Given joint moment generating function, what value of $a$ makes $X + 2Y$ and $2X − Y$ independent?

Question

I am new to joint moment generating functions and their properties, so am a bit stuck on how to begin the following problem:

Given $M_{X,Y}(t,u) = exp[2t+3u+t^2+atu+2u^2]$, what value of $a$ makes $X + 2Y$ and $2X − Y$ independent?

How should I start working on this problem? I have calculated that $M_X(t)=M_{X,Y}(t,0) = e^{2t+t^2}$ and likewise that $M_Y(u) = M_{X,Y}(0,u) = e^{3u+2u^2}$, but this does not seem to get me anywhere. Can I use the fact that $M_{aX+bY}(\theta) = M_{X+Y}(a\theta,b\theta)$?

Any help with solving this problem would be much appreciated. Thank you!

Answer 1

Let $V=X+2Y$ and $W=2X-Y$. Work out the MGF of $V$ and $W$. This is $$M_{U,V}(r,s)=E(\exp(rU+sV))=E(\exp((r+2s)X+(2r-s)Y)) =M_{X,Y}(r+2s,2r-s).$$ I'll leave it to you to make the substitution. The aim is to find the $a$ for which $M_{U,V}(r,s)$ factors in the form $F(r)G(s)$.