Joint MGF of double exponential distribution.

Question

A random variable x has density function;

Fx(X) = 1/2λexp(-λ|x|) for -∞ < x < ∞ , λ > 0

1) If we let you U=P+Q and V=P-Q how do we get the mgfs of Mu(S) and Mv(t) and also how would be we able to find the joint mgf of U and V?

Quite baffled on how to do this. I know how to find the mgf of the density fuction and due to the absolute value, we split it into 2 different integrals and add them up. But quite confused on how to get it for the above mentioned. Any help or tips would be much appreciated. Thanks in advance.

Answer 1

$$M_{U,V}(s,t)=E[\mathrm e^{sU+tV}]=E[\mathrm e^{(s+t)X+(s-t)Y}]=M_X(s+t)M_Y(s-t)$$