Jointly normal distribution

**BERRY** · November 15th 2009, 10:51 AM

Let X and Y be jointly normal, with the mean vectors and covariance matrix below:

$<br /> \mu =\left(\begin{array}{cc}1\\2\end{array}\right)$ $\Sigma = \left(\begin{array}{cc}2&.5\\.5&1\end{array}\right )$

Let $Z_{1} = X + Y \mbox{ and } Z_{2} = X - 2Y$

Find the mean vector and covariance matrix of $Z_{1} \mbox{ and }Z{2}$

I figured out the following:

$\mu_{Z_{1}} = \mu_X + \mu_Y = 3,$

$\mu_{Z_{2}} = \mu_X - 2\mu_Y = -3,$

$\sigma_{Z_{1}}^{2} = \sigma_{X}^{2} + \sigma_{Y}^{2} + 2\rho\sigma_{X}\sigma_{Y} = 3 + \sqrt{2},$

$\sigma_{Z_{2}}^{2} = \sigma_{X}^{2} + 4\sigma_{Y}^{2} - 4\rho\sigma_{X}\sigma_{Y} = 6 - 2\sqrt{2}$

But how do I figure out $Cov(Z_{1}, Z_{2})$ ?

**Moo** · November 15th 2009, 01:41 PM

Hello,

Well, the elements of the covariance matrix are the covariances...
So we have cov(X,Y)=cov(Y,X)=.5

And finding cov(Z1,Z2) is then easy, using the fact that the covariance is a bilinear form (cov(ax,y)=a*cov(x,y) , cov(x+y,z)=cov(x,z)+cov(y,z))

Otherwise, you can note that :
$\begin{pmatrix}Z_1 \\ Z_2\end{pmatrix}=\begin{pmatrix} 1&1 \\ 1&-2\end{pmatrix} \begin{pmatrix}X\\Y\end{pmatrix}$

Let's denote $A=\begin{pmatrix} 1&1 \\ 1&-2\end{pmatrix}$

And we know that $A\begin{pmatrix}X\\Y\end{pmatrix}$ will follow a normal distribution $\mathcal{N}(A\mu,A\Sigma A')$

See a proof in the attached pdf

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