3.13.2 Example
Consider joint-normal random vector X ~ N3(μ,Σ) with
[3.167]
Let Y be a quadratic polynomial of X:
[3.168]
where
[3.169]
[3.170]
[3.171]
We wish to express Y as a linear polynomial of independent chi-squared and normal random variables. To do so, we construct the Cholesky matrix z of Σ,
[3.172]
and a matrix u with rows equal to orthonormal eigenvectors of 
:
[3.173]
We define the change of variables 
~ N3(0,I) for X
[3.174]
and obtain
[3.175]
where
[3.176]
[3.177]
[3.178]
Multiplying [3.175] out:
[3.179]
We complete the squares for terms involving 
to obtain
[3.180]
We have expressed Y as a linear polynomial of three independent random variables:
~ χ2(1,0),
~ χ2(1,4),
~ N(0,1).