Chi-Square Distribution

Overview

The chi-square distribution is commonly used in hypothesis testing, particularly the chi-squared test for goodness of fit.

Parameters

The chi-square distribution uses the following parameter.

ParameterDescriptionSupport
νDegrees of freedomν is a nonnegative integer value

Probability Density Function

The probability density function (pdf) is

y=f(x|ν)=x(ν2)/2ex/22ν2Γ(ν/2)

where Γ( · ) is the Gamma function, ν is the degrees of freedom, and x0.

Cumulative Distribution Function

The cumulative distribution function (cdf) is

p=F(x|ν)=0xt(ν2)/2et/22ν/2Γ(ν/2)dt

where Γ( · ) is the Gamma function, ν is the degrees of freedom, and x0.

Descriptive Statistics

The mean is ν.

The variance is 2ν.

Relationship to Other Distributions

The χ2 distribution is a special case of the gamma distribution where b = 2 in the equation for gamma distribution below.

y=f(x|a,b)=1baΓ(a)xa1exb

The χ2 distribution gets special attention because of its importance in normal sampling theory. If a set of n observations is normally distributed with variance σ2, and s2 is the sample standard deviation, then

(n1)s2σ2χ2(n1)

This relationship is used to calculate confidence intervals for the estimate of the normal parameter σ2 in the function normfit.

Examples

Compute Chi-Square Distribution pdf

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)

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

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