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Noncentral chi distribution
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<h2 id="definition">Definition</h2> <p>If X i {\displaystyle X_{i}} are <i>k</i> independent, <a href="/facts/Normal_distribution/UapjjPyQ">normally distributed</a> random variables with means μ i {\displaystyle \mu _{i}} and variances σ i 2 {\displaystyle \sigma _{i}^{2}} , then the statistic </p> Z = ∑ i = 1 k ( X i σ i ) 2 {\displaystyle Z={\sqrt {\sum _{i=1}^{k}\left({\frac {X_{i}}{\sigma _{i}}}\right)^{2}}}} <p>is distributed according to the noncentral chi distribution. The noncentral chi distribution has two parameters: k {\displaystyle k} which specifies the number of <a href="/facts/Degrees_of_freedom_(statistics)/fvwKCU86">degrees of freedom</a> (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: </p> λ = ∑ i = 1 k ( μ i σ i ) 2 {\displaystyle \lambda ={\sqrt {\sum _{i=1}^{k}\left({\frac {\mu _{i}}{\sigma _{i}}}\right)^{2}}}} <h2 id="properties">Properties</h2> <h3>Probability density function</h3> <p>The probability density function (pdf) is </p> 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)} <p>where I ν ( z ) {\displaystyle I_{\nu }(z)} is a modified <a href="/facts/Bessel_function/hSyvFPqC">Bessel function</a> of the first kind. </p> <h3>Raw moments</h3> <p>The first few raw <a href="/facts/Moment_(mathematics)/GL7vJ1nb">moments</a> are: </p> 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}} μ 3 ′ = 3 π 2 L 3 / 2 ( k / 2 − 1 ) ( − λ 2 2 ) {\displaystyle \mu _{3}^{'}=3{\sqrt {\frac {\pi }{2}}}L_{3/2}^{(k/2-1)}\left({\frac {-\lambda ^{2}}{2}}\right)} μ 4 ′ = ( k + λ 2 ) 2 + 2 ( k + 2 λ 2 ) {\displaystyle \mu _{4}^{'}=(k+\lambda ^{2})^{2}+2(k+2\lambda ^{2})} <p>where L n ( a ) ( z ) {\displaystyle L_{n}^{(a)}(z)} is a <a href="/facts/Laguerre_function/7RM3ypWl">Laguerre function</a>. Note that the 2 n {\displaystyle n} th moment is the same as the n {\displaystyle n} th moment of the <a href="/facts/Noncentral_chi-squared_distribution/saAcjyRh">noncentral chi-squared distribution</a> with λ {\displaystyle \lambda } being replaced by λ 2 {\displaystyle \lambda ^{2}} . </p> <h2 id="bivariate-non-central-chi-distribution">Bivariate non-central chi distribution</h2> <p>Let X j = ( X 1 j , X 2 j ) , j = 1 , 2 , … n {\displaystyle X_{j}=(X_{1j},X_{2j}),j=1,2,\dots n} , be a set of <i>n</i> independent and identically distributed <a href="/facts/Bivariate_normal/2Xfegqz2">bivariate normal</a> random vectors with marginal distributions N ( μ i , σ i 2 ) , i = 1 , 2 {\displaystyle N(\mu _{i},\sigma _{i}^{2}),i=1,2} , correlation ρ {\displaystyle \rho } , and <a href="/facts/Mean_vector/swcEd4Pg">mean vector</a> and <a href="/facts/Covariance_matrix/kLhgjHC6">covariance matrix</a> </p> E ( X j ) = μ = ( μ 1 , μ 2 ) T , Σ = [ σ 11 σ 12 σ 21 σ 22 ] = [ σ 1 2 ρ σ 1 σ 2 ρ σ 1 σ 2 σ 2 2 ] , {\displaystyle 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}},} <p>with Σ {\displaystyle \Sigma } <a href="/facts/Positive_definite_matrix/Hbr6AuPS">positive definite</a>. Define </p> U = [ ∑ j = 1 n X 1 j 2 σ 1 2 ] 1 / 2 , V = [ ∑ j = 1 n X 2 j 2 σ 2 2 ] 1 / 2 . {\displaystyle 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}.} <p>Then the joint distribution of <i>U</i>, <i>V</i> is central or noncentral bivariate chi distribution with <i>n</i> <a href="/facts/Degrees_of_freedom_(statistics)/fvwKCU86">degrees of freedom</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> If either or both μ 1 ≠ 0 {\displaystyle \mu _{1}\neq 0} or μ 2 ≠ 0 {\displaystyle \mu _{2}\neq 0} the distribution is a noncentral bivariate chi distribution. </p> <h2 id="related-distributions">Related distributions</h2> <ul><li>If X {\displaystyle X} is a random variable with the non-central chi distribution, the random variable X 2 {\displaystyle X^{2}} will have the <a href="/facts/Noncentral_chi-squared_distribution/saAcjyRh">noncentral chi-squared distribution</a>. Other related distributions may be seen there.</li> <li>If X {\displaystyle X} is <a href="/facts/Chi_distribution/bV4MEanU">chi</a> distributed: X ∼ χ k {\displaystyle X\sim \chi _{k}} then X {\displaystyle X} is also non-central chi distributed: X ∼ N C χ k ( 0 ) {\displaystyle X\sim NC\chi _{k}(0)} . In other words, the <a href="/facts/Chi_distribution/bV4MEanU">chi distribution</a> is a special case of the non-central chi distribution (i.e., with a non-centrality parameter of zero).</li> <li>A noncentral chi distribution with 2 degrees of freedom is equivalent to a <a href="/facts/Rice_distribution/E9tY1XNJ">Rice distribution</a> with σ = 1 {\displaystyle \sigma =1} .</li> <li>If <i>X</i> follows a noncentral chi distribution with 1 degree of freedom and noncentrality parameter λ, then σ<i>X</i> follows a <a href="/facts/Folded_normal_distribution/6zdCbLYu">folded normal distribution</a> whose parameters are equal to σλ and σ2 for any value of σ.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>J. H. Park (1961). "Moments of the Generalized Rayleigh Distribution". Quarterly of Applied Mathematics. 19 (1): 45–49. doi:10.1090/qam/119222. JSTOR 43634840. <a href="https://doi.org/10.1090%2Fqam%2F119222" target="_blank">https://doi.org/10.1090%2Fqam%2F119222</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Marakatha Krishnan (1967). "The Noncentral Bivariate Chi Distribution". SIAM Review. 9 (4): 708–714. Bibcode:1967SIAMR...9..708K. doi:10.1137/1009111. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>P. R. Krishnaiah, P. Hagis, Jr. and L. Steinberg (1963). "A note on the bivariate chi distribution". SIAM Review. 5 (2): 140–144. Bibcode:1963SIAMR...5..140K. doi:10.1137/1005034. JSTOR 2027477.{{cite journal}}: CS1 maint: multiple names: authors list (link) <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> </ol>