Covariance matrix for multivariate normal random variable

Question

Suppose we have a multivariate normal random variable $X = [X_1, X_2, X_3, X_4]^⊤$. And here $X_1$ and $X_4$ are independent (not correlated). Also $X_2$ and $X_4$ are independent. But $X_1$ and $X_2$ are not independent. Assume that $Y = [Y_1, Y_2]^⊤$ is defined by $$Y_1 = X_1 + X_4,~~ Y_2 = X_2 − X_4.$$ If I know the covariance matrix of $X$, what would be the covariance matrix of $Y$?

Answer 1

You can assume w.l.o.g. that $E[X]=0$. Then $E[Y]=0$ (variances/covariances are not dependent on means).

You need to compute $Var(Y_1),Var(Y_2), E[Y_1 Y_2]$ since the covariance matrix of $Y$ is comprised of these three elements.

Since $X_1,X_4$ are independent, $Var(Y_1)=Var(X_1)+Var(X_4)$ which you should konw from the covariance matrix of $X$.

Similarly for $Var(Y_2)$.

Finally you can compute $E[Y_1 Y_2]= E[ X_1 X_2 - X_4^2] = Cov(X_1,X_2) -Var(X_4)$ which you should know from the covariance matrix of $X$