Given joint moment generating function (mgf), calculate $P(X + 2Y < 2X − Y)$

Question

Given $X+2Y$ and $2X-Y$ are independent, and that $M_{X,Y}(t,u)=\exp\left[2t+3u+t^2+\dfrac{4}{3}tu+2u^2\right]$, how would one calculate $P(X + 2Y < 2X − Y)$?

Answer 1

With the help of @kimchilover, I was able to solve this problem, and hence wanted to post the solution:

Let $U = (2X-Y)-(X+2Y)$. Then, $P(X+2Y<2X-Y) = P(X-3Y>0) = P(U>0)$.

$M_U(t) = M_{X-3Y}(t) =M_{X,Y}(t,-3t)=exp[2t-9t+t^2+\dfrac{4}{3}(t)(-3t)+18t^2]=exp[-7t+15t^2]$. This is the mgf for a $N(-7,30)$ distribution. Therefore, we have $U \sim N(-7,30)$.

Then, $P(X+2Y<2X-Y) = P(U>0) = 0.101$.