// <copyright file="Poisson.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 Poisson distribution.
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/// </summary>
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/// <remarks>
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/// <para>Distribution is described at <a href="http://en.wikipedia.org/wiki/Poisson_distribution"> Wikipedia - Poisson distribution</a>.</para>
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/// <para>Knuth's method is used to generate Poisson distributed random variables.</para>
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/// <para>f(x) = exp(-λ)*λ^x/x!;</para>
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/// </remarks>
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public class Poisson : IDiscreteDistribution
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{
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System.Random _random;
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readonly double _lambda;
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/// <summary>
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/// Initializes a new instance of the <see cref="Poisson"/> class.
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/// </summary>
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/// <param name="lambda">The lambda (λ) parameter of the Poisson distribution. Range: λ > 0.</param>
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/// <exception cref="System.ArgumentOutOfRangeException">If <paramref name="lambda"/> is equal or less then 0.0.</exception>
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public Poisson(double lambda)
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{
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if (!IsValidParameterSet(lambda))
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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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_lambda = lambda;
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}
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/// <summary>
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/// Initializes a new instance of the <see cref="Poisson"/> class.
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/// </summary>
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/// <param name="lambda">The lambda (λ) parameter of the Poisson distribution. Range: λ > 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="System.ArgumentOutOfRangeException">If <paramref name="lambda"/> is equal or less then 0.0.</exception>
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public Poisson(double lambda, System.Random randomSource)
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{
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if (!IsValidParameterSet(lambda))
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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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_lambda = lambda;
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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 $"Poisson(λ = {_lambda})";
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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="lambda">The lambda (λ) parameter of the Poisson distribution. Range: λ > 0.</param>
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public static bool IsValidParameterSet(double lambda)
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{
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return lambda > 0.0;
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}
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/// <summary>
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/// Gets the Poisson distribution parameter λ. Range: λ > 0.
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/// </summary>
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public double Lambda => _lambda;
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/// <summary>
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/// Gets 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 => _lambda;
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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 => _lambda;
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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(_lambda);
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/// <summary>
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/// Gets the entropy of the distribution.
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/// </summary>
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/// <remarks>Approximation, see Wikipedia <a href="http://en.wikipedia.org/wiki/Poisson_distribution">Poisson distribution</a></remarks>
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public double Entropy => (0.5*Math.Log(2*Constants.Pi*Constants.E*_lambda)) - (1.0/(12.0*_lambda)) - (1.0/(24.0*_lambda*_lambda)) - (19.0/(360.0*_lambda*_lambda*_lambda));
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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/Math.Sqrt(_lambda);
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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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/// Gets the mode of the distribution.
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/// </summary>
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public int Mode => (int)Math.Floor(_lambda);
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/// <summary>
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/// Gets the median of the distribution.
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/// </summary>
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/// <remarks>Approximation, see Wikipedia <a href="http://en.wikipedia.org/wiki/Poisson_distribution">Poisson distribution</a></remarks>
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public double Median => Math.Floor(_lambda + (1.0/3.0) - (0.02/_lambda));
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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 Math.Exp(-_lambda + (k*Math.Log(_lambda)) - SpecialFunctions.FactorialLn(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 -_lambda + (k*Math.Log(_lambda)) - SpecialFunctions.FactorialLn(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 1.0 - SpecialFunctions.GammaLowerRegularized(x + 1, _lambda);
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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="lambda">The lambda (λ) parameter of the Poisson distribution. Range: λ > 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 lambda, int k)
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{
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if (!(lambda > 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.Exp(-lambda + (k*Math.Log(lambda)) - SpecialFunctions.FactorialLn(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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/// <param name="lambda">The lambda (λ) parameter of the Poisson distribution. Range: λ > 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 lambda, int k)
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{
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if (!(lambda > 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 -lambda + (k*Math.Log(lambda)) - SpecialFunctions.FactorialLn(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="lambda">The lambda (λ) parameter of the Poisson distribution. Range: λ > 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 lambda, double x)
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{
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if (!(lambda > 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 - SpecialFunctions.GammaLowerRegularized(x + 1, lambda);
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}
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/// <summary>
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/// Generates one sample from the Poisson distribution.
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/// </summary>
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/// <param name="rnd">The random source to use.</param>
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/// <param name="lambda">The lambda (λ) parameter of the Poisson distribution. Range: λ > 0.</param>
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/// <returns>A random sample from the Poisson distribution.</returns>
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static int SampleUnchecked(System.Random rnd, double lambda)
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{
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return (lambda < 30.0) ? DoSampleShort(rnd, lambda) : DoSampleLarge(rnd, lambda);
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}
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static void SamplesUnchecked(System.Random rnd, int[] values, double lambda)
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{
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if (lambda < 30.0)
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{
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var limit = Math.Exp(-lambda);
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for (int i = 0; i < values.Length; i++)
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{
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var count = 0;
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for (var product = rnd.NextDouble(); product >= limit; product *= rnd.NextDouble())
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{
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count++;
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}
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values[i] = count;
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}
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}
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else
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{
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var c = 0.767 - (3.36/lambda);
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var beta = Math.PI/Math.Sqrt(3.0*lambda);
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var alpha = beta*lambda;
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var k = Math.Log(c) - lambda - Math.Log(beta);
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for (int i = 0; i < values.Length; i++)
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{
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for (;;)
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{
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var u = rnd.NextDouble();
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var x = (alpha - Math.Log((1.0 - u)/u))/beta;
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var n = (int)Math.Floor(x + 0.5);
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if (n < 0)
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{
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continue;
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}
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var v = rnd.NextDouble();
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var y = alpha - (beta*x);
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var temp = 1.0 + Math.Exp(y);
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var lhs = y + Math.Log(v/(temp*temp));
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var rhs = k + (n*Math.Log(lambda)) - SpecialFunctions.FactorialLn(n);
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if (lhs <= rhs)
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{
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values[i] = n;
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break;
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}
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}
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}
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}
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}
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static IEnumerable<int> SamplesUnchecked(System.Random rnd, double lambda)
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{
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if (lambda < 30.0)
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{
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while (true)
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{
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yield return DoSampleShort(rnd, lambda);
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}
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}
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else
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{
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while (true)
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{
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yield return DoSampleLarge(rnd, lambda);
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}
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}
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}
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/// <summary>
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/// Generates one sample from the Poisson distribution by Knuth's method.
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/// </summary>
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/// <param name="rnd">The random source to use.</param>
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/// <param name="lambda">The lambda (λ) parameter of the Poisson distribution. Range: λ > 0.</param>
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/// <returns>A random sample from the Poisson distribution.</returns>
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static int DoSampleShort(System.Random rnd, double lambda)
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{
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var limit = Math.Exp(-lambda);
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var count = 0;
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for (var product = rnd.NextDouble(); product >= limit; product *= rnd.NextDouble())
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{
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count++;
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}
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return count;
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}
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/// <summary>
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/// Generates one sample from the Poisson distribution by "Rejection method PA".
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/// </summary>
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/// <param name="rnd">The random source to use.</param>
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/// <param name="lambda">The lambda (λ) parameter of the Poisson distribution. Range: λ > 0.</param>
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/// <returns>A random sample from the Poisson distribution.</returns>
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/// <remarks>"Rejection method PA" from "The Computer Generation of Poisson Random Variables" by A. C. Atkinson,
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/// Journal of the Royal Statistical Society Series C (Applied Statistics) Vol. 28, No. 1. (1979)
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/// The article is on pages 29-35. The algorithm given here is on page 32. </remarks>
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static int DoSampleLarge(System.Random rnd, double lambda)
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{
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var c = 0.767 - (3.36/lambda);
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var beta = Math.PI/Math.Sqrt(3.0*lambda);
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var alpha = beta*lambda;
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var k = Math.Log(c) - lambda - Math.Log(beta);
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for (;;)
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{
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var u = rnd.NextDouble();
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var x = (alpha - Math.Log((1.0 - u)/u))/beta;
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var n = (int)Math.Floor(x + 0.5);
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if (n < 0)
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{
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continue;
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}
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var v = rnd.NextDouble();
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var y = alpha - (beta*x);
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var temp = 1.0 + Math.Exp(y);
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var lhs = y + Math.Log(v/(temp*temp));
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var rhs = k + (n*Math.Log(lambda)) - SpecialFunctions.FactorialLn(n);
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if (lhs <= rhs)
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{
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return n;
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}
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}
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}
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/// <summary>
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/// Samples a Poisson distributed random variable.
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/// </summary>
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/// <returns>A sample from the Poisson distribution.</returns>
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public int Sample()
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{
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return SampleUnchecked(_random, _lambda);
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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, _lambda);
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}
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/// <summary>
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/// Samples an array of Poisson 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, _lambda);
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}
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/// <summary>
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/// Samples a Poisson 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="lambda">The lambda (λ) parameter of the Poisson distribution. Range: λ > 0.</param>
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/// <returns>A sample from the Poisson distribution.</returns>
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public static int Sample(System.Random rnd, double lambda)
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{
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if (!(lambda > 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, lambda);
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}
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/// <summary>
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/// Samples a sequence of Poisson distributed random variables.
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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="lambda">The lambda (λ) parameter of the Poisson distribution. Range: λ > 0.</param>
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/// <returns>a sequence of samples from the distribution.</returns>
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public static IEnumerable<int> Samples(System.Random rnd, double lambda)
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{
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if (!(lambda > 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, lambda);
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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="lambda">The lambda (λ) parameter of the Poisson distribution. Range: λ > 0.</param>
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/// <returns>a sequence of samples from the distribution.</returns>
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public static void Samples(System.Random rnd, int[] values, double lambda)
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{
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if (!(lambda > 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, lambda);
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}
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/// <summary>
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/// Samples a Poisson distributed random variable.
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/// </summary>
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/// <param name="lambda">The lambda (λ) parameter of the Poisson distribution. Range: λ > 0.</param>
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/// <returns>A sample from the Poisson distribution.</returns>
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public static int Sample(double lambda)
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{
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if (!(lambda > 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, lambda);
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}
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/// <summary>
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/// Samples a sequence of Poisson distributed random variables.
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/// </summary>
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/// <param name="lambda">The lambda (λ) parameter of the Poisson distribution. Range: λ > 0.</param>
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/// <returns>a sequence of samples from the distribution.</returns>
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public static IEnumerable<int> Samples(double lambda)
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{
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if (!(lambda > 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, lambda);
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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="lambda">The lambda (λ) parameter of the Poisson distribution. Range: λ > 0.</param>
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/// <returns>a sequence of samples from the distribution.</returns>
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public static void Samples(int[] values, double lambda)
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{
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if (!(lambda > 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, lambda);
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}
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}
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}
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