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[parent] non-central chi-squared random variable (Definition)

Let $ X_1,\ldots,X_n$ be iid random variables, each with the distribution $ N(0,1)$, the standard normal distribution. Define new iid random variables $ Y_i$, $ i=1,\ldots,n$ by

$\displaystyle Y_i=X_i+c_i,$
where $ c_i\in\mathbb{R}$. Then the random variable $ Z$, 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 $ n$ is the degrees of freedom of the random variable, and $ c$, called the non-centrality parameter, is the sum of squares of the $ c_i$'s, or the last term of the rightmost expression above.

Remarks

  1. $ \chi^2(n,c)$ has mean $ n+c$ and variance $ 2n+4c$.
  2. $ \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 $ c=0$.
  3. (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)$.
  4. If $ \boldsymbol{x}$ is a multivariate normally distributed $ n$-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 $ k$ = rank of $ V<n$. 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 $ k$ degrees of freedom with non-centrality parameter $ c=\boldsymbol{\mu^{\operatorname{T}}V^{-}\mu}$.



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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
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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 MSC60E05 (Probability theory and stochastic processes :: Distribution theory :: Distributions: general theory)
 62E99 (Statistics :: Distribution theory :: Miscellaneous)

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