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


Let Y be a quadratic polynomial of X:






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,


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


We define the change of variables  ~ N3(0,I) for X


and obtain






Multiplying [3.175] out:


We complete the squares for terms involving  to obtain


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

  •  ~ χ2(1,0),
  •  ~ χ2(1,4),
  •  ~ N(0,1).