Let $X$ and $Y$ be two independent random variables both have Laplace distribution. What is the moment generating function of $U=X+Y$ and $V=X-Y$?
Initially, I want to work out the $f_{U,V}(u,v)$, and then work out the $M_{U,V}(s,t)=E_{f_{U,V}}(e^{sU+tV})$. But they are hard to compute.
So I try another way: $M_{U,V}(s,t)=E_{f_{U,V}}(e^{sU+tV})=E_{f_{U,V}}(e^{s(X+Y)+t(X-Y)})=E_{f_{U,V}}(e^{(s+t)X+(s-t)Y)})$
But at this point I am not so sure about whether I can make it become $E_{f_{X}}(e^{(s+t)X})E_{f_{Y}}(e^{(s-t)Y})$.
Can I do this? Why? Thanks in advance.