InverseChiSquareDistribution Class

In probability and statistics, the inverse-chi-squared distribution (or inverted-chi-square distribution) is a continuous probability distribution of a positive-valued random variable. It is closely related to the chi-squared distribution and its specific importance is that it arises in the application of Bayesian inference to the normal distribution, where it can be used as the prior and posterior distribution for an unknown variance.

The inverse-chi-squared distribution (or inverted-chi-square distribution) is the probability distribution of a random variable whose multiplicative inverse (reciprocal) has a chi-squared distribution. It is also often defined as the distribution of a random variable whose reciprocal divided by its degrees of freedom is a chi-squared distribution. That is, if X has the chi-squared distribution with v degrees of freedom, then according to the first definition, 1/X has the inverse-chi-squared distribution with v degrees of freedom; while according to the second definition, vX has the inverse-chi-squared distribution with v degrees of freedom. Only the first definition is covered by this class.

References:

• Wikipedia, The Free Encyclopedia. Inverse-chi-square distribution. Available on: http://en.wikipedia.org/wiki/Inverse-chi-squared_distribution

The following example demonstrates how to create a new inverse χ² distribution with the given degrees of freedom.

// Create a new inverse χ² distribution with 7 d.f.
var invchisq = new InverseChiSquareDistribution(degreesOfFreedom: 7);
double mean = invchisq.Mean;     // 0.2
double median = invchisq.Median; // 6.345811068141737
double var = invchisq.Variance;  // 75

double cdf = invchisq.DistributionFunction(x: 6.27);           // 0.50860033566176044
double pdf = invchisq.ProbabilityDensityFunction(x: 6.27);     // 0.0000063457380298844403
double lpdf = invchisq.LogProbabilityDensityFunction(x: 6.27); // -11.967727146795536

double ccdf = invchisq.ComplementaryDistributionFunction(x: 6.27); // 0.49139966433823956
double icdf = invchisq.InverseDistributionFunction(p: cdf);        // 6.2699998329362963

double hf = invchisq.HazardFunction(x: 6.27);            // 0.000012913598625327002
double chf = invchisq.CumulativeHazardFunction(x: 6.27); // 0.71049750196765715

string str = invchisq.ToString(); // "Inv-χ²(x; df = 7)"

Reference