Multivariate Normal Distribution

  • Thread starter gatorain
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Homework Statement


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.

The Attempt at a Solution



I'm pretty sure the solution is normal, but what would be its mean and variance?
 

Answers and Replies

  • #2
D H
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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.
 
  • #3
statdad
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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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