Moment generating function applied in $2t$

Question

I am having some trouble with this problem, adapted from Grimmet&Welsh:

If $X + Y$ and $X - Y$ are independent, show that \begin{align} M\left(2t\right) = M\left(t\right)^{3}M\left(-t\right), \end{align} where $X,Y$ are independet r.v. with mean $0$, variance $1$ and $M(t)$ finite.

How to prove it? Does $X$ and $Y$ needs to have normal distribution? Thank you!

Answer 1

Hints:

• $M(2t) = E[e^{2tX}]$
• $M(t) = E[e^{tX}] = E[e^{tY}]$
• $M(-t) = E[e^{-tY}]$
• $2X = (X+Y) + (X-Y)$
• If $U$ and $V$ are independent random variables, then $E[f(U)g(V)] = E[f(U)] E[g(V)]$.