Chi-square distribution: Definition from Answers.com

This article is about the mathematics of the chi-square distribution. For its uses in statistics, see chi-square test.

"Chi2" redirects here. For the music group, see Chi2 (band).

chi-square
Probability density function
Cumulative distribution function
parameters:	$k > 0\,$ degrees of freedom
support:	$x \in [0; +\infty)\,$
pdf:	$\frac{x^{(k/2) - 1}}{2^{k/2} \Gamma(k/2)} e^{-x/2}\,$
cdf:	$\frac{\gamma(k/2,x/2)}{\Gamma(k/2)}\,$
mean:	$k\,$
median:	approximately $k-\frac{2}{3}+\frac{4}{27k}-\frac{8}{729k^2}$
mode:	$k-2\,$ if $k\geq 2\,$
variance:	$2\,k\,$
skewness:	$\sqrt{8/k}\,$
kurtosis:	$12/k\,$
entropy:	$\frac{k}{2}\!+\!\ln(2\Gamma(k/2))\!+\!(1\!-\!k/2)\psi(k/2)$
mgf:	$(1-2\,t)^{-k/2}$ for $2\,t<1\,$
cf:	$(1-2\,i\,t)^{-k/2}$ ^[1]

In probability theory and statistics, the chi-square distribution (also chi-squared or χ²-distribution) is one of the most widely used theoretical probability distributions in inferential statistics, e.g., in statistical significance tests.^[2]^[3]^[4]^[5] A random variable is said to have a chi-square distribution if it equals the sum of the squares of a set of statistically independent standard Gaussian random variables.

The best-known situations in which the chi-square distribution is 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 $X 1,..., X k$ 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

$Q\sim\chi^2_k.\,$

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

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

Characteristics

Further properties of the chi-square distribution can be found in the box at right.

Probability density function

A probability density function of the chi-square distribution is

$f(x;k)= \begin{cases}\displaystyle \frac{1}{2^{k/2}\Gamma(k/2)}\,x^{(k/2) - 1} e^{-x/2}&\text{for }x>0,\\ 0&\text{for }x\le0, \end{cases}$

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

For derivations of the pdf in the cases of one and two degrees of freedom, see Proofs related to chi-square distribution.

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 $γ(k, z)$ 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.

Additivity

It follows from the definition of the chi-square distribution that the sum of independent chi-square variables is also chi-square distributed. Specifically, if $\{X_i\}_{i=1}^n$ are independent chi-square variables with $\{k_i\}_{i=1}^n$ degrees of freedom, respectively, then $Y = X_1 + \cdots + X_n$ is chi-square distributed with $k_1 + \cdots + k_n$ degrees of freedom.

Information entropy

The information entropy is given by

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

where $ψ(x)$ is the Digamma function.

Noncentral moments

The moments about zero of a chi-square distribution with k degrees of freedom are given by^[6]^[7]

$\begin{align} E(X^m) &= k (k+2) (k+4) \cdots (k+2m-2) \\ &= 2^m \frac{\Gamma(m+k/2)}{\Gamma(k/2)}. \end{align}$

Cumulants

The cumulants are readily obtained by a (formal) power series expansion of the logarithm of the characteristic function:

$\kappa_n = \frac{2^n \, n!\, k}{2n}$

Asymptotic properties

By the central limit theorem, because the chi-square distribution is the sum of k independent random variables, it converges to a normal distribution for large k ( $k > 50$ is "approximately normal" according to ^[8]). Specifically, if $X\sim\chi^2_k$ , then as $k$ tends to infinity, the distribution of $(X-k)/\sqrt{2k}$ tends to a standard normal distribution. However, convergence is slow as the skewness is $\sqrt{8/k}$ and the excess kurtosis is $12 / k$ .

Other functions of the chi-square distribution converge more rapidly to a normal distribution. Some examples are:

If $X\sim\chi^2_k$ then $\sqrt{2X}$ is approximately normally distributed with mean $\sqrt{2k-1}$ and unit variance (result credited to R. A. Fisher).
If $X\sim\chi^2_k$ then $\sqrt[3]{X/k}$ is approximately normally distributed with mean $1 - 2 / (9 k)$ and variance $2 / (9 k)$ (Wilson and Hilferty,1931)

Related distributions

Normal distribution

A chi-square variable with k degrees of freedom is defined as the sum of the squares of k independent standard normal random variables.

More generally, the chi-square distribution is related to any Gaussian random vector of length k as follows. If Y is a Gaussian random vector having mean vector μ and covariance matrix C, then $X = (Y - μ) T C - 1 (Y - μ)$ is chi-square distributed with k degrees of freedom. This is because the subtraction of μ and the multiplication by $C - 1 / 2$ effectively transforms the Gaussian vector to an i.i.d., zero-mean distribution.

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

If Y is a vector of k i.i.d. standard normal random variables and A is a $k \times k$ idempotent matrix with rank $k - n$ then the quadratic form $Y T A Y$ is chi-square distributed with $k - n$ degrees of freedom.

The chi-square distribution is also naturally related to other distributions arising from the Gaussian. In particular,

Y is F-distributed, $Y \sim \mathrm{F}(\nu_1, \nu_2),\,\!$ if $Y = \frac{X_1 / \nu_1}{X_2 / \nu_2}$ where $X_1 \sim \chi_{\nu_1}^2$ and $X_2 \sim \chi_{\nu_2}^2$ are statistically independent.
If $X$ is chi-square distributed, then $\sqrt{X}$ is chi distributed.

Generalizations

The chi-square distribution is obtained from the sum of 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.

Noncentral chi-square distribution

Main article: Noncentral chi-square distribution

The noncentral chi-square distribution is obtained from the sum of the squares of independent Gaussian random variables having unit variance and nonzero means.

Generalized chi-square distribution

Main article: Generalized chi-square distribution

The generalized chi-square distribution is obtained from the quadratic form $z T A z,$ where z is a zero-mean Gaussian vector having an arbitrary covariance matrix, and A is an arbitrary matrix.

Gamma, exponential, and related distributions

The chi-square distribution $X\sim\chi^2_k$ is a special case of the gamma distribution, in that $X \sim \Gamma(\frac{k}{2}, \theta=2).$

Because the exponential distribution is also a special case of the Gamma distribution, we also have that if $X \sim \chi_2^2$ (with 2 degrees of freedom), then $X \sim \mathrm{Exponential}(\lambda = \tfrac{1}{2})$ is an exponential distribution.

The Erlang distribution is also a special case of the Gamma distribution and thus we also have that if $X\sim\chi^2_k$ with even k, then X is Erlang distributed with shape parameter $k / 2$ and scale parameter 1/2.

Other generalizations

If $Z i$ are k independent, complex Gaussian random variables with mean 0 and variance $\sigma_i^2$ , then the random variable

$\tilde{Q} = \sum_{i=1}^k |Z_i|^2$

is a type of generalized chi-square distribution. The differences from the standard chi-square distribution is that $Z i$ are complex and can have different variances. If $\mu=\sigma_i^2$ for all i, then $\tilde{Q}$ becomes a $\chi^2_{2k}$ scaled by $μ / 2$ , also known as the Erlang distribution. If $\sigma_i^2$ have distinct values for all i, then $\tilde{Q}$ has the pdf^[9]

$f(x; k,\sigma_1^2,\ldots,\sigma_k^2) = \sum_{i=1}^{k} \frac{e^{-\frac{x}{\sigma_i^2}}}{\sigma_i^2 \prod_{j=1, j\neq i}^{k} (1- \frac{\sigma_j^2}{\sigma_i^2})} \quad\mbox{for }x\geq0.$

If there are sets of repeated variances among $\sigma_i^2$ , assume that they are divided into M sets, each representing a certain variance value. Denote $\mathbf{r}=(r_1, r_2, \dots, r_M)$ to be the number of repetitions in each group. I.e., the mth set contains $r m$ variables that have variance $\sigma^2_m$ . it represents an arbitrary linear combination of $χ 2$ random variables with different degree of freedoms:

$\tilde{Q} = \sum_{m=1}^M \sigma^2_m Q_m, \quad Q_m \sim \chi^2_{2r_m} \,$

The pdf of $\tilde{Q}$ becomes^[10]

$f(x; \mathbf{r}, \sigma^2_1, \dots \sigma^2_M) = \prod_{m=1}^M \frac{1}{\sigma^{2r_m}_m} \sum_{k=1}^M \sum_{l=1}^{r_k} \frac{\Psi_{k,l,\mathbf{r}}}{(r_k-l)!} (-x)^{r_k-l} e^{-\frac{x}{\sigma^2_k}} \quad\mbox{for }x\geq0.$

where

$\Psi_{k,l,\mathbf{r}} = (-1)^{r_k-1} \sum_{\mathbf{i} \in \Omega_{k,l}} \prod_{j \neq k} \Big( \!\!\! \begin{array}{c} i_j + r_j-1\\ i_j \end{array} \!\!\! \Big) \Big(\frac{1}{\sigma^2_j}\!-\!\frac{1}{\sigma^2_k} \Big)^{-(r_j + i_j)},$

with $\mathbf{i}=[i_1,\ldots,i_M]^T$ from the set $Ω k, l$ of all partitions of $l - 1$ (with $i k = 0$ ) defined as

$\Omega_{k,l} = \Big\{ [i_1,\ldots,i_m]\in \mathbb{Z}^m; \sum_{j=1}^M i_j \!= l-1, i_k=0, i_j\geq 0 \,\, \forall j \Big\}.$

Applications

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.

Following are some of the most common situations in which the chi-square distribution arises from a Gaussian-distributed sample.

if $X_1, \dots, X_n$ are i.i.d. $N (μ,σ 2)$ random variables, then $\sum_{i=1}^n(X_i - \bar X)^2 \sim \sigma^2 \chi^2_{n-1}$ where $\bar X = \frac{1}{n} \sum_{i=1}^n X_i$ .

The box below shows probability distributions with name starting with chi for some statistics based on $X_i\sim \mathrm{Normal}(\mu_i,\sigma^2_i),i=1,\cdots,k,$ independent random variables:

Name	Statistic
chi-square distribution	$\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2$
noncentral chi-square distribution	$\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2$
chi distribution	$\sqrt{\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2}$
noncentral chi distribution	$\sqrt{\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2}$

References

^ M.A. Sanders. "Characteristic function of the central chi-square distribution". http://www.planetmathematics.com/CentralChiDistr.pdf. Retrieved 2009-03-06.
^ Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 26", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, ISBN 0-486-61272-4, http://www.math.sfu.ca/~cbm/aands/page_940.htm .
^ NIST (2006). Engineering Statistics Handbook - Chi-Square Distribution
^ Jonhson, N.L.; S. Kotz, , N. Balakrishnan (1994). Continuous Univariate Distributions (Second Ed., Vol. 1, Chapter 18). John Willey and Sons. ISBN 0-471-58495-9.
^ Mood, Alexander; Franklin A. Graybill, Duane C. Boes (1974). Introduction to the Theory of Statistics (Third Edition, p. 241-246). McGraw-Hill. ISBN 0-07-042864-6.
^ Chi-square distribution, from MathWorld, retrieved Feb. 11, 2009
^ M. K. Simon, Probability Distributions Involving Gaussian Random Variables, New York: Springer, 2002, eq. (2.35), ISBN 978-0-387-34657-1
^ Box, Hunter and Hunter. Statistics for experimenters. Wiley. p. 46.
^ D. Hammarwall, M. Bengtsson, B. Ottersten, Acquiring Partial CSI for Spatially Selective Transmission by Instantaneous Channel Norm Feedback, IEEE Transactions on Signal Processing, vol 56, pp. 1188-1204, March 2008.
^ E. Björnson, D. Hammarwall, B. Ottersten, Exploiting Quantized Channel Norm Feedback through Conditional Statistics in Arbitrarily Correlated MIMO Systems, IEEE Transactions on Signal Processing, vol 57, pp. 4027-4041, October 2009

Wilson, E.B. Hilferty, M.M. (1931) The distribution of chi-square. Procedings of the National Academy of Sciences, Washington, 17, 684–688.

External links

Earliest Uses of Some of the Words of Mathematics: entry on Chi square has a brief history
Comparison of noncentral and central distributions Density plot, critical value, cumulative probability, etc., online calculator based on R embedded in Mediawiki.
Course notes on Chi-Square Goodness of Fit Testing from Yale University Stats 101 class. Example includes hypothesis testing and parameter estimation.
On-line calculator for the significance of chi-square, in Richard Lowry's statistical website at Vassar College.
Distribution Calculator Calculates probabilities and critical values for normal, t-, chi2- and F-distribution
Chi-Square Calculator for critical values of Chi-Square in R. Webster West's applet website at University of South Carolina
Chi-Square Calculator from GraphPad
Table of Chi-squared distribution
Mathematica demonstration showing the chi-squared sampling distribution of various statistics, e.g. Σx², for a normal population
Simple algorithm for approximating cdf and inverse cdf for the chi-square distribution with a pocket calculator

Probability distributions

Discrete univariate with finite support

Benford · Bernoulli · binomial · categorical · hypergeometric · Rademacher · discrete uniform · Zipf · Zipf-Mandelbrot

Discrete univariate with infinite support

Boltzmann · Conway–Maxwell–Poisson · compound Poisson · discrete phase-type · extended negative binomial · Gauss–Kuzmin · geometric · logarithmic · negative binomial · parabolic fractal · Poisson · Skellam · Yule–Simon · zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Beta · Irwin–Hall · Kumaraswamy · raised cosine · triangular · U-quadratic · uniform · Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,∞)

Beta prime · Bose–Einstein · Burr · chi-square · chi · Coxian · Erlang · exponential · F · Fermi–Dirac · folded normal · Fréchet · Gamma · generalized extreme value · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-square · hyper-exponential · hypoexponential · inverse chi-square (scaled inverse chi-square) · inverse Gaussian · inverse gamma · Lévy · log-normal · log-logistic · Maxwell–Boltzmann · Maxwell speed · Nakagami · noncentral chi-square · Pareto · phase-type · Rayleigh · relativistic Breit–Wigner · Rice · Rosin–Rammler · shifted Gompertz · truncated normal · type-2 Gumbel · Weibull · Wilks' lambda

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy · extreme value · exponential power · Fisher's z · generalized normal · generalized hyperbolic · Gumbel · hyperbolic secant · Landau · Laplace · logistic · normal (Gaussian) · normal inverse Gaussian · skew normal · slash · Student's t · type-1 Gumbel · Variance-Gamma · Voigt

Multivariate (joint)

Discrete: Ewens · Beta-binomial · multinomial · multivariate Polya Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · multivariate Student · normal-scaled inverse gamma · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart

Directional, degenerate, and singular

Directional: Kent · von Mises · von Mises–Fisher Degenerate: discrete degenerate · Dirac delta function Singular: Cantor

Families

exponential · natural exponential · location-scale · maximum entropy · mixture · Pearson · stable · Tweedie

v • d • e Some common univariate probability distributions

Continuous	beta • Cauchy • chi-square • exponential • F • gamma • Laplace • log-normal • normal • Pareto • Student's t • uniform • Weibull

Discrete	Bernoulli • binomial • discrete uniform • geometric • hypergeometric • negative binomial • Poisson

List of probability distributions

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Donate to Wikimedia

Chi-square distribution

Contents