Let $X, Y$ be IID $\sim N(\mu, \sigma^2)$.
$$M = \frac12(X + Y),\qquad V = (X - M)^2 + (Y - M)^2$$
Consider the joint moment generating function of $(M, X - M, Y - M)$, show that $M$ and $V$ are independent.
We haven't learn about stuff like Cochran's Theorm or multivariable normal distribution.
I'm confused in that how do you find the MGF of something whos pmf is not give? Or should I be able to work out the pmf? Any help appreciated, thanks!