// <copyright file="Chi.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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/// Continuous Univariate Chi distribution.
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/// This distribution is a continuous probability distribution. The distribution usually arises when a k-dimensional vector's orthogonal
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/// components are independent and each follow a standard normal distribution. The length of the vector will
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/// then have a chi distribution.
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/// <a href="http://en.wikipedia.org/wiki/Chi_distribution">Wikipedia - Chi distribution</a>.
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/// </summary>
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public class Chi : IContinuousDistribution
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{
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System.Random _random;
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readonly double _freedom;
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/// <summary>
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/// Initializes a new instance of the <see cref="Chi"/> class.
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/// </summary>
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/// <param name="freedom">The degrees of freedom (k) of the distribution. Range: k > 0.</param>
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public Chi(double freedom)
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{
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if (!IsValidParameterSet(freedom))
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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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_freedom = freedom;
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}
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/// <summary>
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/// Initializes a new instance of the <see cref="Chi"/> class.
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/// </summary>
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/// <param name="freedom">The degrees of freedom (k) of the distribution. Range: k > 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 Chi(double freedom, System.Random randomSource)
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{
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if (!IsValidParameterSet(freedom))
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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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_freedom = freedom;
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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 $"Chi(k = {_freedom})";
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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="freedom">The degrees of freedom (k) of the distribution. Range: k > 0.</param>
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public static bool IsValidParameterSet(double freedom)
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{
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return freedom > 0.0;
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}
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/// <summary>
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/// Gets the degrees of freedom (k) of the Chi distribution. Range: k > 0.
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/// </summary>
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public double DegreesOfFreedom => _freedom;
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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 => Constants.Sqrt2*(SpecialFunctions.Gamma((_freedom + 1.0)/2.0)/SpecialFunctions.Gamma(_freedom/2.0));
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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 => _freedom - (Mean*Mean);
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/// <summary>
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/// Gets the standard deviation of the distribution.
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/// </summary>
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public double StdDev => Math.Sqrt(Variance);
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/// <summary>
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/// Gets the entropy of the distribution.
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/// </summary>
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public double Entropy => SpecialFunctions.GammaLn(_freedom/2.0) + ((_freedom - Math.Log(2) - ((_freedom - 1.0)*SpecialFunctions.DiGamma(_freedom/2.0)))/2.0);
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/// <summary>
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/// Gets the skewness of the distribution.
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/// </summary>
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public double Skewness
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{
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get
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{
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var sigma = StdDev;
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return (Mean*(1.0 - (2.0*(sigma*sigma))))/(sigma*sigma*sigma);
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}
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}
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/// <summary>
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/// Gets the mode of the distribution.
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/// </summary>
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public double Mode
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{
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get
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{
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if (_freedom < 1)
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{
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throw new NotSupportedException();
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}
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return Math.Sqrt(_freedom - 1.0);
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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 => throw new NotSupportedException();
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/// <summary>
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/// Gets the minimum of the distribution.
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/// </summary>
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public double Minimum => 0.0;
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/// <summary>
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/// Gets the maximum of the distribution.
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/// </summary>
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public double Maximum => double.PositiveInfinity;
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/// <summary>
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/// Computes the probability density of the distribution (PDF) at x, i.e. ∂P(X ≤ x)/∂x.
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/// </summary>
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/// <param name="x">The location at which to compute the density.</param>
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/// <returns>the density at <paramref name="x"/>.</returns>
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/// <seealso cref="PDF"/>
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public double Density(double x)
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{
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return PDF(_freedom, x);
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}
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/// <summary>
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/// Computes the log probability density of the distribution (lnPDF) at x, i.e. ln(∂P(X ≤ x)/∂x).
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/// </summary>
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/// <param name="x">The location at which to compute the log density.</param>
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/// <returns>the log density at <paramref name="x"/>.</returns>
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/// <seealso cref="PDFLn"/>
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public double DensityLn(double x)
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{
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return PDFLn(_freedom, x);
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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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/// <seealso cref="CDF"/>
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public double CumulativeDistribution(double x)
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{
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return CDF(_freedom, x);
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}
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/// <summary>
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/// Generates a sample from the Chi distribution.
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/// </summary>
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/// <returns>a sample from the distribution.</returns>
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public double Sample()
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{
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return SampleUnchecked(_random, (int)_freedom);
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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(double[] values)
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{
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SamplesUnchecked(_random, values, (int)_freedom);
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}
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/// <summary>
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/// Generates a sequence of samples from the Chi distribution.
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/// </summary>
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/// <returns>a sequence of samples from the distribution.</returns>
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public IEnumerable<double> Samples()
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{
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return SamplesUnchecked(_random, (int)_freedom);
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}
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/// <summary>
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/// Samples 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="freedom">The degrees of freedom (k) of the distribution. Range: k > 0.</param>
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/// <returns>a random number from the distribution.</returns>
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static double SampleUnchecked(System.Random rnd, int freedom)
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{
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double sum = 0;
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for (var i = 0; i < freedom; i++)
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{
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sum += Math.Pow(Normal.Sample(rnd, 0.0, 1.0), 2);
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}
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return Math.Sqrt(sum);
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}
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static void SamplesUnchecked(System.Random rnd, double[] values, int freedom)
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{
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var standard = new double[values.Length*freedom];
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Normal.SamplesUnchecked(rnd, standard, 0.0, 1.0);
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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*freedom;
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double sum = 0;
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for (int j = 0; j < freedom; j++)
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{
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sum += standard[k + j]*standard[k + j];
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}
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values[i] = Math.Sqrt(sum);
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}
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});
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}
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static IEnumerable<double> SamplesUnchecked(System.Random rnd, int freedom)
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{
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while (true)
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{
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yield return SampleUnchecked(rnd, freedom);
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}
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}
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/// <summary>
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/// Computes the probability density of the distribution (PDF) at x, i.e. ∂P(X ≤ x)/∂x.
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/// </summary>
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/// <param name="freedom">The degrees of freedom (k) of the distribution. Range: k > 0.</param>
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/// <param name="x">The location at which to compute the density.</param>
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/// <returns>the density at <paramref name="x"/>.</returns>
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/// <seealso cref="Density"/>
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public static double PDF(double freedom, double x)
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{
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if (freedom <= 0.0)
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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if (double.IsPositiveInfinity(freedom) || double.IsPositiveInfinity(x) || x == 0.0)
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{
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return 0.0;
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}
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if (freedom > 160.0)
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{
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return Math.Exp(PDFLn(freedom, x));
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}
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return (Math.Pow(2.0, 1.0 - (freedom/2.0))*Math.Pow(x, freedom - 1.0)*Math.Exp(-x*x/2.0))/SpecialFunctions.Gamma(freedom/2.0);
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}
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/// <summary>
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/// Computes the log probability density of the distribution (lnPDF) at x, i.e. ln(∂P(X ≤ x)/∂x).
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/// </summary>
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/// <param name="freedom">The degrees of freedom (k) of the distribution. Range: k > 0.</param>
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/// <param name="x">The location at which to compute the density.</param>
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/// <returns>the log density at <paramref name="x"/>.</returns>
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/// <seealso cref="DensityLn"/>
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public static double PDFLn(double freedom, double x)
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{
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if (freedom <= 0.0)
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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if (double.IsPositiveInfinity(freedom) || double.IsPositiveInfinity(x) || x == 0.0)
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{
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return double.NegativeInfinity;
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}
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return ((1.0 - (freedom/2.0))*Math.Log(2.0)) + ((freedom - 1.0)*Math.Log(x)) - (x*x/2.0) - SpecialFunctions.GammaLn(freedom/2.0);
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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="freedom">The degrees of freedom (k) of the distribution. Range: k > 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 freedom, double x)
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{
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if (freedom <= 0.0)
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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if (double.IsPositiveInfinity(x))
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{
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return 1.0;
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}
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if (double.IsPositiveInfinity(freedom))
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{
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return 1.0;
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}
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return SpecialFunctions.GammaLowerRegularized(freedom/2.0, x*x/2.0);
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}
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/// <summary>
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/// Generates a sample 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="freedom">The degrees of freedom (k) of the distribution. Range: k > 0.</param>
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/// <returns>a sample from the distribution.</returns>
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public static double Sample(System.Random rnd, int freedom)
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{
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if (freedom <= 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, freedom);
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}
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/// <summary>
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/// Generates a sequence of samples 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="freedom">The degrees of freedom (k) of the distribution. Range: k > 0.</param>
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/// <returns>a sequence of samples from the distribution.</returns>
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public static IEnumerable<double> Samples(System.Random rnd, int freedom)
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{
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if (freedom <= 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, freedom);
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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="freedom">The degrees of freedom (k) of the distribution. Range: k > 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, double[] values, int freedom)
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{
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if (freedom <= 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, freedom);
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}
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/// <summary>
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/// Generates a sample from the distribution.
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/// </summary>
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/// <param name="freedom">The degrees of freedom (k) of the distribution. Range: k > 0.</param>
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/// <returns>a sample from the distribution.</returns>
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public static double Sample(int freedom)
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{
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if (freedom <= 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, freedom);
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}
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/// <summary>
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/// Generates a sequence of samples from the distribution.
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/// </summary>
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/// <param name="freedom">The degrees of freedom (k) of the distribution. Range: k > 0.</param>
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/// <returns>a sequence of samples from the distribution.</returns>
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public static IEnumerable<double> Samples(int freedom)
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{
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if (freedom <= 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, freedom);
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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="freedom">The degrees of freedom (k) of the distribution. Range: k > 0.</param>
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/// <returns>a sequence of samples from the distribution.</returns>
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public static void Samples(double[] values, int freedom)
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{
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if (freedom <= 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, freedom);
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}
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}
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}
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