# Covariance matrix for multivariate normal random variable

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 $$Y$$?

• By using properties of variance and covariance, you can write $\text{Var}(Y_1)$, $\text{Var}(Y_2)$, and $\text{Cov}(Y_1, Y_2)$ in terms of variances/covariances involving the $X_i$. Nov 21, 2021 at 23:28

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 $$X$$

• How does var Y1, var y2 and E [y1y2] lead to Cov(Y)? Nov 21, 2021 at 23:33
• The covariance matrix of $Y$ is comprised of the variances in the diagonal and covariances in the off-diagonal. If you assume that $E[X]=0$ then $E[Y]=0$ which means that $Cov(Y_1,Y_2)=E[Y_1,Y_2]$
– Cris
Nov 21, 2021 at 23:35
• I see. But how will this be expressed in matrix form. Nov 21, 2021 at 23:39
• The X has a covariance matrix 4x4 so there are many off diagonal elements. I know that we can find the variance Ford y1, y2 by looking at the diagonal elements. I’m not so sure what we are doing with the expectation [y1, y2] And if this covariance will be a 3 x 3 because X3 is not part of equations in y1, y2 Nov 21, 2021 at 23:41
• You need the covariance matrix of $Y$ which is $2x2$, not that of $X$. See the definition of the covariance matrix here: en.wikipedia.org/wiki/Covariance_matrix
– Cris
Nov 21, 2021 at 23:43