# Noncentral chi-square distribution

Parameters Probability density function325px Cumulative distribution function325px $k>0\,$ degrees of freedom $\lambda >0\,$ non-centrality parameter $x\in [0;+\infty )\,$ ${\frac {1}{2}}e^{-(x+\lambda )/2}\left({\frac {x}{\lambda }}\right)^{k/4-1/2}I_{k/2-1}({\sqrt {\lambda x}})$ :$\sum _{j=0}^{\infty }e^{-\lambda /2}{\frac {(\lambda /2)^{j}}{j!}}{\frac {\gamma (j+k/2,x/2)}{\Gamma (j+k/2)}}\,$ $k+\lambda \,$ $2(k+2\lambda )\,$ ${\frac {2^{3/2}(k+3\lambda )}{(k+2\lambda )^{3/2}}}$ ${\frac {12(k+4\lambda )}{(k+2\lambda )^{2}}}$ ${\frac {\exp \left({\frac {\lambda t}{1-2t}}\right)}{(1-2t)^{k/2}}}$ ${\frac {\exp \left({\frac {i\lambda t}{1-2it}}\right)}{(1-2it)^{k/2}}}$ 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

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 density 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

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

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 distributions

• If $Z$ is chi-square distributed $Z\sim \chi _{k}^{2}$ then $Z$ is also non-central chi-square distributed: $Z\sim {\chi '}_{k}^{2}(0)$ • If $J\sim Poisson(\lambda /2)$ , 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 ${\sqrt {\sum _{1}^{k}\left({\frac {X_{i}}{\sigma _{i}}}\right)^{2}}}$  