The Chi Square Distribution

The chi square (χ²) distribution with n degrees of freedom models the distribution of the sum of the squares of n independent normal variables. It is best known for its use in the Testing Goodness-Of-Fit, and for the one sample Testing Variances of a sample. The chi square distribution is a special case of the The Gamma Distribution.

The chi square distribution has one parameter: the degrees of freedom. This value is usually an integer, but this is not an absolute requirement. The probability density function (PDF) is:

where n is the degrees of freedom.

The chi square distribution is a special case of the gamma distribution, with scale parameter 2 and shape parameter n/2.

The chi square distribution is implemented by the ChiSquareDistribution class. It has one constructor which takes the degrees of freedom as its only argument. The following constructs a chi square distribution with 10 degrees of freedom:

var chiSquare = new ChiSquareDistribution(10);

Dim chiSquare = New ChiSquareDistribution(10)

let chiSquare = ChiSquareDistribution(10.0)

The ChiSquareDistribution class has one specific property, DegreesOfFreedom, that returns the degrees of freedom of the distribution.

ChiSquareDistribution has one static (Shared in Visual Basic) method, Sample, which generates a random sample using a user-supplied uniform random number generator.

var random = new MersenneTwister();
double sample = ChiSquareDistribution.Sample(random, 10);

Dim random = New MersenneTwister()
Dim sample = ChiSquareDistribution.Sample(random, 10)

let random = MersenneTwister()
let sample = ChiSquareDistribution.Sample(random, 10.0)

The above example uses the MersenneTwister to generate uniform random numbers.

For details of the properties and methods common to all continuous distribution classes, see the topic on continuous distributions..