# Multivariate Normal Distribution

1. Feb 10, 2009

### gatorain

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. Feb 10, 2009

### D H

Staff Emeritus
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. Feb 11, 2009

If $$Y$$ is a random vector with mean vector $$\mu$$, the mean of $$A Y$$ is
$$E(A Y)$$
The covariance matrix of $$AY$$ can also be easily simplified. (Hint: this is where you'll use your fact about $$A$$