non-central chi-squared random variable

Let X1,…,Xn be IID random variablesMathworldPlanetmath, each with the standard normal distributionMathworldPlanetmath. Then, for any πβˆˆβ„n, the random variable X defined by


is called a non-central chi-squared random variable. Its distributionPlanetmathPlanetmath depends only on the number of degrees of freedom n and non-centrality parameter λ≑βˆ₯𝝁βˆ₯. This is denoted by Ο‡2⁒(n,Ξ») and has moment generating function

MX⁑(t)≑𝔼⁒[et⁒X]=(1-2⁒t)-n2⁒exp⁑(λ⁒t1-2⁒t), (1)

which is defined for all tβˆˆβ„‚ with real part less than 1/2. More generally, for any n,Ξ»β‰₯0, not necessarily integers, a random variable has the non-central chi-squared distribution, Ο‡2⁒(n,Ξ»), if its moment generating function is given by (1).

A non-central chi-squared random variable for any n,Ξ»β‰₯0 can be constructed as follows. Let Y be a (central) chi-squared variable with degree n, Z1,Z2,… be standard normals, and N have the Poisson⁒(Ξ»/2) distribution. If these are all independentPlanetmathPlanetmath then


has the Ο‡2⁒(n,Ξ») distribution. Correspondingly, the probability density function for X is

fX⁒(x)=βˆ‘k=0∞λk2k⁒k!⁒e-Ξ»/2⁒fn+2⁒k⁒(x), (2)

where x>0 and fk is the probability density of the Ο‡(k)2 distribution. Alternatively, this can be expressed as


where IΞ½ is a modified Bessel function of the first kind,

FigureΒ 1: DensitiesPlanetmathPlanetmath of the non-central chi-squared distribution Ο‡2⁒(n,Ξ»).


  1. 1.

    Ο‡2⁒(n,Ξ») has mean n+Ξ» and varianceMathworldPlanetmath 2⁒n+4⁒λ.

  2. 2.

    Ο‡2⁒(n,0)=Ο‡(n)2. The (central) chi-squared random variable is a special case of the non-central chi-squared random variable, when the non-centrality parameter Ξ»=0.

  3. 3.

    (The reproductive property of chi-squared distributions). If Z1,…,Zm are non-central chi-squared random variables such that each ZiβˆΌΟ‡2⁒(ni,Ξ»i), then their total Z=βˆ‘Zi is also a non-central chi-squared random variable with distribution Ο‡2⁒(βˆ‘ni,βˆ‘Ξ»i).

  4. 4.

    If n>0 then the Ο‡2⁒(n,Ξ») distribution is restricted to the domain (0,∞) with probability density function (2). On the other hand, if n=0, then there is also an atom at 0,

  5. 5.

    If 𝒙 is a multivariate normally distributed n-dimensional random vector with distribution 𝑡⁒(𝝁,𝑽) where 𝝁 is the mean vector and 𝑽 is the nΓ—n covariance matrixMathworldPlanetmath. Suppose that 𝑽 is singularPlanetmathPlanetmath, with k = rank of V<n. Then 𝒙𝐓⁒𝑽-⁒𝒙 is a non-central chi-squared random variable, where 𝑽- is a generalized inverse of 𝑽. Its distribution has k degrees of freedom with non-centrality parameter Ξ»=𝝁𝐓⁒𝑽-⁒𝝁.

Title non-central chi-squared random variable
Canonical name NoncentralChisquaredRandomVariable
Date of creation 2013-03-22 14:56:16
Last modified on 2013-03-22 14:56:16
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 11
Author CWoo (3771)
Entry type Definition
Classification msc 62E99
Classification msc 60E05
Synonym non-central chi-squared distribution
Defines non-centrality parameter