# Noncentral chi-square distribution

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 Probability density function Cumulative distribution function Parameters $k > 0\,$ degrees of freedom $\lambda > 0\,$ non-centrality parameter Support $x \in [0; +\infty)\,$ pdf $\frac{1}{2}e^{-(x+\lambda)/2}\left (\frac{x}{\lambda} \right)^{k/4-1/2} I_{k/2-1}(\sqrt{\lambda x})$ cdf :$\sum_{j=0}^\infty e^{-\lambda/2} \frac{(\lambda/2)^j}{j!} \frac{\gamma(j+k/2,x/2)}{\Gamma(j+k/2)}\,$ Mean $k+\lambda\,$ Median Mode Variance $2(k+2\lambda)\,$ Skewness $\frac{2^{3/2}(k+3\lambda)}{(k+2\lambda)^{3/2}}$ Kurtosis $\frac{12(k+4\lambda)}{(k+2\lambda)^2}$ Entropy mgf $\frac{\exp\left(\frac{ \lambda t}{1-2t }\right)}{(1-2 t)^{k/2}}$ Char. func. $\frac{\exp\left(\frac{i\lambda t}{1-2it}\right)}{(1-2it)^{k/2}}$

In probability theory and statistics, the noncentral chi-square or noncentral $\chi^2$ distribution is a generalization of the chi-square distribution. If $X_i$ are k independent, normally distributed random variables with means $\mu_i$ and variances $\sigma_i^2$, then the random variable

$\sum_1^k \left(\frac{X_i}{\sigma_i}\right)^2$

is distributed according to the noncentral chi-square distribution. The noncentral chi-square distribution has two parameters: $k$ which specifies the number of degrees of freedom (i.e. the number of $X_i$), and $\lambda$ which is related to the mean of the random variables $X_i$ by:

$\lambda=\sum_1^k \left(\frac{\mu_i}{\sigma_i}\right)^2$.

Note that some references define $\lambda$ as one half of the above sum.

## Properties Edit

The noncentral chi-square distribution is equivalent to a (central) chi-square distribution with $k + 2P$ degrees of freedom, where $P$ is a Poisson random variable with parameter $\lambda/2$. Thus, the probability distribution function is given by

$f_X(x; k,\lambda) = \sum_{i=0}^\infty \frac{e^{-\lambda/2} (\lambda/2)^i}{i!} f_{Y_{k+2i}}(x),$

where $Y_q$ is distributed as chi-square with $q$ degrees of freedom.

Alternatively, the pdf can be written as

$f_X(x;k,\lambda)=\frac{1}{2} e^{-(x+\lambda)/2} \left (\frac{x}{\lambda}\right)^{k/4-1/2} I_{k/2-1}(\sqrt{\lambda x})$

where $I_\nu(z)$ is a modified Bessel function of the first kind given by

$I_a(y) := (y/2)^a \sum_{j=0}^\infty \frac{ (y^2/4)^j}{j! \Gamma(a+j+1)}$

The moment generating function is given by:

$M(t;k,\lambda)=\frac{\exp\left(\frac{ \lambda t}{1-2t }\right)}{(1-2 t)^{k/2}}$

The first few raw moments are:

$\mu^'_1=k+\lambda$
$\mu^'_2=(k+\lambda)^2 + 2(k + 2\lambda)$
$\mu^'_3=(k+\lambda)^3 + 6(k+\lambda)(k+2\lambda)+8(k+3\lambda)$
$\mu^'_4=(k+\lambda)^4+12(k+\lambda)^2(k+2\lambda)+4(11k^2+44k\lambda+36\lambda^2)+48(k+4\lambda)$

The first few central moments are:

$\mu_2=2(k+2\lambda)\,$
$\mu_3=8(k+3\lambda)\,$
$\mu_4=12(k+2\lambda)^2+48(k+4\lambda)\,$

The nth cumulant is :

$K_n=2^{n-1}(n-1)!(k+n\lambda)\,$

Hence

$\mu^'_n = 2^{n-1}(n-1)!(k+n\lambda)+\sum_{j=1}^{n-1} \frac{(n-1)!2^{j-1}}{(n-j)!}(k+j\lambda )\mu^'_{n-j}$

Again using the relation between the central and noncentral chi-square distributions, the cumulative distribution function (cdf) can be written as

$P(x; k, \lambda ) = \sum_{j=0}^\infty e^{-\lambda/2} \frac{(\lambda/2)^j}{j!} Q(x; k+2j)$

where $Q(x; k)$ is the cumulative density of the central chi-squared distribution which is given by

$Q(x;k)=\frac{\gamma(k/2,x/2)}{\Gamma(k/2)}\,$

where $\gamma(k,z)$ is the lower incomplete Gamma function.

## Derivation of the pdfEdit

The derivation of the probability density function is most easily done by performing the following steps:

1. Start with the joint PDF of two independent non-zero mean Gaussian distributions, $X$ and $Y$.
2. Convert the joint density $f(X, Y)$ to polar: $f(R, A)$ where $R^2 = (X^2+Y^2)$, $tan(A) = Y/X$.
3. Integrate over the angular variable $A$.
4. Convert from R to r where $r^2 = R$. This will yield a series expansion in r one factor of which matches the modified Bessel function $I_0$.
5. Take the Laplace (Fourier) transform term-by-term and the special case K = 2 and the MGF will result.
6. For the general case, take the K = 2 MGF and raise it to the $K/2$ power.
7. The final trick to hide the K-dependence in the numerator of the MGF is to note that $\lambda$ is a function of K; that is,
$\lambda_2=\sum_1^2 \left(\frac{\mu_i}{\sigma_i}\right)^2$
$\lambda_K=\sum_1^k \left(\frac{\mu_i}{\sigma_i}\right)^2 = \lambda$
and therefore, $\lambda$ is not explicitly a function of K in the above table.

## Related distributionsEdit

• If $X$ is chi-square distributed $X \sim \chi_k^2$ then $X^2$ is also non-central chi-square distributed: $X^2 \sim NC\chi^2_k(0)$
• If $J \sim Poisson(\lambda)$, then $\chi_k^2(\lambda) \sim \chi_{k+2J}^2$
Various chi and chi-square distributions
Name Statistic
chi-square distribution $\sum_1^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2$
noncentral chi-square distribution $\sum_1^k \left(\frac{X_i}{\sigma_i}\right)^2$
chi distribution $\sqrt{\sum_1^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2}$
noncentral chi distribution $\sqrt{\sum_1^k \left(\frac{X_i}{\sigma_i}\right)^2}$
Probability distributions [[[:Template:Tnavbar-plain-nodiv]]]
Univariate Multivariate
Discrete: BernoullibinomialBoltzmanncompound PoissondegeneratedegreeGauss-Kuzmingeometrichypergeometriclogarithmicnegative binomialparabolic fractalPoissonRademacherSkellamuniformYule-SimonzetaZipfZipf-Mandelbrot Ewensmultinomial
Continuous: BetaBeta primeCauchychi-squareDirac delta functionErlangexponentialexponential powerFfadingFisher's zFisher-TippettGammageneralized extreme valuegeneralized hyperbolicgeneralized inverse GaussianHotelling's T-squarehyperbolic secanthyper-exponentialhypoexponentialinverse chi-squareinverse gaussianinverse gammaKumaraswamyLandauLaplaceLévyLévy skew alpha-stablelogisticlog-normalMaxwell-BoltzmannMaxwell speednormal (Gaussian)ParetoPearsonpolarraised cosineRayleighrelativistic Breit-WignerRiceStudent's ttriangulartype-1 Gumbeltype-2 GumbeluniformVoigtvon MisesWeibullWigner semicircle DirichletKentmatrix normalmultivariate normalvon Mises-FisherWigner quasiWishart
Miscellaneous: Cantorconditionalexponential familyinfinitely divisiblelocation-scale familymarginalmaximum entropy phase-typeposterior priorquasisampling
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## ReferencesEdit

• Abramowitz, M. and Stegun, I.A. (1972), Handbook of Mathematical Functions, Dover. Section 26.4.25.
• Johnson, N. L. and Kotz, S., (1970), Continuous Univariate Distributions, vol. 2, Houghton-Mifflin.