Describe the range of a random vector with multivariate normal distribution

**tttcomrader** · October 15th 2012, 11:09 AM

Let $X = \begin{bmatrix} X_1 \\ X_2 \\ X_3 \end{bmatrix}$ have covariance matrix $\Sigma = \begin{bmatrix}3 & 1 & 1 \\ 1 & 4 & -7 \\ 1 & -7 & 15 \end{bmatrix}$ .

Note that $\Sigma \cdot \begin{bmatrix} 1 \\ -2 \\ -1 \end{bmatrix} = 0$ , so $\Sigma$ have rank 2. If $\mu _X = \begin{bmatrix} 10 \\ 20 \\ 30 \end{bmatrix}$ and $X$ has a multivariate normal distribution, describe the range of X.

Solution so far:

Now, since $\Sigma \cdot \begin{bmatrix} 1 \\ -2 \\ -1 \end{bmatrix} = 0$ , I have:

$\begin{bmatrix} 1 & -2 & -1 \end{bmatrix} \Sigma \cdot \begin{bmatrix} 1 \\ -2 \\ -1 \end{bmatrix} = 0$

$\begin{bmatrix} 1 & -2 & -1 \end{bmatrix} Cov(X,X) \begin{bmatrix} 1 \\ -2 \\ -1 \end{bmatrix} = 0$

$Cov( \begin{bmatrix} 1 & -2 & -1 \end{bmatrix} \begin{bmatrix} X_1 \\ X_2 \\ X_3 \end{bmatrix} ) = 0$

$Var( X_1-2X_2-X_3) = 0$

So I need to find out when does that happens, but I'm kind of stuck at the moment, any help please? Thanks!

**chiro** · October 15th 2012, 06:51 PM

Hey tttcomrader.

What part of R^3 describes this linear equation involving X1, X2, and X3? (Hint: It will be on a plane in 3 dimensional space with two independent parameters).

**tttcomrader** · October 17th 2012, 01:24 PM

So since $Var(X_1-2X_2-X_3)=0$ , then we must have $X_1-2X_2-X_3 = k$ for some constant k.

That also means that $E[X_1-2X_2-X_3] = k$ , which is $10-40-30 = -60$ , so $X_1-2X_2-X_3 = -60$ ?

**chiro** · October 17th 2012, 05:52 PM

Looks pretty good.

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