Chi-Square Distribution

Overview

The chi-square (χ²) distribution is a one-parameter family of curves. The chi-square distribution is commonly used in hypothesis testing, particularly the chi-square test for goodness of fit.

Statistics and Machine Learning Toolbox™ offers multiple ways to work with the chi-square distribution.

Use distribution-specific functions (chi2cdf,chi2inv,chi2pdf,chi2rnd,chi2stat) with specified distribution parameters. The distribution-specific functions can accept parameters of multiple chi-square distributions.
Use generic distribution functions (cdf,icdf,pdf,random) with a specified distribution name ('Chisquare') and parameters.

Parameters

The chi-square distribution uses the following parameter.

Parameter	Description	金宝app
nu (ν)	Degrees of freedom	ν`= 1, 2, 3,...`

The degrees of freedom parameter is typically an integer, but chi-square functions accept any positive value.

The sum of two chi-square random variables with degrees of freedomν₁andν₂is a chi-square random variable with degrees of freedomν=ν₁+ν₂。

Probability Density Function

The probability density function (pdf) of the chi-square distribution is

$y = f (x | ν) = \frac{x^{(ν - 2) / 2} e^{- x / 2}}{2^{\frac{ν}{2}} Γ (ν / 2)},$

whereνis the degrees of freedom and Γ( · ) is the Gamma function.

For an example, seeCompute Chi-Square Distribution pdf。

Cumulative Distribution Function

The cumulative distribution function (cdf) of the chi-square distribution is

$p = F (x | ν) = \int_{0}^{x} \frac{t^{(ν - 2) / 2} e^{- t / 2}}{2^{ν / 2} Γ (ν / 2)} d t,$

whereνis the degrees of freedom and Γ( · ) is the Gamma function. The resultpis the probability that a single observation from the chi-square distribution withνdegrees of freedom falls in the interval[0,x]。

For an example, seeCompute Chi-Square Distribution cdf。

Inverse Cumulative Distribution Function

The inverse cumulative distribution function (icdf) of the chi-square distribution is

$x = F^{- 1} (p | ν) = {x : F (x | ν) = p},$

where

$p = F (x | ν) = \int_{0}^{x} \frac{t^{(ν - 2) / 2} e^{- t / 2}}{2^{ν / 2} Γ (ν / 2)} d t,$

ν自由度,Γγ(·)function. The resultpis the probability that a single observation from the chi-square distribution withνdegrees of freedom falls in the interval[0,x]。

Descriptive Statistics

The mean of the chi-square distribution isν。

The variance of the chi-square distribution is2ν。

Examples

Compute Chi-Square Distribution pdf

Open Live Script

Compute the pdf of a chi-square distribution with 4 degrees of freedom.

x = 0:0.2:15; y = chi2pdf(x,4);

Plot the pdf.

figure; plot(x,y) xlabel('Observation') ylabel('Probability Density')

Figure contains an axes object. The axes object contains an object of type line.

The chi-square distribution is skewed to the right, especially for few degrees of freedom.

Compute Chi-Square Distribution cdf

Open Live Script

Compute the cdf of a chi-square distribution with 4 degrees of freedom.

x = 0:0.2:15; y = chi2cdf(x,4);

Plot the cdf.

figure; plot(x,y) xlabel('Observation') ylabel('Cumulative Probability')

Figure contains an axes object. The axes object contains an object of type line.

Related Distributions

F Distribution— TheFdistribution is a two-parameter distribution that has parametersν₁(numerator degrees of freedom) andν₂(denominator degrees of freedom). TheFdistribution can be defined as the ratio $F = \frac{\frac{χ_{1}^{2}}{ν_{1}}}{\frac{χ_{2}^{2}}{ν_{2}}}$ , whereχ²₁andχ²₂are both chi-square distributed withν₁andν₂degrees of freedom, respectively.
Gamma Distribution— The gamma distribution is a two-parameter continuous distribution that has parametersa(shape) andb(scale). The chi-square distribution is equal to the gamma distribution with2a=νandb=2。
Noncentral Chi-Square Distribution— The noncentral chi-square distribution is a two-parameter continuous distribution that has parametersν(degrees of freedom) andδ(noncentrality). The noncentral chi-square distribution is equal to the chi-square distribution whenδ=0。
Normal Distribution— The normal distribution is a two-parameter continuous distribution that has parametersμ(mean) andσ(standard deviation). The standard normal distribution occurs whenμ=0andσ=1。
IfZ₁,Z₂, …,Z_nare standard normal random variables, then $\sum_{i = 1}^{n} Z_{i}^{2}$ has a chi-square distribution with degrees of freedomν=n– 1。
If a set ofnobservations is normally distributed with varianceσ²and sample variances², then $\frac{(n - 1) s^{2}}{σ^{2}}$ has a chi-square distribution with degrees of freedomν=n– 1。This relationship is used to calculate confidence intervals for the estimate of the normal parameterσ²in the functionnormfit。
Student's t Distribution— The Student'stdistribution is a one-parameter continuous distribution that has parameterν(degrees of freedom). IfZhas a standard normal distribution andχ²has a chi-square distribution with degrees of freedomν, then $t = \frac{Z}{\sqrt{χ^{2} / ν}}$ has a Student'stdistribution with degrees of freedomν。
Wishart Distribution— The Wishart distribution is a higher dimensional analog of the chi-square distribution.

References

[1] Abramowitz, Milton, and Irene A. Stegun, eds.Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables。9. Dover print.; [Nachdr. der Ausg. von 1972]. Dover Books on Mathematics. New York, NY: Dover Publ, 2013.

[2] Devroye, Luc.Non-Uniform Random Variate Generation。New York, NY: Springer New York, 1986.https://doi.org/10.1007/978-1-4613-8643-8

[3] Evans, M., N. Hastings, and B. Peacock.Statistical Distributions。2nd ed., Hoboken, NJ: John Wiley & Sons, Inc., 1993.

[4] Kreyszig, Erwin.Introductory Mathematical Statistics: Principles and Methods。纽约:威利,1970年。