# 3.13 Quadratic Polynomials of Joint-Normal Random Vectors

Formulas [3.27] and [3.28] provide general expressions for the mean and variance of a linear polynomial of a random vector. What about the mean and variance of a quadratic polynomial of a random vector? Unfortunately, no general formulas exist. However, if we restrict our attention to quadratic polynomials of a joint-normal random vector, there are expressions for the mean, variance, and all moments. To obtain these, we generalize an earlier result regarding chi-squared distributions.

###### 3.13.1 Simplified Representation

In Section 3.10.3, we saw that the specific form [3.112] of quadratic polynomial of a joint standard normal random vector has a chi-squared distribution. Generalizing this, we shall demonstrate that *any* quadratic polynomial of *any* joint-normal random vector can be expressed as a linear polynomial of independent chi-squared and normal random variables. Specifically, let ** X** ~

*N*(

_{m}**μ**,

**Σ**) with

**Σ**positive definite.11 Define

*Y*as a quadratic polynomial of

**:**

*X*[3.147]

Let ** z** be the Cholesky matrix of

**Σ**, and define

**as a square matrix whose rows comprise orthonormal eigenvectors of . By construction,**

*u***is orthogonal:**

*u*

*u*^{–1}=

*′. Define the change of variables*

**u**[3.148]

Then, by [3.30] and [3.31], is joint-normal with mean vector

[3.149]

[3.150]

[3.151]

[3.152]

and covariance matrix

[3.153]

[3.154]

[3.155]

[3.156]

Accordingly, ~ *N _{m}*(

**0**,

*). Applying our change of variables [3.148]:*

**I**[3.157]

[3.158]

[3.159]

so *Y* has form

[3.160]

where

[3.161]

[3.162]

[3.163]

Recall that we defined ** u** as a matrix whose rows comprise orthonormal eigenvectors of . This means, by the spectral theorem of linear algebra, that the matrix is diagonal with diagonal elements equal to the eigenvalues of Consequently,

*Y*depends upon no cross terms of the form . We can write [3.159] as

[3.164]

and conclude that *Y* depends upon each of the variables in one of four ways:

- No dependence: = 0 and = 0.
- Linear dependence: = 0 and ≠ 0, so
*Y*depends upon a term . - Central quadratic dependence: ≠ 0 and = 0, so
*Y*depends upon a term . - Noncentral quadratic dependence: ≠ 0 and ≠ 0, so
*Y*depends upon a term + .

In the last case, “completing the squares” results in a dependence of the form

[3.165]

*Y* is a linear polynomial of independent random variables, each of which is standard normal, central chi-squared with one degree of freedom, or noncentral chi-squared with one degree of freedom and non centrality parameter .

Since a linear polynomial of independent normal random variables is itself normal, all normal terms can be combined into one. A general expression for *Y* is

[3.166]

where the *Q _{k}* are chi-squared with one degree of freedom, noncentrality parameters are obtainable from [3.165], and

*Q*

_{0}is standard normal. The constants γ

*, β, and α can be calculated directly from the terms , , and .*

_{k}