Chi-square distribution

Probability distribution
name =chi-square
type =density
pdf_

cdf_

parameters =$k\; >\; 0,$ degrees of freedom
support =$x\; in\; [0;\; +infty),$
pdf =$frac\{(1/2)^\{k/2${Gamma(k/2)} x^{k/2 - 1} e^{-x/2},
cdf =$frac\{gamma(k/2,x/2)\}\{Gamma(k/2)\},$
mean =$k,$
median =approximately $k-2/3,$
mode =$k-2,$ if $kgeq\; 2,$
variance =$2,k,$
skewness =$sqrt\{8/k\},$
kurtosis =$12/k,$
entropy =$frac\{k\}\{2\}!+!ln(2Gamma(k/2))!+!(1!-!k/2)psi(k/2)$
mgf =$(1-2,t)^\{-k/2\}$ for
char =$(1-2,i,t)^\{-k/2\},$

In probability theory and statistics, the chi-square distribution (also chi-squared or $chi^2$ distribution) is one of the most widely used theoretical probability distributions in inferential statistics, e.g., in statistical significance tests.Abramowitz_Stegun_ref|26|940] [NIST (2006). [http://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm Engineering Statistics Handbook - Chi-Square Distribution] ] [cite book
last = Jonhson
first = N.L.
coauthors = S. Kotz, , N. Balakrishnan
title = Continuous Univariate Distributions (Second Ed., Vol. 1, Chapter 18)
publisher = John Willey and Sons
date = 1994
isbn = 0-471-58495-9] [cite book
last = Mood
first = Alexander
coauthors = Franklin A. Graybill, Duane C. Boes
title = Introduction to the Theory of Statistics (Third Edition, p. 241-246)
publisher = McGraw-Hill
date = 1974
isbn = 0-07-042864-6] It is useful because, under reasonable assumptions, easily calculated quantities can be proven to have distributions that approximate to the chi-square distribution if the null hypothesis is true.

The best-known situations in which the chi-square distribution are used are the common chi-square tests for goodness of fit of an observed distribution to a theoretical one, and of the independence of two criteria of classification of qualitative data. Many other statistical tests also lead to a use of this distribution, like Friedman's analysis of variance by ranks.

Definition

If $X\_i$ are "k" independent, normally distributed random variables with mean 0 and variance 1, then the random variable

:$Q\; =\; sum\_\{i=1\}^k\; X\_i^2$

is distributed according to the chi-square distribution with $k$ degrees of freedom. This is usually written

:$Qsimchi^2\_k.,$

The chi-square distribution has one parameter: $k$ - a positive integer that specifies the number of degrees of freedom (i.e. the number of $X\_i$)

The chi-square distribution is a special case of the gamma distribution.

Characteristics

Probability density function

A probability density function of the chi-square distribution is

:

where $Gamma$ denotes the Gamma function, which has closed-form values at the half-integers.

Cumulative distribution function

Its cumulative distribution function is:

:$F(x;k)=frac\{gamma(k/2,x/2)\}\{Gamma(k/2)\}\; =\; P(k/2,\; x/2)$

where $gamma(k,z)$ is the lower incomplete Gamma function and $P(k,\; z)$ is the regularized Gamma function.

Tables of this distribution — usually in its cumulative form — are widely available and the function is included in many spreadsheets and all statistical packages.

Characteristic function

The characteristic function of the Chi-square distribution is

:$chi(t;k)=(1-2it)^\{-k/2\}.,$

Expected value and variance

If $Xsimchi^2\_k$ then:$mathrm\{E\}(X)=k$ :$mathrm\{Var\}(X)=2k$

Median

The median of $Xsimchi^2\_k$ is given approximately by

:$k-frac\{2\}\{3\}+frac\{4\}\{27k\}-frac\{8\}\{729k^2\}.$

Information entropy

The information entropy is given by

:$H=int\_\{-infty\}^infty\; f(x;k)ln(f(x;k))\; dx=frac\{k\}\{2\}+ln\; left(\; 2\; Gamma\; left(\; frac\{k\}\{2\}\; ight)\; ight)+left(1\; -\; frac\{k\}\{2\}\; ight)psi(k/2).$

where $psi(x)$ is the Digamma function.

Related distributions and properties

The chi-square distribution has numerous applications in inferential statistics, for instance in chi-square tests and in estimating variances.It enters the problem of estimating the mean of a normally distributed population and the problem of estimating the slope of a regression line via its role in Student's t-distribution.It enters all analysis of variance problems via its role in the F-distribution, which is the distribution of the ratio of two independent chi-squared random variables divided by their respective degrees of freedom.

*If $Xsimchi^2\_k$, then as $k$ tends to infinity, the distribution of $X$ tends to a normal distribution with mean $k$ and variance $2k$ (convergence is slow as the skewness is $sqrt\{8/k\}$ and the excess kurtosis is $12/k$)
*If $Xsimchi^2\_k$ then $sqrt\{2X\}$ is approximately normally distributed with mean $sqrt\{2k-1\}$ and unit variance (result credited to R. A. Fisher).
* If $Xsimchi^2\_k$ then $sqrt\; [3]\; \{X/k\}$ is approximately normally distributed with mean $1-2/(9k)$ and variance $2/(9k)$ (Wilson and Hilferty,1931)
*$X\; sim\; mathrm\{Exponential\}(lambda\; =\; frac\{1\}\{2\})$ is an exponential distribution if $X\; sim\; chi\_2^2$ (with 2 degrees of freedom).
*$Y\; sim\; chi\_\{\; u\}^2$ is a chi-square distribution if $Y\; =\; sum\_\{m=1\}^\{\; u\}\; X\_m^2$ for $X\_i\; sim\; N(0,1)$ independent that are normally distributed.
*If , where the $Z\_i$s are independent $m\{Normal\}(0,sigma^2)$ random variables or and is an $n\; imes\; n$ idempotent matrix with rank $n-k$ then the quadratic form .
*If the $X\_isim\; N(mu\_i,1)$ have nonzero means, then $Y\; =\; sum\_\{m=1\}^k\; X\_m^2$ is drawn from a noncentral chi-square distribution.
*The chi-square distribution $Xsimchi^2\_\; u$ is a special case of the gamma distribution, in that $X\; sim\; \{Gamma\}(frac\{\; u\}\{2\},\; heta=2)$.
*$Y\; sim\; mathrm\{F\}(\; u\_1,\; u\_2)$ is an F-distribution if $Y\; =\; frac\{X\_1\; /\; u\_1\}\{X\_2\; /\; u\_2\}$ where $X\_1\; sim\; chi\_\{\; u\_1\}^2$ and $X\_2\; sim\; chi\_\{\; u\_2\}^2$ are independent with their respective degrees of freedom.
* is a chi-square distribution if $Y\; =\; sum\_\{m=1\}^N\; X\_m$ where $X\_m\; sim\; chi^2(\; u\_m)$ are independent and .
*if $X$ is chi-square distributed, then $sqrt\{X\}$ is chi distributed.
*in particular, if $X\; sim\; chi\_2^2$ (chi-square with 2 degrees of freedom), then $sqrt\{X\}$ is Rayleigh distributed.
*if $X\_1,\; dots,\; X\_n$ are i.i.d. $N(mu,sigma^2)$ random variables, then where .
*if $X\; sim\; mathrm\{SkewLogistic\}(\; frac\{1\}\{2\}),$, then $mathrm\{log\}(1\; +\; e^\{-X\})\; sim\; chi\_2^2,$
*The box below shows probability distributions with name starting with chi for some statistics based on $X\_isim\; mathrm\{Normal\}(mu\_i,sigma^2\_i),i=1,cdots,k,$ independent random variables:

ee also

*Cochran's theorem
*Inverse-chi-square distribution
*Degrees of freedom (statistics)
*Fisher's method for combining independent tests of significance
*Noncentral chi-square distribution
*Normal distribution
*Normalised Innovation Squared

External links

* [http://www.stat.yale.edu/Courses/1997-98/101/chigf.htm Course notes on Chi-Square Goodness of Fit Testing] from Yale University Stats 101 class. Example includes hypothesis testing and parameter estimation.
* [http://faculty.vassar.edu/lowry/tabs.html#csq On-line calculator for the significance of chi-square] , in Richard Lowry's statistical website at Vassar College.
* [http://www.vias.org/simulations/simusoft_distcalc.html Distribution Calculator] Calculates probabilities and critical values for normal, t-, chi2- and F-distribution
* [http://www.stat.sc.edu/~west/applets/chisqdemo.html Chi-Square Calculator for critical values of Chi-Square] in R. Webster West's applet website at University of South Carolina
* [http://graphpad.com/quickcalcs/chisquared2.cfm Chi-Square Calculator from GraphPad]
* [http://www.medcalc.be/manual/chi-square-table.php Table of Chi-squared distribution]

References