PlanetMath: non-central chi-squared random variable

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non-central chi-squared random variable

(Definition)

Let $X_1,\ldots,X_n$ be iid random variables, each with the distribution , the standard normal distribution. Define new iid random variables , $i=1,\ldots,n$ by

$\displaystyle Y_i=X_i+c_i,$

where $c_i\in\mathbb{R}$ . Then the random variable

, defined by

$\displaystyle Z:=\sum_{i=1}^{n}Y_i^2=\sum_{i=1}^{n}X_i^2+2\sum_{i=1}^{n}X_ic_i+\sum_{i=1}^{n}c_i^2$

is called a non-central chi-squared random variable, whose distribution, called the non-central chi-squared distribution, is denoted by $\chi^2(n,c)$ , where

is the degrees of freedom of the random variable, and

, called the non-centrality parameter, is the sum of squares of the

's, or the last term of the rightmost expression above.

Remarks

$\chi^2(n,c)$ has mean and variance .
$\chi^2(n,0)=\chi^2(n)$ . The (central) chi-squared random variable is a special case of the non-central chi-squared random variable, when the non-centrality parameter .
(The reproductive property of chi-squared distributions). If $Z_1,\ldots,Z_m$ are non-central chi-squared random variables such that each $Z_i\sim\chi^2(n_i,c_i)$ , then their total $Z=\sum Z_i$ is also a non-central chi-squared random variable with distribution $\chi^2(\sum n_i, \sum c_i)$ .
If $\boldsymbol{x}$ is a multivariate normally distributed -dimensional random vector with distribution $\boldsymbol{N(\mu,V)}$ where $\boldsymbol{\mu}$ is the mean vector and $\boldsymbol{V}$ is the $n\times n$ covariance matrix. Suppose that $\boldsymbol{V}$ is singular, with = rank of . Then $\boldsymbol{x^{\operatorname{T}}V^{-}x}$ is a non-central chi-squared random variable, where $\boldsymbol{V^{-}}$ is a generalized inverse of $\boldsymbol{V}$ . Its distribution has degrees of freedom with non-centrality parameter $c=\boldsymbol{\mu^{\operatorname{T}}V^{-}\mu}$ .

"non-central chi-squared random variable" is owned by CWoo.

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Other names:

non-central chi-squared distribution

Also defines:

non-centrality parameter

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Cross-references: generalized inverse, rank, singular, covariance matrix, mean vector, random vector, property, chi-squared random variable, variance, mean, expression, term, squares, sum, degrees of freedom, standard normal distribution, distribution, random variables, iid
There are 2 references to this entry.

This is version 2 of non-central chi-squared random variable, born on 2005-01-07, modified 2005-01-07.
Object id is 6628, canonical name is NonCentralChiSquaredRandomVariable.
Accessed 6968 times total.

Classification:

AMS MSC:	60E05 (Probability theory and stochastic processes :: Distribution theory :: Distributions: general theory)
	62E99 (Statistics :: Distribution theory :: Miscellaneous)

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