Given:
- and jointly distributed normal RV
- [] = $\alpha$
- () = () = 1
- [] = [] = 0
- = +
- = −
I need to compute the joint and marginal distributions of , and say whether and are independent. For the second part, I said that and are independent because we know for multivariate random variables Cov(,) = 0 implies independence, and Cov(,) = $\frac{1}{2}$Var(+)-$\frac{1}{2}$Var()-$\frac{1}{2}$Var(B)=0. Is that correct? And how do I find the joint and marginal distributions?
Yes, $A$ and $B$ are independent because they are jointly normal and their covariance is $0$.
$A$ and $B$ are normal with mean $0$ and variances $EX^{2}+EY^{2}+2EXY-E(A+B)=2+2\alpha$ and $EX^{2}+EY^{2}-2EXY-E(A+B)=2-2\alpha$.
Their joint distribution is normal with means $0$ and variance covariance matrix $\begin{bmatrix} (2+2\alpha) \,\,\,\,\,0\\0\, \,\,\,\,\,\,\,(2-2\alpha)\end{bmatrix} $