Multivariate Normal Distribution

[cse63146]

Homework Statement

http://img16.imageshack.us/img16/7703/ass1lx.jpg

Homework Equations

The Attempt at a Solution

I know that $$f(x_1, x_2, x_3) = \frac{1}{(2 \pi)^{3/2}|\Sigma|^{1/2}}exp(-\frac{1}{2}x \Sigma^{-1} x)$$ since n = 3 and mu = 0.

I've never used the multivariate normal distribution. My prof just derived it, but never taught us how to use it.

so does X₁~N(mu,sigma₁₁)?

 

[Billy Bob]

Yes, although you don't necessarily need it. Here is the useful property you will need for this problem.

Let $$\bold X$$ be multivariate normal with $$\bold \mu=E(\bold X)$$ and $$\bold \Sigma=Var(\bold X)$$.

Then any linear combination $$\bold a^T\bold X$$ is univariate normal with $$E(\bold a^T\bold X)=\bold a^T E(\bold X)$$ and $$Var(\bold a^T\bold X)=\bold a^T Var(\bold X) \bold a$$.