# Can I conclude the following about bivariate normal RV?

If $(X,Y)$ is bivariate normal with mean $[0, 0]$ and variance-covariance matrix $\left[ \begin{array}{ccc}1 & \rho \\ \rho & 1 \end{array} \right]$ and $Z=-X$ then is it true that $(Z,Y)$ is bivariate normal with mean $[0, 0]$ and variance-covariance matrix $\left[ \begin{array}{ccc}1 & -\rho \\ -\rho & 1 \end{array} \right]$? I think the answer is yes since bivariate normaility of $(X,Y)$ is defined as being expressable as $X=aU+bV$, $Y=cU+dV$, where $U$ and $V$ are independent and normal. Therefore $-X=-aU-bV$ and by definition $(-X,Y)$ are bivariate normal. Now for the variance-covariance matrix, since $Var(X)=Var(-X)$ the main diagonal will be the same. The covariance is $-\rho$ because of the linearity of epectation operator. I am not very experienced in joint distributions, so if somebody could confirm/correct me that would be very much appreciated. Thanks.

## 1 Answer

If $(X,Y)$ is bivariate normal, every collection of linear combinations of $X$ and $Y$ is normal. Your case is a special case of $(Z,T)^t=A(X,Y)^t$ for some matrix $A$. If $(X,Y)$ is normal with mean $(0,0)$ and variance-covariance $C$, then $(Z,T)$ is normal with mean $(0,0)$ and variance-covariance $D$, where $$D=E[(Z,T)^t(Z,T)]=AE[(X,Y)^t(X,Y)]A^t=ACA^t.$$

• thanks. In what textbook can I read more about this? By more I mean proofs. Commented Jan 2, 2014 at 20:12
• – Did
Commented Jan 2, 2014 at 20:42