The Chi-Square Distribution - Maple Programming Help

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The Chi-Square Distribution

Main Concept

The Chi-Square ( χ2) distribution is the distribution of the sum of squares of a number of independent normal random variables.  The Chi-Square distribution can be used to test for goodness of fit of observed sample data to theoretical models, as well as in estimating variances. It is also commonly used in the analysis of contingency tables.

 

The Chi-Square distribution depends on one parameter, ν, which is the number of degrees of freedom.

 

The probability density function of the Chi-Square distribution is given by:

 

χ2ν=xν21ⅇx22ν2Γν2,

 

where the parameter ν is the number of degrees of freedom and Γ denotes the Gamma function.

 

The cumulative probability function is given by:

 

1Γ12ν,12xΓ12ν

Properties

The Chi-Squared distribution has the following properties:

PDF

xν21ⅇx22ν2Γν2

The probability density function.

CDF

1Γ12ν,12xΓ12ν

The cumulative distribution function.

Mean μ

ν

The mean of the distribution μ is equal to the number of degrees of freedom, μ = ν .

Variance σ2

The variance, σ2, is two times the number of degrees of freedom, σ2 =  2 ν .

 

Additionally:

• 

When the number of degrees of freedom is greater or equal to 2, the maximum value occurs when χ2 = ν 2.

• 

As the number of degrees of freedom increases, the chi-square distribution approaches a normal distribution.

 

Change the value for the number of degrees of freedom n to see how the Chi-Squared Distribution changes:

 

 

ν =

μx =

σ2 =

 

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