# Covariance matrix for multivariate normal random variable

Suppose we have a multivariate normal random variable $X=\left[{X}_{1},{X}_{2},{X}_{3},{X}_{4}{\right]}^{\top }$ . 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 $Y=\left[{Y}_{1},{Y}_{2}{\right]}^{\top }$ is defined by

If I know the covariance matrix of X, what would be the covariance matrix of Y?
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Step 1
You can assume w.l.o.g. that $E\left[X\right]=0$ . Then $E\left[Y\right]=0$ (variances/covariances are not dependent on means).
You need to compute $Var\left({Y}_{1}\right),Var\left({Y}_{2}\right),E\left[{Y}_{1}{Y}_{2}\right]$ since the covariance matrix of Y is comprised of these three elements.
Since ${X}_{1},{X}_{4}$ are independent, $Var\left({Y}_{1}\right)=Var\left({X}_{1}\right)+Var\left({X}_{4}\right)$ which you should konw from the covariance matrix of X.
Similarly for $Var\left({Y}_{2}\right)$ .
Finally you can compute $E\left[{Y}_{1}{Y}_{2}\right]=E\left[{X}_{1}{X}_{2}-{X}_{4}^{2}\right]=Cov\left({X}_{1},{X}_{2}\right)-Var\left({X}_{4}\right)$ which you should know from the covariance matrix of X