Chi-square distribution

chi-square
Probability density function
Cumulative distribution function
Parameters	$k > 0\,$ degrees of freedom
Support	$x \in [0; +\infty)\,$
Probability density function (pdf)	$\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,$
Cumulative distribution function (cdf)	$\frac{\gamma(k/2,x/2)}{\Gamma(k/2)}\,$
Mean	$k\,$
Median	approximately $k-2/3\,$
Mode	$k-2\,$ if $k\geq 2\,$
Variance	$2\,k\,$
Skewness	$\sqrt{8/k}\,$
Excess kurtosis	$12/k\,$
Entropy	$\frac{k}{2}\!+\!\ln(2\Gamma(k/2))\!+\!(1\!-\!k/2)\psi(k/2)$
Moment-generating function (mgf)	$(1-2\,t)^{-k/2}$ for $2\,t<1\,$
Characteristic function	$(1-2\,i\,t)^{-k/2}\,$

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.^[1]^[2]^[3]^[4] 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.

1 Definition
2 Characteristics
3 Related distributions and properties
4 See also
5 References
6 External links

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

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

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

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

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

where $\Gamma$ 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 $X\sim\chi^2_k$ then

$\mathrm{E}(X)=k$

$\mathrm{Var}(X)=2k$

Median

The median of $X\sim\chi^2_k$ is given approximately by

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

Information entropy

The information entropy is given by

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

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

Derivation of the pdf for one degree of freedom

Let $Y = X^2$ where $X \sim N(0,1)$

then $P(Y<y) = P(X^2<y)=P(X<|\sqrt{y}|)=F_x(\sqrt{y})-F_x(-\sqrt{y})$

Then $Y = X^2 \sim \chi^2(1)$

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 $X\sim\chi^2_k$ , 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 $X\sim\chi^2_k$ 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 $1-2/(9k)$ and variance $2/(9k)$ (Wilson and Hilferty,1931)
$X \sim \mathrm{Exponential}(\lambda = \tfrac{1}{2})$ is an exponential distribution if $X \sim \chi_2^2$ (with 2 degrees of freedom).
$Y \sim \chi_{\nu}^2$ is a chi-square distribution if $Y = \sum_{m=1}^{\nu} X_m^2$ for $X_i \sim N(0,1)$ independent that are normally distributed.
If $\boldsymbol{z}'=[Z_1,Z_2,\cdots,Z_n]$ , where the $Z_i$ s are independent $\rm{Normal}(0,\sigma^2)$ random variables or $\boldsymbol{z}\sim N_p(\boldsymbol{0},\sigma^2 \mathrm{I})$ and $\boldsymbol{A}$ is an $n\times n$ idempotent matrix with rank $n-k$ then the quadratic form $\frac{\boldsymbol{z}'\boldsymbol{A}\boldsymbol{z}}{\sigma^2}\sim \chi^2_{n-k}$ .
If the $X_i\sim N(\mu_i,1)$ have nonzero means, then $Y = \sum_{m=1}^k X_m^2$ is drawn from a noncentral chi-square distribution.
The chi-square distribution $X\sim\chi^2_\nu$ is a special case of the gamma distribution, in that $X \sim {\Gamma}(\frac{\nu}{2}, \theta=2)$ .
$Y \sim \mathrm{F}(\nu_1, \nu_2)$ is an F-distribution 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 independent with their respective degrees of freedom.
$Y \sim \chi^2(\bar{\nu})$ is a chi-square distribution if $Y = \sum_{m=1}^N X_m$ where $X_m \sim \chi^2(\nu_m)$ are independent and $\bar{\nu} = \sum_{m=1}^N \nu_m$ .
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 $\sum_{i=1}^n(X_i - \bar X)^2 \sim \sigma^2 \chi^2_{n-1}$ where $\bar X = \frac{1}{n} \sum_{i=1}^n X_i$ .
if $X \sim \mathrm{SkewLogistic}(\tfrac{1}{2})\,$ , 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 $X_i\sim \mathrm{Normal}(\mu_i,\sigma^2_i),i=1,\cdots,k,$ independent random variables:

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

↑ 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 .
↑ 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.

External links

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 sampling distribution (chi-squared) of various statistics, e.g. Σx², for a normal population

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