// <copyright file="Binomial.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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using IStation.Numerics.Threading;
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namespace IStation.Numerics.Distributions
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{
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/// <summary>
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/// Discrete Univariate Binomial distribution.
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/// For details about this distribution, see
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/// <a href="http://en.wikipedia.org/wiki/Binomial_distribution">Wikipedia - Binomial distribution</a>.
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/// </summary>
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/// <remarks>
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/// The distribution is parameterized by a probability (between 0.0 and 1.0).
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/// </remarks>
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public class Binomial : IDiscreteDistribution
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{
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System.Random _random;
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readonly double _p;
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readonly int _trials;
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/// <summary>
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/// Initializes a new instance of the Binomial class.
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/// </summary>
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/// <param name="p">The success probability (p) in each trial. Range: 0 ≤ p ≤ 1.</param>
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/// <param name="n">The number of trials (n). Range: n ≥ 0.</param>
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/// <exception cref="ArgumentOutOfRangeException">If <paramref name="p"/> is not in the interval [0.0,1.0].</exception>
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/// <exception cref="ArgumentOutOfRangeException">If <paramref name="n"/> is negative.</exception>
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public Binomial(double p, int n)
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{
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if (!IsValidParameterSet(p, 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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_p = p;
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_trials = n;
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}
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/// <summary>
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/// Initializes a new instance of the Binomial class.
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/// </summary>
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/// <param name="p">The success probability (p) in each trial. Range: 0 ≤ p ≤ 1.</param>
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/// <param name="n">The number of trials (n). Range: n ≥ 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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/// <exception cref="ArgumentOutOfRangeException">If <paramref name="p"/> is not in the interval [0.0,1.0].</exception>
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/// <exception cref="ArgumentOutOfRangeException">If <paramref name="n"/> is negative.</exception>
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public Binomial(double p, int n, System.Random randomSource)
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{
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if (!IsValidParameterSet(p, 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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_p = p;
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_trials = 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 $"Binomial(p = {_p}, n = {_trials})";
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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="p">The success probability (p) in each trial. Range: 0 ≤ p ≤ 1.</param>
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/// <param name="n">The number of trials (n). Range: n ≥ 0.</param>
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public static bool IsValidParameterSet(double p, int n)
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{
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return p >= 0.0 && p <= 1.0 && n >= 0;
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}
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/// <summary>
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/// Gets the success probability in each trial. Range: 0 ≤ p ≤ 1.
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/// </summary>
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public double P => _p;
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/// <summary>
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/// Gets the number of trials. Range: n ≥ 0.
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/// </summary>
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public int N => _trials;
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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 => _p*_trials;
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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(_p*(1.0 - _p)*_trials);
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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 => _p*(1.0 - _p)*_trials;
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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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if (_p == 0.0 || _p == 1.0)
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{
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return 0.0;
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}
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var e = 0.0;
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for (var i = 0; i <= _trials; i++)
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{
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var p = Probability(i);
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e -= p*Math.Log(p);
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}
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return e;
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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 => (1.0 - (2.0*_p))/Math.Sqrt(_trials*_p*(1.0 - _p));
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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 => _trials;
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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
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{
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get
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{
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if (_p == 1.0)
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{
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return _trials;
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}
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if (_p == 0.0)
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{
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return 0;
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}
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return (int)Math.Floor((_trials + 1)*_p);
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}
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}
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/// <summary>
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/// Gets all modes of the distribution.
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/// </summary>
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public int[] Modes
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{
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get
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{
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if (_p == 1.0)
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{
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return new[] { _trials };
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}
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if (_p == 0.0)
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{
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return new[] { 0 };
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}
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double td = (_trials + 1)*_p;
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int t = (int)Math.Floor(td);
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return t != td ? new[] { t } : new[] { t, t - 1 };
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}
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}
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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 => Math.Floor(_p*_trials);
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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(_p, _trials, 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(_p, _trials, 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(_p, _trials, 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="k">The location in the domain where we want to evaluate the probability mass function.</param>
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/// <param name="p">The success probability (p) in each trial. Range: 0 ≤ p ≤ 1.</param>
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/// <param name="n">The number of trials (n). Range: n ≥ 0.</param>
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/// <returns>the probability mass at location <paramref name="k"/>.</returns>
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public static double PMF(double p, int n, int k)
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{
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if (!(p >= 0.0 && p <= 1.0 && n >= 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 (k < 0 || k > n)
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{
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return 0.0;
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}
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if (p == 0.0)
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{
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return k == 0 ? 1.0 : 0.0;
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}
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if (p == 1.0)
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{
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return k == n ? 1.0 : 0.0;
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}
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return Math.Exp(SpecialFunctions.BinomialLn(n, k) + (k*Math.Log(p)) + ((n - k)*Math.Log(1.0 - p)));
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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="p">The success probability (p) in each trial. Range: 0 ≤ p ≤ 1.</param>
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/// <param name="n">The number of trials (n). Range: n ≥ 0.</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 p, int n, int k)
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{
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if (!(p >= 0.0 && p <= 1.0 && n >= 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 (k < 0 || k > n)
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{
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return double.NegativeInfinity;
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}
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if (p == 0.0)
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{
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return k == 0 ? 0.0 : double.NegativeInfinity;
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}
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if (p == 1.0)
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{
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return k == n ? 0.0 : double.NegativeInfinity;
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}
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return SpecialFunctions.BinomialLn(n, k) + (k*Math.Log(p)) + ((n - k)*Math.Log(1.0 - p));
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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="p">The success probability (p) in each trial. Range: 0 ≤ p ≤ 1.</param>
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/// <param name="n">The number of trials (n). Range: n ≥ 0.</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 p, int n, double x)
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{
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if (!(p >= 0.0 && p <= 1.0 && n >= 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 < 0.0)
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{
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return 0.0;
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}
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if (x > n)
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{
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return 1.0;
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}
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double k = Math.Floor(x);
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return SpecialFunctions.BetaRegularized(n - k, k + 1, 1 - p);
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}
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/// <summary>
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/// Generates a sample from the Binomial 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="p">The success probability (p) in each trial. Range: 0 ≤ p ≤ 1.</param>
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/// <param name="n">The number of trials (n). Range: n ≥ 0.</param>
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/// <returns>The number of successful trials.</returns>
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internal static int SampleUnchecked(System.Random rnd, double p, int n)
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{
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var k = 0;
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for (var i = 0; i < n; i++)
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{
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k += rnd.NextDouble() < p ? 1 : 0;
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}
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return k;
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}
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static void SamplesUnchecked(System.Random rnd, int[] values, double p, int n)
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{
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var uniform = rnd.NextDoubles(values.Length*n);
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CommonParallel.For(0, values.Length, 4096, (a, b) =>
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{
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for (int i = a; i < b; i++)
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{
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int k = i*n;
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int sum = 0;
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for (int j = 0; j < n; j++)
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{
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sum += uniform[k + j] < p ? 1 : 0;
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}
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values[i] = sum;
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}
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});
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}
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static IEnumerable<int> SamplesUnchecked(System.Random rnd, double p, int n)
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{
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while (true)
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{
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yield return SampleUnchecked(rnd, p, n);
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}
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}
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/// <summary>
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/// Samples a Binomially distributed random variable.
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/// </summary>
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/// <returns>The number of successes in N trials.</returns>
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public int Sample()
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{
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return SampleUnchecked(_random, _p, _trials);
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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, _p, _trials);
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}
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/// <summary>
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/// Samples an array of Binomially distributed random variables.
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/// </summary>
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/// <returns>a sequence of successes in N trials.</returns>
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public IEnumerable<int> Samples()
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{
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return SamplesUnchecked(_random, _p, _trials);
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}
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/// <summary>
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/// Samples a binomially 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="p">The success probability (p) in each trial. Range: 0 ≤ p ≤ 1.</param>
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/// <param name="n">The number of trials (n). Range: n ≥ 0.</param>
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/// <returns>The number of successes in <paramref name="n"/> trials.</returns>
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public static int Sample(System.Random rnd, double p, int n)
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{
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if (!(p >= 0.0 && p <= 1.0 && n >= 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, p, n);
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}
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/// <summary>
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/// Samples a sequence of binomially 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="p">The success probability (p) in each trial. Range: 0 ≤ p ≤ 1.</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 successes in <paramref name="n"/> trials.</returns>
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public static IEnumerable<int> Samples(System.Random rnd, double p, int n)
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{
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if (!(p >= 0.0 && p <= 1.0 && n >= 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, p, 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="p">The success probability (p) in each trial. Range: 0 ≤ p ≤ 1.</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 successes in <paramref name="n"/> trials.</returns>
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public static void Samples(System.Random rnd, int[] values, double p, int n)
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{
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if (!(p >= 0.0 && p <= 1.0 && n >= 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, p, n);
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}
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/// <summary>
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/// Samples a binomially distributed random variable.
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/// </summary>
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/// <param name="p">The success probability (p) in each trial. Range: 0 ≤ p ≤ 1.</param>
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/// <param name="n">The number of trials (n). Range: n ≥ 0.</param>
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/// <returns>The number of successes in <paramref name="n"/> trials.</returns>
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public static int Sample(double p, int n)
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{
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if (!(p >= 0.0 && p <= 1.0 && n >= 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, p, n);
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}
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/// <summary>
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/// Samples a sequence of binomially distributed random variable.
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/// </summary>
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/// <param name="p">The success probability (p) in each trial. Range: 0 ≤ p ≤ 1.</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 successes in <paramref name="n"/> trials.</returns>
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public static IEnumerable<int> Samples(double p, int n)
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{
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if (!(p >= 0.0 && p <= 1.0 && n >= 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, p, 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="p">The success probability (p) in each trial. Range: 0 ≤ p ≤ 1.</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 successes in <paramref name="n"/> trials.</returns>
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public static void Samples(int[] values, double p, int n)
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{
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if (!(p >= 0.0 && p <= 1.0 && n >= 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, p, n);
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}
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}
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}
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