// <copyright file="Zipf.cs" company="Math.NET">
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// Math.NET Numerics, part of the Math.NET Project
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// http://numerics.mathdotnet.com
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// http://github.com/mathnet/mathnet-numerics
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//
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// Copyright (c) 2009-2014 Math.NET
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//
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// Permission is hereby granted, free of charge, to any person
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// obtaining a copy of this software and associated documentation
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// files (the "Software"), to deal in the Software without
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// restriction, including without limitation the rights to use,
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// copy, modify, merge, publish, distribute, sublicense, and/or sell
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// copies of the Software, and to permit persons to whom the
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// Software is furnished to do so, subject to the following
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// conditions:
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//
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// The above copyright notice and this permission notice shall be
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// included in all copies or substantial portions of the Software.
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//
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// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
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// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
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// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
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// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
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// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
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// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
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// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
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// OTHER DEALINGS IN THE SOFTWARE.
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// </copyright>
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using System;
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using System.Collections.Generic;
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using IStation.Numerics.Random;
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namespace IStation.Numerics.Distributions
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{
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/// <summary>
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/// Discrete Univariate Zipf distribution.
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/// Zipf's law, an empirical law formulated using mathematical statistics, refers to the fact
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/// that many types of data studied in the physical and social sciences can be approximated with
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/// a Zipfian distribution, one of a family of related discrete power law probability distributions.
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/// For details about this distribution, see
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/// <a href="http://en.wikipedia.org/wiki/Zipf%27s_law">Wikipedia - Zipf distribution</a>.
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/// </summary>
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public class Zipf : IDiscreteDistribution
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{
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System.Random _random;
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/// <summary>
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/// The s parameter of the distribution.
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/// </summary>
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readonly double _s;
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/// <summary>
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/// The n parameter of the distribution.
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/// </summary>
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readonly int _n;
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/// <summary>
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/// Initializes a new instance of the <see cref="Zipf"/> class.
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/// </summary>
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/// <param name="s">The s parameter of the distribution.</param>
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/// <param name="n">The n parameter of the distribution.</param>
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public Zipf(double s, int n)
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{
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if (!IsValidParameterSet(s, n))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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_random = SystemRandomSource.Default;
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_s = s;
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_n = n;
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}
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/// <summary>
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/// Initializes a new instance of the <see cref="Zipf"/> class.
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/// </summary>
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/// <param name="s">The s parameter of the distribution.</param>
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/// <param name="n">The n parameter of the distribution.</param>
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/// <param name="randomSource">The random number generator which is used to draw random samples.</param>
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public Zipf(double s, int n, System.Random randomSource)
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{
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if (!IsValidParameterSet(s, n))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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_random = randomSource ?? SystemRandomSource.Default;
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_s = s;
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_n = n;
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}
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/// <summary>
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/// A string representation of the distribution.
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/// </summary>
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/// <returns>a string representation of the distribution.</returns>
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public override string ToString()
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{
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return $"Zipf(S = {_s}, N = {_n})";
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}
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/// <summary>
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/// Tests whether the provided values are valid parameters for this distribution.
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/// </summary>
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/// <param name="s">The s parameter of the distribution.</param>
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/// <param name="n">The n parameter of the distribution.</param>
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public static bool IsValidParameterSet(double s, int n)
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{
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return n > 0 && s > 0.0;
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}
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/// <summary>
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/// Gets or sets the s parameter of the distribution.
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/// </summary>
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public double S => _s;
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/// <summary>
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/// Gets or sets the n parameter of the distribution.
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/// </summary>
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public int N => _n;
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/// <summary>
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/// Gets or sets the random number generator which is used to draw random samples.
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/// </summary>
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public System.Random RandomSource
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{
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get => _random;
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set => _random = value ?? SystemRandomSource.Default;
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}
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/// <summary>
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/// Gets the mean of the distribution.
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/// </summary>
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public double Mean => SpecialFunctions.GeneralHarmonic(_n, _s - 1.0)/SpecialFunctions.GeneralHarmonic(_n, _s);
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/// <summary>
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/// Gets the variance of the distribution.
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/// </summary>
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public double Variance
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{
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get
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{
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if (_s <= 3)
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{
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throw new NotSupportedException();
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}
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var ghns = SpecialFunctions.GeneralHarmonic(_n, _s);
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return (SpecialFunctions.GeneralHarmonic(_n, _s - 2)*SpecialFunctions.GeneralHarmonic(_n, _s))
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- (Math.Pow(SpecialFunctions.GeneralHarmonic(_n, _s - 1), 2)/(ghns*ghns));
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}
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}
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/// <summary>
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/// Gets the standard deviation of the distribution.
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/// </summary>
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public double StdDev => Math.Sqrt(Variance);
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/// <summary>
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/// Gets the entropy of the distribution.
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/// </summary>
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public double Entropy
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{
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get
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{
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double sum = 0;
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for (var i = 0; i < _n; i++)
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{
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sum += Math.Log(i + 1)/Math.Pow(i + 1, _s);
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}
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return ((_s/SpecialFunctions.GeneralHarmonic(_n, _s))*sum) + Math.Log(SpecialFunctions.GeneralHarmonic(_n, _s));
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}
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}
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/// <summary>
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/// Gets the skewness of the distribution.
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/// </summary>
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public double Skewness
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{
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get
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{
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if (_s <= 4)
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{
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throw new NotSupportedException();
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}
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return ((SpecialFunctions.GeneralHarmonic(_n, _s - 3)*Math.Pow(SpecialFunctions.GeneralHarmonic(_n, _s), 2)) - (SpecialFunctions.GeneralHarmonic(_n, _s - 1)*((3*SpecialFunctions.GeneralHarmonic(_n, _s - 2)*SpecialFunctions.GeneralHarmonic(_n, _s)) - Math.Pow(SpecialFunctions.GeneralHarmonic(_n, _s - 1), 2))))/Math.Pow((SpecialFunctions.GeneralHarmonic(_n, _s - 2)*SpecialFunctions.GeneralHarmonic(_n, _s)) - Math.Pow(SpecialFunctions.GeneralHarmonic(_n, _s - 1), 2), 1.5);
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}
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}
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/// <summary>
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/// Gets the mode of the distribution.
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/// </summary>
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public int Mode => 1;
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/// <summary>
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/// Gets the median of the distribution.
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/// </summary>
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public double Median => throw new NotSupportedException();
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/// <summary>
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/// Gets the smallest element in the domain of the distributions which can be represented by an integer.
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/// </summary>
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public int Minimum => 1;
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/// <summary>
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/// Gets the largest element in the domain of the distributions which can be represented by an integer.
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/// </summary>
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public int Maximum => _n;
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/// <summary>
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/// Computes the probability mass (PMF) at k, i.e. P(X = k).
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/// </summary>
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/// <param name="k">The location in the domain where we want to evaluate the probability mass function.</param>
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/// <returns>the probability mass at location <paramref name="k"/>.</returns>
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public double Probability(int k)
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{
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return (1.0/Math.Pow(k, _s))/SpecialFunctions.GeneralHarmonic(_n, _s);
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}
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/// <summary>
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/// Computes the log probability mass (lnPMF) at k, i.e. ln(P(X = k)).
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/// </summary>
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/// <param name="k">The location in the domain where we want to evaluate the log probability mass function.</param>
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/// <returns>the log probability mass at location <paramref name="k"/>.</returns>
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public double ProbabilityLn(int k)
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{
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return Math.Log(Probability(k));
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}
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/// <summary>
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/// Computes the cumulative distribution (CDF) of the distribution at x, i.e. P(X ≤ x).
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/// </summary>
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/// <param name="x">The location at which to compute the cumulative distribution function.</param>
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/// <returns>the cumulative distribution at location <paramref name="x"/>.</returns>
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public double CumulativeDistribution(double x)
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{
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if (x < 1)
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{
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return 0.0;
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}
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return SpecialFunctions.GeneralHarmonic((int)x, _s)/SpecialFunctions.GeneralHarmonic(_n, _s);
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}
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/// <summary>
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/// Computes the probability mass (PMF) at k, i.e. P(X = k).
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/// </summary>
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/// <param name="k">The location in the domain where we want to evaluate the probability mass function.</param>
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/// <param name="s">The s parameter of the distribution.</param>
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/// <param name="n">The n parameter of the distribution.</param>
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/// <returns>the probability mass at location <paramref name="k"/>.</returns>
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public static double PMF(double s, int n, int k)
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{
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if (!(n > 0 && s > 0.0))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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return (1.0/Math.Pow(k, s))/SpecialFunctions.GeneralHarmonic(n, s);
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}
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/// <summary>
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/// Computes the log probability mass (lnPMF) at k, i.e. ln(P(X = k)).
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/// </summary>
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/// <param name="k">The location in the domain where we want to evaluate the log probability mass function.</param>
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/// <param name="s">The s parameter of the distribution.</param>
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/// <param name="n">The n parameter of the distribution.</param>
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/// <returns>the log probability mass at location <paramref name="k"/>.</returns>
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public static double PMFLn(double s, int n, int k)
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{
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if (!(n > 0 && s > 0.0))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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return Math.Log(PMF(s, n, k));
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}
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/// <summary>
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/// Computes the cumulative distribution (CDF) of the distribution at x, i.e. P(X ≤ x).
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/// </summary>
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/// <param name="x">The location at which to compute the cumulative distribution function.</param>
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/// <param name="s">The s parameter of the distribution.</param>
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/// <param name="n">The n parameter of the distribution.</param>
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/// <returns>the cumulative distribution at location <paramref name="x"/>.</returns>
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/// <seealso cref="CumulativeDistribution"/>
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public static double CDF(double s, int n, double x)
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{
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if (!(n > 0 && s > 0.0))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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if (x < 1)
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{
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return 0.0;
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}
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return SpecialFunctions.GeneralHarmonic((int)x, s)/SpecialFunctions.GeneralHarmonic(n, s);
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}
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/// <summary>
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/// Generates a sample from the Zipf distribution without doing parameter checking.
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/// </summary>
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/// <param name="rnd">The random number generator to use.</param>
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/// <param name="s">The s parameter of the distribution.</param>
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/// <param name="n">The n parameter of the distribution.</param>
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/// <returns>a random number from the Zipf distribution.</returns>
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static int SampleUnchecked(System.Random rnd, double s, int n)
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{
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var r = 0.0;
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while (r == 0.0)
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{
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r = rnd.NextDouble();
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}
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var p = 1.0/SpecialFunctions.GeneralHarmonic(n, s);
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int i;
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var sum = 0.0;
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for (i = 1; i <= n; i++)
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{
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sum += p/Math.Pow(i, s);
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if (sum >= r)
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{
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break;
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}
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}
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return i;
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}
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static void SamplesUnchecked(System.Random rnd, int[] values, double s, int n)
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{
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for (int i = 0; i < values.Length; i++)
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{
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values[i] = SampleUnchecked(rnd, s, n);
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}
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}
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static IEnumerable<int> SamplesUnchecked(System.Random rnd, double s, int n)
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{
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while (true)
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{
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yield return SampleUnchecked(rnd, s, n);
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}
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}
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/// <summary>
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/// Draws a random sample from the distribution.
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/// </summary>
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/// <returns>a sample from the distribution.</returns>
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public int Sample()
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{
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return SampleUnchecked(_random, _s, _n);
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}
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/// <summary>
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/// Fills an array with samples generated from the distribution.
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/// </summary>
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public void Samples(int[] values)
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{
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SamplesUnchecked(_random, values, _s, _n);
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}
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/// <summary>
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/// Samples an array of zipf distributed random variables.
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/// </summary>
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/// <returns>a sequence of samples from the distribution.</returns>
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public IEnumerable<int> Samples()
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{
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return SamplesUnchecked(_random, _s, _n);
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}
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/// <summary>
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/// Samples a random variable.
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/// </summary>
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/// <param name="rnd">The random number generator to use.</param>
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/// <param name="s">The s parameter of the distribution.</param>
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/// <param name="n">The n parameter of the distribution.</param>
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public static int Sample(System.Random rnd, double s, int n)
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{
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if (!(n > 0 && s > 0.0))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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return SampleUnchecked(rnd, s, n);
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}
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/// <summary>
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/// Samples a sequence of this random variable.
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/// </summary>
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/// <param name="rnd">The random number generator to use.</param>
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/// <param name="s">The s parameter of the distribution.</param>
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/// <param name="n">The n parameter of the distribution.</param>
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public static IEnumerable<int> Samples(System.Random rnd, double s, int n)
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{
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if (!(n > 0 && s > 0.0))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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return SamplesUnchecked(rnd, s, n);
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}
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/// <summary>
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/// Fills an array with samples generated from the distribution.
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/// </summary>
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/// <param name="rnd">The random number generator to use.</param>
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/// <param name="values">The array to fill with the samples.</param>
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/// <param name="s">The s parameter of the distribution.</param>
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/// <param name="n">The n parameter of the distribution.</param>
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public static void Samples(System.Random rnd, int[] values, double s, int n)
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{
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if (!(n > 0 && s > 0.0))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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SamplesUnchecked(rnd, values, s, n);
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}
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/// <summary>
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/// Samples a random variable.
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/// </summary>
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/// <param name="s">The s parameter of the distribution.</param>
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/// <param name="n">The n parameter of the distribution.</param>
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public static int Sample(double s, int n)
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{
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if (!(n > 0 && s > 0.0))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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return SampleUnchecked(SystemRandomSource.Default, s, n);
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}
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/// <summary>
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/// Samples a sequence of this random variable.
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/// </summary>
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/// <param name="s">The s parameter of the distribution.</param>
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/// <param name="n">The n parameter of the distribution.</param>
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public static IEnumerable<int> Samples(double s, int n)
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{
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if (!(n > 0 && s > 0.0))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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return SamplesUnchecked(SystemRandomSource.Default, s, n);
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}
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/// <summary>
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/// Fills an array with samples generated from the distribution.
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/// </summary>
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/// <param name="values">The array to fill with the samples.</param>
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/// <param name="s">The s parameter of the distribution.</param>
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/// <param name="n">The n parameter of the distribution.</param>
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public static void Samples(int[] values, double s, int n)
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{
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if (!(n > 0 && s > 0.0))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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SamplesUnchecked(SystemRandomSource.Default, values, s, n);
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}
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}
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}
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