Sample Mean & Variance 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!
 A: $X-M$ is the same thing as $(X-Y)/2$ and $Y-M$ is the same as $(Y-X)/2$.
The requested mgf is:
$$
\begin{align}
& {}\qquad M_{(M,X-M,Y-M)}(s,t,u) = \mathbb{E}\left( e^{sM+t(X-M)+u(Y-M)} \right) = \mathbb{E}\left( e^{s(X+Y)+t(X-Y)/2 + u(Y-X)/2} \right) \\  \\  \\
& = \mathbb{E}\left(e^{(s+t/2-u/2)X+(s-t/2+u/2)Y} \right) =M_{(X,Y)}\left(s+\frac {t-u}{2},s-\frac {t-u}{2}\right) \\  \\  \\
& = M_X\left(s+\frac {t-u}{2}\right) M_Y\left(s-\frac {t-u}{2}\right) = M_X\left(s+\frac {t-u}{2}\right) M_X\left(s-\frac {t-u}{2}\right) \\  \\  \\
& = \exp\left( \left(s+\frac{t-u}{2}\right)\mu +  \left(s+\frac{t-u}{2}\right)^2\frac{\sigma^2}{2} \right)\cdot \exp\left( \left(s-\frac{t-u}{2}\right)\mu +  \left(s+\frac{t-u}{2}\right)^2\frac{\sigma^2}{2} \right) \\  \\
& = \exp\left( s(2\mu) + s^2\frac{2\sigma^2}{2} \right)\cdot\exp \left( (t-u)^2\frac{\sigma^2/2}{2} \right) = M_{X+Y}(s) M_{N(0,\sigma^2/2)} (t-u)
\end{align}
$$
Conclude that the thing that originally got multiplied by $s$ is independent of the thing that originally got multiplied by $t-u$.  The former is $X+Y$.  The latter is $(X-Y)/2$.  You can also conclude that the former is distributed as $N(2\mu,2\sigma^2)$ and the latter as $N(0,\sigma^2/2)$.
