Editing Chi-squared distribution (section)

== Related distributions ==
{{More citations needed section|date=September 2011}}
* As <math>k\to\infty</math>, <math> (\chi^2_k-k)/\sqrt{2k} ~ \xrightarrow{d}\ N(0,1) \,</math> ([[normal distribution]])
* <math> \chi_k^2 \sim {\chi'}^2_k(0)</math> ([[noncentral chi-squared distribution]] with non-centrality parameter <math> \lambda = 0 </math>)
* If <math>Y \sim \mathrm{F}(\nu_1, \nu_2)</math> then <math>X = \lim_{\nu_2 \to \infty} \nu_1 Y</math> has the chi-squared distribution <math>\chi^2_{\nu_{1}}</math>
:*As a special case, if <math>Y \sim \mathrm{F}(1, \nu_2)\,</math> then <math>X = \lim_{\nu_2 \to \infty} Y\,</math> has the chi-squared distribution <math>\chi^2_{1}</math>
* <math> \|\boldsymbol{N}_{i=1,\ldots,k} (0,1) \|^2 \sim \chi^2_k </math> (The squared [[Norm (mathematics)|norm]] of ''k'' standard normally distributed variables is a chi-squared distribution with ''k'' [[degrees of freedom (statistics)|degrees of freedom]])
* If <math>X \sim \chi^2_\nu\,</math> and <math>c>0 \,</math>, then <math>cX \sim \Gamma(k=\nu/2, \theta=2c)\,</math>. ([[gamma distribution]])
* If <math>X \sim \chi^2_k</math> then <math>\sqrt{X} \sim \chi_k</math> ([[chi distribution]])
* If <math>X \sim \chi^2_2</math>, then <math>X \sim \operatorname{exp}(1/2)</math> is an [[exponential distribution]]. (See [[gamma distribution]] for more.)
* If <math>X \sim \chi^2_{2k}</math>, then <math>X \sim \operatorname{Erlang}(k, 1/2)</math> is an [[Erlang distribution]].
* If <math> X \sim \operatorname{Erlang}(k,\lambda)</math>, then <math> 2\lambda X\sim \chi^2_{2k}</math>
* If <math>X \sim \operatorname{Rayleigh}(1)\,</math> ([[Rayleigh distribution]]) then <math>X^2 \sim \chi^2_2\,</math>
* If <math>X \sim \operatorname{Maxwell}(1)\,</math> ([[Maxwell distribution]]) then <math>X^2 \sim \chi^2_3\,</math>
* If <math>X \sim \chi^2_\nu</math> then <math>\tfrac{1}{X} \sim \operatorname{Inv-}\chi^2_\nu\, </math> ([[Inverse-chi-squared distribution]])
* The chi-squared distribution is a special case of type III [[Pearson distribution]]
* If <math>X \sim \chi^2_{\nu_1}\,</math> and <math>Y \sim \chi^2_{\nu_2}\,</math> are independent then <math>\tfrac{X}{X+Y} \sim \operatorname{Beta}(\tfrac{\nu_1}{2}, \tfrac{\nu_2}{2})\,</math> ([[beta distribution]])
* If <math> X \sim \operatorname{U}(0,1)\, </math> ([[Uniform distribution (continuous)|uniform distribution]]) then <math> -2\log(X) \sim \chi^2_2\,</math>
* If <math>X_i \sim \operatorname{Laplace}(\mu,\beta)\,</math> then <math>\sum_{i=1}^n \frac{2 |X_i-\mu|}{\beta} \sim \chi^2_{2n}\,</math>
* If <math>X_i</math> follows the [[generalized normal distribution]] (version 1) with parameters <math>\mu,\alpha,\beta</math> then <math>\sum_{i=1}^n \frac{2 |X_i-\mu|^\beta}{\alpha} \sim \chi^2_{2n/\beta}\,</math> <ref>{{cite journal |last= Bäckström |first= T. |author2=Fischer, J. |date=January 2018|title= Fast Randomization for Distributed Low-Bitrate Coding of Speech and Audio|journal= IEEE/ACM Transactions on Audio, Speech, and Language Processing |volume= 26|issue= 1|pages= 19–30|doi= 10.1109/TASLP.2017.2757601|s2cid= 19777585 |url= https://research.aalto.fi/files/27158975/ELEC_backstrom_et_al_Fast_randomization.pdf }}</ref>
* The chi-squared distribution is a transformation of [[Pareto distribution]]
* [[Student's t-distribution]] is a transformation of chi-squared distribution
* [[Student's t-distribution]] can be obtained from chi-squared distribution and [[normal distribution]]
* The [[noncentral beta distribution]] can be obtained as a transformation of chi-squared distribution and [[noncentral chi-squared distribution]]
* The [[noncentral t-distribution]] can be obtained from normal distribution and chi-squared distribution

A chi-squared variable with <math>k</math> degrees of freedom is defined as the sum of the squares of <math>k</math> independent [[standard normal]] random variables.

If <math>Y</math> is a <math>k</math>-dimensional Gaussian random vector with mean vector <math>\mu</math> and rank <math>k</math> covariance matrix <math>C</math>, then <math>X = (Y-\mu )^{T}C^{-1}(Y-\mu)</math> is chi-squared distributed with <math>k</math> degrees of freedom.

The sum of squares of [[statistically independent]] unit-variance Gaussian variables which do ''not'' have mean zero yields a generalization of the chi-squared distribution called the [[noncentral chi-squared distribution]].

If <math>Y</math> is a vector of <math>k</math> [[i.i.d.]] standard normal random variables and <math>A</math> is a <math>k\times k</math> [[symmetric matrix|symmetric]], [[idempotent matrix]] with [[rank (linear algebra)|rank]] <math>k-n</math>, then the [[quadratic form]] <math>Y^TAY</math> is chi-square distributed with <math>k-n</math> degrees of freedom.

If <math>\Sigma</math> is a <math>p\times p</math> positive-semidefinite covariance matrix with strictly positive diagonal entries, then for <math>X\sim N(0,\Sigma)</math> and <math>w</math> a random <math>p</math>-vector independent of <math>X</math> such that <math>w_1+\cdots+w_p=1</math> and <math>w_i\geq 0, i=1,\ldots,p,</math> then

: <math>\frac{1}{\left(\frac{w_1}{X_1},\ldots,\frac{w_p}{X_p}\right)\Sigma\left(\frac{w_1}{X_1},\ldots,\frac{w_p}{X_p}\right)^\top} \sim \chi_1^2.</math><ref name="Pillai-2016" />

The chi-squared distribution is also naturally related to other distributions arising from the Gaussian. In particular,
* <math>Y</math> is [[F-distribution|F-distributed]], <math>Y \sim F(k_1, k_2)</math> if <math>Y = \frac{ {X_1}/{k_1} }{ {X_2}/{k_2} }</math>, where <math>X_1 \sim \chi^2_{k_1}</math> and <math>X_2 \sim \chi^2_{k_2}</math> are statistically independent.
* If <math>X_1 \sim \chi^2_{k_1}</math> and <math>X_2 \sim \chi^2_{k_2}</math> are statistically independent, then <math>X_1 + X_2\sim \chi^2_{k_1+k_2}</math>. If <math>X_1</math> and <math>X_2</math> are not independent, then <math>X_1+X_2</math> is not chi-square distributed.

=== Generalizations ===
The chi-squared distribution is obtained as the sum of the squares of {{mvar|k}} independent, zero-mean, unit-variance Gaussian random variables. Generalizations of this distribution can be obtained by summing the squares of other types of Gaussian random variables. Several such distributions are described below.

=== Linear combination ===
If <math>X_1,\ldots,X_n</math> are chi square random variables and <math>a_1,\ldots,a_n\in\mathbb{R}_{>0}</math>, then the distribution of <math>X=\sum_{i=1}^n a_i X_i</math> is a special case of a [[Generalized chi-squared distribution|Generalized Chi-squared Distribution]].
A closed expression for this distribution is not known. It may be, however, approximated efficiently using the [[Characteristic function (probability theory)#Properties|property of characteristic functions]] of chi-square random variables.<ref>{{cite journal
|first=J.
|last=Bausch
|title=On the Efficient Calculation of a Linear Combination of Chi-Square Random Variables with an Application in Counting String Vacua
|journal=J. Phys. A: Math. Theor.
|volume=46
|issue=50
|year=2013
|pages=505202
|doi=10.1088/1751-8113/46/50/505202 |bibcode=2013JPhA...46X5202B
|arxiv=1208.2691
|s2cid=119721108
}}</ref>

=== Chi-squared distributions ===

==== Noncentral chi-squared distribution ====
{{Main|Noncentral chi-squared distribution}}
The noncentral chi-squared distribution is obtained from the sum of the squares of independent Gaussian random variables having unit variance and ''nonzero'' means.

==== Generalized chi-squared distribution ====
{{Main|Generalized chi-squared distribution}}
The generalized chi-squared distribution is obtained from the quadratic form {{math|z'Az}} where {{mvar|z}} is a zero-mean Gaussian vector having an arbitrary covariance matrix, and {{mvar|A}} is an arbitrary matrix.

=== Gamma, exponential, and related distributions ===
The chi-squared distribution <math>X \sim \chi_k^2</math> is a special case of the [[gamma distribution]], in that <math>X \sim \Gamma \left(\frac{k}2,\frac{1}2\right)</math> using the rate parameterization of the gamma distribution (or
<math>X \sim \Gamma \left(\frac{k}2,2 \right)</math> using the scale parameterization of the gamma distribution)
where {{mvar|k}} is an integer.

Because the [[exponential distribution]] is also a special case of the gamma distribution, we also have that if <math>X \sim \chi_2^2</math>, then <math>X\sim \operatorname{exp}\left(\frac 1 2\right)</math> is an [[exponential distribution]].

The [[Erlang distribution]] is also a special case of the gamma distribution and thus we also have that if <math>X \sim\chi_k^2</math> with even <math>k</math>, then <math>X</math> is Erlang distributed with shape parameter <math>k/2</math> and scale parameter <math>1/2</math>.