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| {{Refimprove|date=December 2012}}
| | The name of the writer is Jayson. My working day occupation is an info officer but I've already utilized for an additional one. One of the very very best issues in the world for him is doing ballet and he'll be beginning some thing else alongside with it. For a while I've been in Mississippi but now I'm contemplating other options.<br><br>Here is my web page psychic readings ([http://kard.dk/?p=24252 http://kard.dk]) |
| {{Probability distribution|
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| name =Noncentral chi|
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| type =density|
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| pdf_image =|
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| cdf_image =|
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| parameters =<math>k > 0\,</math> degrees of freedom<br />
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| <math>\lambda > 0\,</math>|
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| support =<math>x \in [0; +\infty)\,</math>|
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| pdf =<math>\frac{e^{-(x^2+\lambda^2)/2}x^k\lambda}
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| {(\lambda x)^{k/2}} I_{k/2-1}(\lambda x)</math>|
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| cdf =|
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| mean =<math>\sqrt{\frac{\pi}{2}}L_{1/2}^{(k/2-1)}\left(\frac{-\lambda^2}{2}\right)\,</math>|
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| median =|
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| mode =|
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| variance =<math>k+\lambda^2-\mu^2\,</math>|
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| skewness =|
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| kurtosis =|
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| entropy =|
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| mgf =|
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| char =
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| }}
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| In [[probability theory]] and [[statistics]], the '''noncentral chi distribution''' is a generalization of the [[chi distribution]]. If <math>X_i</math> are ''k'' independent, [[normal distribution|normally distributed]] random variables with means <math>\mu_i</math> and variances <math>\sigma_i^2</math>, then the statistic
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| :<math>Z = \sqrt{\sum_1^k \left(\frac{X_i}{\sigma_i}\right)^2}</math>
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| is distributed according to the noncentral chi distribution. The noncentral chi distribution has two parameters: <math>k</math> which specifies the number of [[Degrees of freedom (statistics)|degrees of freedom]] (i.e. the number of <math>X_i</math>), and <math>\lambda</math> which is related to the mean of the random variables <math>X_i</math> by:
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| :<math>\lambda=\sqrt{\sum_1^k \left(\frac{\mu_i}{\sigma_i}\right)^2}</math>
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| ==Properties==
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| The probability density function is
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| :<math>f(x;k,\lambda)=\frac{e^{-(x^2+\lambda^2)/2}x^k\lambda}
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| {(\lambda x)^{k/2}} I_{k/2-1}(\lambda x)</math>
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| where <math>I_\nu(z)</math> is a modified [[Bessel function]] of the first kind.
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| The first few raw [[moment (mathematics)|moments]] are:
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| :<math>\mu^'_1=\sqrt{\frac{\pi}{2}}L_{1/2}^{(k/2-1)}\left(\frac{-\lambda^2}{2}\right)</math>
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| :<math>\mu^'_2=k+\lambda^2</math>
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| :<math>\mu^'_3=3\sqrt{\frac{\pi}{2}}L_{3/2}^{(k/2-1)}\left(\frac{-\lambda^2}{2}\right)</math>
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| :<math>\mu^'_4=(k+\lambda^2)^2+2(k+2\lambda^2)</math>
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| where <math>L_n^{(a)}(z)</math> is the [[Laguerre polynomial#Generalized Laguerre polynomials|generalized Laguerre polynomial]]. Note that the 2''n''th moment is the same as the ''n''th moment of the [[noncentral chi-squared distribution]] with <math>\lambda</math> being replaced by <math>\lambda^2</math>.
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| ==Bivariate non-central chi distribution==
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| Let <math>X_j = (X_{1j}, X_{2j}), j = 1, 2, \dots n</math>, be a set of ''n'' independent and identically distributed [[bivariate normal]] random vectors with marginal distributions <math>N(\mu_i,\sigma_i^2), i=1,2</math>, correlation <math>\rho</math>, and [[mean vector]] and [[covariance matrix]]
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| :<math> E(X_j)= \mu=(\mu_1, \mu_2)^T, \qquad
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| \Sigma =
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| \begin{bmatrix}
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| \sigma_{11} & \sigma_{12} \\
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| \sigma_{21} & \sigma_{22}
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| \end{bmatrix}
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| = \begin{bmatrix}
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| \sigma_1^2 & \rho \sigma_1 \sigma_2 \\
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| \rho \sigma_1 \sigma_2 & \sigma_2^2
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| \end{bmatrix},
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| </math>
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| with <math>\Sigma</math> [[positive definite matrix|positive definite]]. Define
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| :<math> | |
| U = \left[ \sum_{j=1}^n \frac{X_{1j}^2}{\sigma_1^2} \right]^{1/2}, \qquad
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| V = \left[ \sum_{j=1}^n \frac{X_{2j}^2}{\sigma_2^2} \right]^{1/2}.
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| </math>
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| Then the joint distribution of ''U'', ''V'' is central or noncentral bivariate chi distribution with ''n'' [[Degrees of freedom (statistics)|degrees of freedom]].<ref>{{cite journal|author=Marakatha Krishnan|title= The Noncentral Bivariate Chi Distribution|journal= SIAM Review |volume=9|issue=4|year=1967|pages=708–714|doi=10.1137/1009111}}</ref><ref>{{cite journal|author=P. R. Krishnaiah, P. Hagis, Jr. and L. Steinberg |title= A note on the bivariate chi distribution|journal=SIAM Review|volume= 5|year=1963|pages= 140–144|jstor=2027477}}</ref>
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| If either or both <math>\mu_1 \neq 0</math> or <math>\mu_2 \neq 0</math> the distribution is a noncentral bivariate chi distribution.
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| ==Related distributions==
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| *If <math>X</math> is a random variable with the non-central chi distribution, the random variable <math>X^2</math> will have the [[noncentral chi-squared distribution]]. Other related distributions may be seen there.
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| *If <math>X</math> is [[chi distribution|chi]] distributed: <math>X \sim \chi_k</math> then <math>X</math> is also non-central chi distributed: <math>X \sim NC\chi_k(0)</math>. In other words, the [[chi distribution]] is a special case of the non-central chi distribution (i.e., with a non-centrality parameter of zero).
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| *A noncentral chi distribution with 2 degrees of freedom is equivalent to a [[Rice distribution]] with <math>\sigma=1</math>.
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| *If ''X'' follows a noncentral chi distribution with 1 degree of freedom and noncentrality parameter λ, then σ''X'' follows a [[folded normal distribution]] whose parameters are equal to σλ and σ<sup>2</sup> for any value of σ.
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| ==Applications==
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| The [[Euclidean norm]] of a [[normal random vector|multivariate normally distributed random vector]] follows a noncentral chi distribution.
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| ==References==
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| {{Reflist}}
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| {{ProbDistributions|continuous-semi-infinite}}
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| {{DEFAULTSORT:Noncentral Chi Distribution}}
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| [[Category:Continuous distributions]]
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| [[Category:Probability distributions]]
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Here is my web page psychic readings (http://kard.dk)