Prove independence using the fact that a joint distribution is multivariate normal? We have the random variables $X$ and $Y$, and each is distributed normally according to $N(\mu,\sigma^2)$.
We wish to show that the sum $(X+Y)$ and the difference $(X-Y)$ are independent by invoking the fact that their distribution is multivariate normal. Reading around a little, I've found that with this distribution, pairwise independence implies independence. How can I go about showing pairwise independence?
I then need to find the variance–covariance matrix for two transformed
variables, $Z=2X+3Y$ and $V=3X-2Y$. I'm not familiar with this process, but I can only find results online for this process using Excel, R, etc. Is there a simple way to do this hand calculation?
 A: Several key facts are involved:


*

*If two normal random variables are independent, then they are jointly follow a multivariate normal with a diagonal covariance matrix. Here I presume $X$ and $Y$ are independent in this context
So $X, Y$ will jointly follows a multivariate normal

*Any affine transformation of multivariate normal is still multivariate normal. Therefore
$$ \begin{bmatrix} X + Y \\ X - Y \end{bmatrix}
= \begin{bmatrix} 1 & 1 \\ 1 & - 1\end{bmatrix}
\begin{bmatrix}X \\ Y\end{bmatrix}$$
is just another multivariate normal.

*If two normal random variables jointly follow a multivariate normal, they are independent if and only if they are uncorrelated. So you merely need to show that their covariance is equal to zero, which is easy by linearity: 
$$ \begin{align*} Cov[X+Y, X-Y] 
& = Cov[X, X] - Cov[Y, Y] + Cov[Y, X] - Cov[X, Y]\\
& = Var[X] - Var[Y] + 0 - 0  \\
& = 0 \end{align*}$$
For the remaining part, you can again use the linearity of covariance to answer the question. Maybe just left that for you to try to find out the covariance of
$Z, V$ first.
A: If $P(x,y)$ is the joint probability distribution of $x$ and $y$, then the joint distribution of $z(x,y)$ and $w(x,y)$ is given by $$Q(Z,W)=\int\int P(x,y)\delta(z(x,y)-Z)\delta(w(x,y)-W)dxdy$$
Use this to compute the jpd for your variables. Then you can check if they are independent or not, and compute variances, covariances, etc.
