# Noncentral chi distribution

{{ safesubst:#invoke:Unsubst||$N=Refimprove |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} Template:Probability distribution

$Z={\sqrt {\sum _{i=1}^{k}\left({\frac {X_{i}}{\sigma _{i}}}\right)^{2}}}$ is distributed according to the noncentral chi distribution. The noncentral chi 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:

$\lambda ={\sqrt {\sum _{i=1}^{k}\left({\frac {\mu _{i}}{\sigma _{i}}}\right)^{2}}}$ ## Properties

### Probability density function

The probability density function (pdf) is

$f(x;k,\lambda )={\frac {e^{-(x^{2}+\lambda ^{2})/2}x^{k}\lambda }{(\lambda x)^{k/2}}}I_{k/2-1}(\lambda x)$ ### Raw moments

The first few raw moments are:

$\mu _{1}^{'}={\sqrt {\frac {\pi }{2}}}L_{1/2}^{(k/2-1)}\left({\frac {-\lambda ^{2}}{2}}\right)$ $\mu _{2}^{'}=k+\lambda ^{2}$ $\mu _{3}^{'}=3{\sqrt {\frac {\pi }{2}}}L_{3/2}^{(k/2-1)}\left({\frac {-\lambda ^{2}}{2}}\right)$ $\mu _{4}^{'}=(k+\lambda ^{2})^{2}+2(k+2\lambda ^{2})$ ### Differential equation

The pdf of the noncentral chi distribution is a solution to the following differential equation:

$\left\{{\begin{array}{l}x^{2}f''(x)+\left(-kx+2x^{3}+x\right)f'(x)+f(x)\left(-x^{2}\left(\lambda ^{2}+k-2\right)+k+x^{4}-1\right)=0,\\f(1)=e^{-{\frac {\lambda ^{2}}{2}}-{\frac {1}{2}}}\lambda ^{1-{\frac {k}{2}}}I_{\frac {k-2}{2}}(\lambda ),f'(1)=e^{-{\frac {\lambda ^{2}}{2}}-{\frac {1}{2}}}\lambda ^{2-{\frac {k}{2}}}I_{\frac {k-4}{2}}(\lambda )\end{array}}\right\}$ ## Bivariate non-central chi distribution

$E(X_{j})=\mu =(\mu _{1},\mu _{2})^{T},\qquad \Sigma ={\begin{bmatrix}\sigma _{11}&\sigma _{12}\\\sigma _{21}&\sigma _{22}\end{bmatrix}}={\begin{bmatrix}\sigma _{1}^{2}&\rho \sigma _{1}\sigma _{2}\\\rho \sigma _{1}\sigma _{2}&\sigma _{2}^{2}\end{bmatrix}},$ $U=\left[\sum _{j=1}^{n}{\frac {X_{1j}^{2}}{\sigma _{1}^{2}}}\right]^{1/2},\qquad V=\left[\sum _{j=1}^{n}{\frac {X_{2j}^{2}}{\sigma _{2}^{2}}}\right]^{1/2}.$ Then the joint distribution of U, V is central or noncentral bivariate chi distribution with n degrees of freedom. If either or both $\mu _{1}\neq 0$ or $\mu _{2}\neq 0$ the distribution is a noncentral bivariate chi distribution.

## Related distributions

• If X follows a noncentral chi distribution with 1 degree of freedom and noncentrality parameter λ, then σX follows a folded normal distribution whose parameters are equal to σλ and σ2 for any value of σ.

## Applications

The Euclidean norm of a multivariate normally distributed random vector follows a noncentral chi distribution.