Products of Random Variables

Abstract

Let ${\rm X}^{\left( 1 \right)} \in {\rm N}_n \left( {0,\sigma _1^2 } \right)$ and ${\rm X}^{\left( 2 \right)} \in {\rm N}_n \left( {0,\sigma _2^2 } \right)$ be independent Gaussian vectors. Then, the inner product of these vectors, namely,

$$ X = \left( {X^{\left( 1 \right)} ,X^{\left( 2 \right)} } \right) = \sum\limits_{k = 1}^n {X_k^{\left( 1 \right)} X_k^{\left( 2 \right)} } $$

(6.1)

has the following statistical properties.

A large number of the PDF and statistical moment results in this section come from Ref. 5.