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Multivariate Normal Distribution

  1. Feb 10, 2009 #1
    1. The problem statement, all variables and given/known data
    Z = (Z1, Z2, ... Zd) is a d-dimensional normal variable with distribution N(0, E).

    Let A be invertible matrix such that AA' = E. (E = sigma = covariance matrix).

    Find the distribution of Y = (A^-1)*Z.

    3. The attempt at a solution

    I'm pretty sure the solution is normal, but what would be its mean and variance?
     
  2. jcsd
  3. Feb 10, 2009 #2

    D H

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    Staff Emeritus
    Science Advisor

    A couple of hints.

    1. What are the definitions of the mean and variance (you omitted the Relevant equations)? Hint: They involve integrals.

    2. The matrix A, and thus its inverse, is a constant. Hint: You can take constants outside of the integral.
     
  4. Feb 11, 2009 #3

    statdad

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    Homework Helper

    You don't need integration for this if you know how the mean vector and covariance matrix for multivariate distributions work.

    If [tex] Y [/tex] is a random vector with mean vector [tex] \mu [/tex], the mean of [tex] A Y [/tex] is

    [tex]
    E(A Y)
    [/tex]

    How can you simplify that? (This may be what the other poster meant by "integration" - if so, I apologize)

    The covariance matrix of [tex] AY [/tex] can also be easily simplified. (Hint: this is where you'll use your fact about [tex] A [/tex]
     
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