Multivariate Normal Distribution

[gatorain]

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?

 

[D H]

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.

 

[statdad]

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

If $$Y$$ is a random vector with mean vector $$\mu$$, the mean of $$A Y$$ is

$$E(A Y)$$

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

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