// <copyright file="Cauchy.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) 2009-2013 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 Cauchy distribution.
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/// The Cauchy distribution is a symmetric continuous probability distribution. For details about this distribution, see
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/// <a href="http://en.wikipedia.org/wiki/Cauchy_distribution">Wikipedia - Cauchy distribution</a>.
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/// </summary>
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public class Cauchy : IContinuousDistribution
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{
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System.Random _random;
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readonly double _location;
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readonly double _scale;
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/// <summary>
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/// Initializes a new instance of the <see cref="Cauchy"/> class with the location parameter set to 0 and the scale parameter set to 1
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/// </summary>
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public Cauchy() : this(0, 1)
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{
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}
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/// <summary>
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/// Initializes a new instance of the <see cref="Cauchy"/> class.
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/// </summary>
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/// <param name="location">The location (x0) of the distribution.</param>
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/// <param name="scale">The scale (γ) of the distribution. Range: γ > 0.</param>
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public Cauchy(double location, double scale)
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{
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if (!IsValidParameterSet(location, scale))
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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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_location = location;
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_scale = scale;
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}
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/// <summary>
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/// Initializes a new instance of the <see cref="Cauchy"/> class.
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/// </summary>
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/// <param name="location">The location (x0) of the distribution.</param>
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/// <param name="scale">The scale (γ) of the distribution. Range: γ > 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 Cauchy(double location, double scale, System.Random randomSource)
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{
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if (!IsValidParameterSet(location, scale))
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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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_location = location;
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_scale = scale;
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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 $"Cauchy(x0 = {_location}, γ = {_scale})";
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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="location">The location (x0) of the distribution.</param>
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/// <param name="scale">The scale (γ) of the distribution. Range: γ > 0.</param>
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public static bool IsValidParameterSet(double location, double scale)
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{
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return scale > 0.0 && !double.IsNaN(location);
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}
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/// <summary>
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/// Gets the location (x0) of the distribution.
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/// </summary>
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public double Location => _location;
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/// <summary>
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/// Gets the scale (γ) of the distribution. Range: γ > 0.
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/// </summary>
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public double Scale => _scale;
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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 => throw new NotSupportedException();
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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 => throw new NotSupportedException();
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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 => throw new NotSupportedException();
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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 => Math.Log(4.0*Constants.Pi*_scale);
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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 => throw new NotSupportedException();
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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 => _location;
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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 => _location;
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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 => double.NegativeInfinity;
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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 1.0/(Constants.Pi*_scale*(1.0 + (((x - _location)/_scale)*((x - _location)/_scale))));
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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 -Math.Log(Constants.Pi*_scale*(1.0 + (((x - _location)/_scale)*((x - _location)/_scale))));
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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 ((1.0/Constants.Pi)*Math.Atan((x - _location)/_scale)) + 0.5;
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}
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/// <summary>
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/// Computes the inverse of the cumulative distribution function (InvCDF) for the distribution
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/// at the given probability. This is also known as the quantile or percent point function.
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/// </summary>
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/// <param name="p">The location at which to compute the inverse cumulative density.</param>
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/// <returns>the inverse cumulative density at <paramref name="p"/>.</returns>
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/// <seealso cref="InvCDF"/>
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public double InverseCumulativeDistribution(double p)
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{
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return p <= 0.0 ? double.NegativeInfinity : p >= 1.0 ? double.PositiveInfinity
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: _location + _scale*Math.Tan((p - 0.5)*Constants.Pi);
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}
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/// <summary>
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/// Draws a random sample from the distribution.
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/// </summary>
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/// <returns>A random number from this distribution.</returns>
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public double Sample()
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{
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return SampleUnchecked(_random, _location, _scale);
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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, _location, _scale);
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}
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/// <summary>
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/// Generates a sequence of samples from the Cauchy 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, _location, _scale);
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}
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static double SampleUnchecked(System.Random rnd, double location, double scale)
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{
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return location + scale*Math.Tan(Constants.Pi*(rnd.NextDouble() - 0.5));
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}
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static void SamplesUnchecked(System.Random rnd, double[] values, double location, double scale)
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{
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rnd.NextDoubles(values);
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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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values[i] = location + scale*Math.Tan(Constants.Pi*(values[i] - 0.5));
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}
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});
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}
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static IEnumerable<double> SamplesUnchecked(System.Random rnd, double location, double scale)
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{
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while (true)
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{
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yield return location + scale*Math.Tan(Constants.Pi*(rnd.NextDouble() - 0.5));
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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="location">The location (x0) of the distribution.</param>
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/// <param name="scale">The scale (γ) of the distribution. Range: γ > 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 location, double scale, double x)
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{
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if (scale <= 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/(Constants.Pi*scale*(1.0 + (((x - location)/scale)*((x - location)/scale))));
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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="location">The location (x0) of the distribution.</param>
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/// <param name="scale">The scale (γ) of the distribution. Range: γ > 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 location, double scale, double x)
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{
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if (scale <= 0.0)
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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return -Math.Log(Constants.Pi*scale*(1.0 + (((x - location)/scale)*((x - location)/scale))));
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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="location">The location (x0) of the distribution.</param>
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/// <param name="scale">The scale (γ) of the distribution. Range: γ > 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 location, double scale, double x)
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{
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if (scale <= 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.Atan((x - location)/scale)/Constants.Pi + 0.5;
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}
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/// <summary>
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/// Computes the inverse of the cumulative distribution function (InvCDF) for the distribution
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/// at the given probability. This is also known as the quantile or percent point function.
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/// </summary>
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/// <param name="p">The location at which to compute the inverse cumulative density.</param>
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/// <param name="location">The location (x0) of the distribution.</param>
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/// <param name="scale">The scale (γ) of the distribution. Range: γ > 0.</param>
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/// <returns>the inverse cumulative density at <paramref name="p"/>.</returns>
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/// <seealso cref="InverseCumulativeDistribution"/>
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public static double InvCDF(double location, double scale, double p)
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{
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if (scale <= 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 p <= 0.0 ? double.NegativeInfinity : p >= 1.0 ? double.PositiveInfinity
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: location + scale*Math.Tan((p - 0.5)*Constants.Pi);
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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="location">The location (x0) of the distribution.</param>
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/// <param name="scale">The scale (γ) of the distribution. Range: γ > 0.</param>
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/// <returns>a sample from the distribution.</returns>
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public static double Sample(System.Random rnd, double location, double scale)
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{
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if (scale <= 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, location, scale);
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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="location">The location (x0) of the distribution.</param>
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/// <param name="scale">The scale (γ) of the distribution. Range: γ > 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, double location, double scale)
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{
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if (scale <= 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, location, scale);
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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="location">The location (x0) of the distribution.</param>
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/// <param name="scale">The scale (γ) of the distribution. Range: γ > 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, double location, double scale)
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{
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if (scale <= 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, location, scale);
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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="location">The location (x0) of the distribution.</param>
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/// <param name="scale">The scale (γ) of the distribution. Range: γ > 0.</param>
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/// <returns>a sample from the distribution.</returns>
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public static double Sample(double location, double scale)
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{
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if (scale <= 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, location, scale);
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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="location">The location (x0) of the distribution.</param>
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/// <param name="scale">The scale (γ) of the distribution. Range: γ > 0.</param>
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/// <returns>a sequence of samples from the distribution.</returns>
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public static IEnumerable<double> Samples(double location, double scale)
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{
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if (scale <= 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, location, scale);
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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="location">The location (x0) of the distribution.</param>
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/// <param name="scale">The scale (γ) of the distribution. Range: γ > 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, double location, double scale)
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{
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if (scale <= 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, location, scale);
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}
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}
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}
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