// <copyright file="BetaBinomial.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) 2009-2020 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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// <contribution>
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// Andrew J. Willshire
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// </contribution>
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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 Beta-Binomial distribution.
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/// The beta-binomial distribution is a family of discrete probability distributions on a finite support of non-negative integers arising
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/// when the probability of success in each of a fixed or known number of Bernoulli trials is either unknown or random.
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/// The beta-binomial distribution is the binomial distribution in which the probability of success at each of n trials is not fixed but randomly drawn from a beta distribution.
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/// It is frequently used in Bayesian statistics, empirical Bayes methods and classical statistics to capture overdispersion in binomial type distributed data.
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/// <a href="https://en.wikipedia.org/wiki/Beta-binomial_distribution">Wikipedia - Beta-Binomial distribution</a>.
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/// </summary>
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public class BetaBinomial : IDiscreteDistribution
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{
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System.Random _random;
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readonly int _n;
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readonly double _a;
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readonly double _b;
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/// <summary>
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/// Initializes a new instance of the <see cref="BetaBinomial"/> class.
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/// </summary>
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/// <param name="n">The number of Bernoulli trials n - n is a positive integer </param>
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/// <param name="a">Shape parameter alpha of the Beta distribution. Range: a > 0.</param>
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/// <param name="b">Shape parameter beta of the Beta distribution. Range: b > 0.</param>
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public BetaBinomial(int n, double a, double b)
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{
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if (!IsValidParameterSet(n, a, b))
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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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_n = n;
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_a = a;
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_b = b;
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}
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/// <summary>
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/// Initializes a new instance of the <see cref="BetaBinomial"/> class.
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/// </summary>
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/// <param name="n">The number of Bernoulli trials n - n is a positive integer </param>
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/// <param name="a">Shape parameter alpha of the Beta distribution. Range: a > 0.</param>
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/// <param name="b">Shape parameter beta of the Beta distribution. Range: b > 0.</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 BetaBinomial(int n, double a, double b, System.Random randomSource)
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{
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if (!IsValidParameterSet(n,a,b))
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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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_n = n;
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_a = a;
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_b = b;
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}
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/// <summary>
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/// Returns a <see cref="System.String"/> that represents this instance.
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/// </summary>
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/// <returns>
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/// A <see cref="System.String"/> that represents this instance.
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/// </returns>
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public override string ToString()
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{
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return $"BetaBinomial(n = {_n}, a = {_a}, b = {_b})";
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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="n">The number of Bernoulli trials n - n is a positive integer </param>
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/// <param name="a">Shape parameter alpha of the Beta distribution. Range: a > 0.</param>
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/// <param name="b">Shape parameter beta of the Beta distribution. Range: b > 0.</param>
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public static bool IsValidParameterSet(int n, double a, double b)
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{
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return n >= 1.0 && a > 0.0 && b > 0.0;
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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="n">The number of Bernoulli trials n - n is a positive integer </param>
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/// <param name="a">Shape parameter alpha of the Beta distribution. Range: a > 0.</param>
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/// <param name="b">Shape parameter beta of the Beta distribution. Range: b > 0.</param>
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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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public static bool IsValidParameterSet(int n, double a, double b, int k)
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{
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return n >= 1.0 && a > 0.0 && b > 0.0 && k >=0 && k <=n;
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}
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public int N => _n;
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public double A => _a;
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public double B => _b;
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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 => (_n * _a) / (_a + _b);
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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 => (_n*_a*_b*(_a+_b+_n))/(Math.Pow((_a+_b),2) * (_a+_b+1));
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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((_n * _a * _b * (_a + _b + _n)) / (Math.Pow((_a + _b), 2) * (_a + _b + 1)));
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/// <summary>
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/// Gets the entropy of the distribution.
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/// </summary>
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double IUnivariateDistribution.Entropy => throw new NotSupportedException();
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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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(_a + _b + 2 * _n) * (_b - _a) / (_a + _b + 2) * Math.Sqrt((1 + _a + _b) / (_n * _a * _b * (_n + _a + _b)));
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/// <summary>
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/// Gets the mode of the distribution
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/// </summary>
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int IDiscreteDistribution.Mode => throw new NotSupportedException();
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/// <summary>
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/// Gets the median of the distribution.
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/// </summary>
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double IUnivariateDistribution.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 => 0;
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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 => int.MaxValue;
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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 PMF(_n, _a, _b, k);
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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 PMFLn(_n, _a, _b, 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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return CDF(_n, _a, _b, (int)Math.Floor(x));
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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="n">The number of Bernoulli trials n - n is a positive integer </param>
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/// <param name="a">Shape parameter alpha of the Beta distribution. Range: a > 0.</param>
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/// <param name="b">Shape parameter beta of the Beta distribution. Range: b > 0.</param>
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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 static double PMF(int n, double a, double b, int k)
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{
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if (!IsValidParameterSet(n, a, b, k))
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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 (k > n)
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{
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return 0.0;
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}
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else
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{
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return Math.Exp(PMFLn(n, a, b, k));
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}
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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="n">The number of Bernoulli trials n - n is a positive integer </param>
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/// <param name="a">Shape parameter alpha of the Beta distribution. Range: a > 0.</param>
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/// <param name="b">Shape parameter beta of the Beta distribution. Range: b > 0.</param>
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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 log probability mass at location <paramref name="k"/>.</returns>
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public static double PMFLn(int n, double a, double b, int k)
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{
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if (!IsValidParameterSet(n, a, b, k))
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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 SpecialFunctions.BinomialLn((n), k)
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+ SpecialFunctions.BetaLn(k + a, n - k + b)
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- SpecialFunctions.BetaLn(a, b);
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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="n">The number of Bernoulli trials n - n is a positive integer </param>
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/// <param name="a">Shape parameter alpha of the Beta distribution. Range: a > 0.</param>
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/// <param name="b">Shape parameter beta of the Beta distribution. Range: b > 0.</param>
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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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/// <seealso cref="CumulativeDistribution"/>
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public static double CDF(int n, double a, double b, int x)
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{
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if (!IsValidParameterSet(n,a,b,x))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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double accumulator = 0;
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for (int i = 0; i<=x; i++)
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{
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accumulator += PMF(n, a, b, i);
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}
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return accumulator;
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}
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/// <summary>
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/// Samples BetaBinomial distributed random variables by sampling a Beta distribution then passing to a Binomial 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="a">The α shape parameter of the Beta distribution. Range: α ≥ 0.</param>
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/// <param name="b">The β shape parameter of the Beta distribution. Range: β ≥ 0.</param>
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/// <param name="n">The number of trials (n). Range: n ≥ 0.</param>
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/// <returns>a random number from the BetaBinomial distribution.</returns>
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static int SampleUnchecked(System.Random rnd, int n, double a, double b)
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{
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var p = Beta.SampleUnchecked(rnd, a, b);
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var x = Binomial.SampleUnchecked(rnd, p, n);
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return x;
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}
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static void SamplesUnchecked(System.Random rnd, int[] values, int n, double a, double b)
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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, n, a, b);
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}
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}
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static IEnumerable<int> SamplesUnchecked(System.Random rnd, int n, double a, double b)
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{
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while (true)
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{
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yield return SampleUnchecked(rnd, n, a, b);
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}
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}
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/// <summary>
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/// Samples a <c>BetaBinomial</c> distributed random variable.
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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, _n, _a, _b);
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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, _n, _a, _b);
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}
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/// <summary>
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/// Samples an array of <c>BetaBinomial</c> 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, _n, _a, _b);
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}
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/// <summary>
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/// Samples a <c>BetaBinomial</c> distributed 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="a">The α shape parameter of the Beta distribution. Range: α ≥ 0.</param>
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/// <param name="b">The β shape parameter of the Beta distribution. Range: β ≥ 0.</param>
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/// <param name="n">The number of trials (n). Range: n ≥ 0.</param>
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/// <returns>a sample from the distribution.</returns>
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public int Sample(System.Random rnd, int n, double a, double b)
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{
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if (!IsValidParameterSet(n,a,b))
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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, n, a, b);
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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="a">The α shape parameter of the Beta distribution. Range: α ≥ 0.</param>
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/// <param name="b">The β shape parameter of the Beta distribution. Range: β ≥ 0.</param>
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/// <param name="n">The number of trials (n). Range: n ≥ 0.</param>
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public void Samples(System.Random rnd, int[] values, int n, double a, double b)
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{
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if (!IsValidParameterSet(n, a, b))
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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, n, a, b);
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}
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/// <summary>
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/// Samples an array of <c>BetaBinomial</c> distributed random variables.
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/// </summary>
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/// <param name="a">The α shape parameter of the Beta distribution. Range: α ≥ 0.</param>
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/// <param name="b">The β shape parameter of the Beta distribution. Range: β ≥ 0.</param>
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/// <param name="n">The number of trials (n). Range: n ≥ 0.</param>
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/// <returns>a sequence of samples from the distribution.</returns>
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public IEnumerable<int> Samples(int n, double a, double b)
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{
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if (!IsValidParameterSet(n, a, b))
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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(_random, n, a, b);
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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="a">The α shape parameter of the Beta distribution. Range: α ≥ 0.</param>
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/// <param name="b">The β shape parameter of the Beta distribution. Range: β ≥ 0.</param>
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/// <param name="n">The number of trials (n). Range: n ≥ 0.</param>
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public void Samples(int[] values, int n, double a, double b)
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{
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if (!IsValidParameterSet(n, a, b))
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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(_random, values, n, a, b);
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}
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}
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}
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