exp(2t_1t_2 )[/math]
• [math]\exp(t_1^2 + t_2^2 )[/math]

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: E

[[math]] \begin{align*} M(t_1,t_2) = E[e^{t_1W + t_2Z}] = E[e^{t_1(X+Y) + t_2(Y-X)}] &= E[e^{(t_1-t_2)X}e^{(t_1 + t_2)Y}] \\ &=E[e^{(t_1-t_2)X}]E[e^{(t_1 + t_2)Y}] \\ &= e^{\frac{1}{2}(t_1-t_2)^2}e^{\frac{1}{2}(t_1+t_2)^2} \\ &= e^{\frac{1}{2}(t_1^2 - 2t_1t_2 +t_2^2)} e^{\frac{1}{2}(t_1^2 + 2t_1t_2 + t_2^2)} \\ &= e^{t_1^2 + t_2^2} \end{align*} [[/math]]

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.