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Multivariate Normal Distribution

  1. Oct 6, 2009 #1
    1. The problem statement, all variables and given/known data

    http://img16.imageshack.us/img16/7703/ass1lx.jpg [Broken]

    2. Relevant equations



    3. The attempt at a solution

    I know that [tex]f(x_1, x_2, x_3) = \frac{1}{(2 \pi)^{3/2}|\Sigma|^{1/2}}exp(-\frac{1}{2}x \Sigma^{-1} x)[/tex] 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)?
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Oct 7, 2009 #2
    Yes, although you don't necessarily need it. Here is the useful property you will need for this problem.

    Let [tex]\bold X[/tex] be multivariate normal with [tex]\bold \mu=E(\bold X)[/tex] and [tex]\bold \Sigma=Var(\bold X)[/tex].

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