ZipfDistribution Class

public class ZipfDistribution : DiscreteDistribution

Public Class ZipfDistribution
	Inherits DiscreteDistribution

public ref class ZipfDistribution : public DiscreteDistribution

type ZipfDistribution = 
    class
        inherit DiscreteDistribution
    end

Inheritance

Object  →  Distribution  →  DiscreteDistribution  →  ZipfDistribution

The Zipf distribution is a discrete probability distribution commonly used to model the frequency of occurrence of distinct objects in a collection, where items are ordered from most to least frequent. It is characterized by its shape parameter a > 1, which controls how quickly the probabilities decay.

The probability mass function of the Zipf distribution is: $$P(X=n)=(n^{(}-a))/\zeta (a)$$ where $\zeta (a)$ is the Riemann zeta function that serves as the normalization constant.

The Zipf distribution is sometimes called the zeta distribution.

The Zipf distribution is the limit distribution of the ZipfianDistribution where the second (integer) parameter approaches infinity. For the distribution to exist, the shape parameter of the distribution must be greater than one.

This distribution is often used to model:

• Word frequency in natural languages
• City population sizes
• Website traffic patterns
• Income distribution rankings
• Scientific citation counts

When the parameter a approaches 1 from above, the distribution becomes increasingly heavy-tailed. For values of a ≤ 1, the distribution is not properly normalized (the sum of probabilities would diverge).

The following example demonstrates creating a Zipf distribution with a non-standard parameter and using its properties and methods:

// Create a Zipf distribution with shape parameter 2.5
var zipf = new ZipfDistribution(2.5);

// Distribution parameters
Console.WriteLine($"Distribution: Zipf(a = 2.5)");

// Distribution functions
Console.WriteLine("\nDistribution functions:");
Console.WriteLine($"P(X = 1) = {zipf.Probability(1)}");
Console.WriteLine($"P(X = 2) = {zipf.Probability(2)}");
Console.WriteLine($"P(X = 5) = {zipf.Probability(5)}");
Console.WriteLine($"Log P(X = 10) = {zipf.LogProbability(10)}");
Console.WriteLine($"CDF(5) = {zipf.DistributionFunction(5)}");

// Moments
Console.WriteLine("\nMoments:");
Console.WriteLine($"Mean = {zipf.Mean}");
Console.WriteLine($"Variance = {zipf.Variance}");

Reference