Variable $(X, Y)$ has Gaussian distribution and a given covariance matrix. Find $a$ such that the variables $X +Y$ and $X +aY$ are independent.

Question

The random variable $(X, Y)$ has a two-dimensional Gaussian distribution with mean $(0, 0)$ and a given covariance matrix:

\begin{bmatrix} 6 & -2 \\ -2 & 1 \end{bmatrix}

Determine the number $a$ such that the random variables $U = X +Y$ and $V = X +aY$ are independent.

I know that for $2$ variables to be independent, their join PDF must be possible to break into $2$ PDFs of individual variables, but I don't know how to apply that here. Any help would be much appreciated.

Answer 1

If $U$ and $V$ are independent, then $\mathrm{Cov}(U,V)=0$. The solution proceeds as follows: $$0=\mathrm{Cov}(U,V)=\mathrm{Cov}(X,X)+(a+1)\mathrm{Cov}(X,Y)+a\mathrm{Cov}(Y,Y)$$ $$=6-2(a+1)+a=4-a$$ $$\implies\boxed{a=4}$$