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 
~ N3(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).