3.13.2 Example: Simplified Representation of a Quadratic Polynomial of a Multivariate Normal Random Vector

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).