# Exercise Solution 3.42

Construct the Cholesky matrix ** z** of

**Σ**,

[s1]

and a matrix ** u** with rows equal to orthonormal eigenvectors of :

[s2]

We define the change of variables ~ *N*_{3}(**0**,* I*) for

*X*[s3]

and obtain

[s4]

where

[s5]

[s6]

[s7]

Multiplying [s4] out:

[s8]

We complete the squares for terms involving and to obtain

[s9]

We have expressed *Y* as a linear polynomial of three independent random variables:

- ,
- ,
- ~
*N*(0,1).