Like in a previous question, I'm considering the correlated bivariate normal variables
$$\left(\begin{array}{c}X\\Y\end{array}\right)\sim\mathcal{N}\left[\left(\begin{array}{c}4\\2\end{array}\right),\left(\begin{array}{cc}1&\frac12\\\frac12&1\end{array}\right)\right].$$
I know that $U=X+Y$ and $V=X-Y$ are uncorrelated, and know how to show this. However, I could not tell from first sight that this selection of linear combinations of $X$ and $Y$ were uncorrelated at first. I know it is possible to construct correlated variables from uncorrelated ones. My question is therefore, given a set of jointly normal correlated variables, is it possible to construct a set of jointly normal uncorrelated variables?
Best Answer
Basically you are looking for $a,b,c,d\in\mathbb R$ such that \begin{align*}0&=\text{cov}(aX+bY,cX+dY)\\&=ac\,\sigma^2_X+(ad+bc)\text{cov}(X,Y)+bd\,\sigma^2_Y\\&=ac+\frac{ad+bc}2+bd\\&=a\left(c+\frac d2\right)+b\left(d+\frac c2\right).\end{align*} So $(a,b,c,d)=(1,1,1,-1)$ works. You can also take $(a,b,c,d)=(1,0,1,-2)$, giving that $X$ and $X-2Y$ are independent.