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).