# non-central chi-squared random variable

Let $X_{1},\ldots,X_{n}$ be IID random variables, each with the standard normal distribution. Then, for any $\boldsymbol{\mu}\in\mathbb{R}^{n}$, the random variable $X$ defined by

 $X=\sum_{i=1}^{n}(X_{i}+\mu_{i})^{2}$

is called a non-central chi-squared random variable. Its distribution depends only on the number of degrees of freedom $n$ and non-centrality parameter $\lambda\equiv\|\boldsymbol{\mu}\|$. This is denoted by $\chi^{2}(n,\lambda)$ and has moment generating function

 $\operatorname{M}_{X}(t)\equiv\mathbb{E}\left[e^{tX}\right]=\left(1-2t\right)^{% -\frac{n}{2}}\exp\left(\frac{\lambda t}{1-2t}\right),$ (1)

which is defined for all $t\in\mathbb{C}$ with real part less than $1/2$. More generally, for any $n,\lambda\geq 0$, not necessarily integers, a random variable has the non-central chi-squared distribution, $\chi^{2}(n,\lambda)$, if its moment generating function is given by (1).

A non-central chi-squared random variable for any $n,\lambda\geq 0$ can be constructed as follows. Let $Y$ be a (central) chi-squared variable with degree $n$, $Z_{1},Z_{2},\ldots$ be standard normals, and $N$ have the $\textrm{Poisson}(\lambda/2)$ distribution. If these are all independent then

 $X\equiv Y+\sum_{k=1}^{2N}Z_{k}^{2}.$

has the $\chi^{2}(n,\lambda)$ distribution. Correspondingly, the probability density function for $X$ is

 $f_{X}(x)=\sum_{k=0}^{\infty}\frac{\lambda^{k}}{2^{k}k!}e^{-\lambda/2}f_{n+2k}(% x),$ (2)

where $x>0$ and $f_{k}$ is the probability density of the $\chi^{2}_{(k)}$ distribution. Alternatively, this can be expressed as

 $f_{X}(x)=\frac{1}{2}e^{-(x+\lambda)/2}(x/\lambda)^{n/4-1/2}I_{n/2-1}\left(% \sqrt{\lambda x}\right).$

where $I_{\nu}$ is a modified Bessel function of the first kind,

 $I_{\nu}(x)=\sum_{k=0}^{\infty}\frac{\left(x/2\right)^{\nu+2k}}{k!\,\Gamma\left% (\nu+k+1\right)}.$

Remarks

1. 1.

$\chi^{2}(n,\lambda)$ has mean $n+\lambda$ and variance $2n+4\lambda$.

2. 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 $\lambda=0$.

3. 3.

(The reproductive property of chi-squared distributions). If $Z_{1},\ldots,Z_{m}$ are non-central chi-squared random variables such that each $Z_{i}\sim\chi^{2}(n_{i},\lambda_{i})$, then their total $Z=\sum Z_{i}$ is also a non-central chi-squared random variable with distribution $\chi^{2}(\sum n_{i},\sum\lambda_{i})$.

4. 4.

If $n>0$ then the $\chi^{2}(n,\lambda)$ distribution is restricted to the domain $(0,\infty)$ with probability density function (2). On the other hand, if $n=0$, then there is also an atom at $0$,

 $\mathbb{P}(X=0)=\lim_{t\rightarrow-\infty}\operatorname{M}_{X}(t)=e^{-\lambda/% 2}.$
5. 5.

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. 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 $\lambda=\boldsymbol{\mu^{\operatorname{T}}V^{-}\mu}$.

Title non-central chi-squared random variable NoncentralChisquaredRandomVariable 2013-03-22 14:56:16 2013-03-22 14:56:16 CWoo (3771) CWoo (3771) 11 CWoo (3771) Definition msc 62E99 msc 60E05 non-central chi-squared distribution non-centrality parameter