In probability theory and statistics, the noncentral chi distribution is a noncentral generalization of the chi distribution. It is also known as the generalized Rayleigh distribution.
Definition If X i {\displaystyle X_{i}} are k independent, normally distributed random variables with means μ i {\displaystyle \mu _{i}} and variances σ i 2 {\displaystyle \sigma _{i}^{2}} , then the statistic
Z = ∑ i = 1 k ( X i σ i ) 2 {\displaystyle 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 {\displaystyle k} which specifies the number of degrees of freedom (i.e. the number of X i {\displaystyle X_{i}} ), and λ {\displaystyle \lambda } which is related to the mean of the random variables X i {\displaystyle X_{i}} by:
λ = ∑ i = 1 k ( μ i σ i ) 2 {\displaystyle \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 , λ ) = e − ( x 2 + λ 2 ) / 2 x k λ ( λ x ) k / 2 I k / 2 − 1 ( λ x ) {\displaystyle f(x;k,\lambda )={\frac {e^{-(x^{2}+\lambda ^{2})/2}x^{k}\lambda }{(\lambda x)^{k/2}}}I_{k/2-1}(\lambda x)}
where I ν ( z ) {\displaystyle I_{\nu }(z)} is a modified Bessel function of the first kind.
Raw moments The first few raw moments are:
μ 1 ′ = π 2 L 1 / 2 ( k / 2 − 1 ) ( − λ 2 2 ) {\displaystyle \mu _{1}^{'}={\sqrt {\frac {\pi }{2}}}L_{1/2}^{(k/2-1)}\left({\frac {-\lambda ^{2}}{2}}\right)}
μ 2 ′ = k + λ 2 {\displaystyle \mu _{2}^{'}=k+\lambda ^{2}}
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