# Thread: Describe the range of a random vector with multivariate normal distribution

1. ## Describe the range of a random vector with multivariate normal distribution

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

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

Solution so far:

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

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

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

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

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

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

2. ## Re: Describe the range of a random vector with multivariate normal distribution

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).

3. ## Re: Describe the range of a random vector with multivariate normal distribution

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

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

4. ## Re: Describe the range of a random vector with multivariate normal distribution

Looks pretty good.