// <copyright file="Pareto.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 System.Linq;
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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 Pareto distribution.
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/// The Pareto distribution is a power law probability distribution that coincides with social,
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/// scientific, geophysical, actuarial, and many other types of observable phenomena.
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/// For details about this distribution, see
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/// <a href="http://en.wikipedia.org/wiki/Pareto_distribution">Wikipedia - Pareto distribution</a>.
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/// </summary>
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public class Pareto : IContinuousDistribution
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{
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System.Random _random;
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readonly double _scale;
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readonly double _shape;
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/// <summary>
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/// Initializes a new instance of the <see cref="Pareto"/> class.
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/// </summary>
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/// <param name="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
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/// <param name="shape">The shape (α) of the distribution. Range: α > 0.</param>
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/// <exception cref="ArgumentException">If <paramref name="scale"/> or <paramref name="shape"/> are negative.</exception>
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public Pareto(double scale, double shape)
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{
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if (!IsValidParameterSet(scale, shape))
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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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_scale = scale;
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_shape = shape;
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}
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/// <summary>
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/// Initializes a new instance of the <see cref="Pareto"/> class.
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/// </summary>
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/// <param name="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
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/// <param name="shape">The shape (α) of the distribution. Range: α > 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="ArgumentException">If <paramref name="scale"/> or <paramref name="shape"/> are negative.</exception>
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public Pareto(double scale, double shape, System.Random randomSource)
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{
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if (!IsValidParameterSet(scale, shape))
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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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_scale = scale;
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_shape = shape;
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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 $"Pareto(xm = {_scale}, α = {_shape})";
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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="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
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/// <param name="shape">The shape (α) of the distribution. Range: α > 0.</param>
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public static bool IsValidParameterSet(double scale, double shape)
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{
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return scale > 0.0 && shape > 0.0;
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}
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/// <summary>
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/// Gets the scale (xm) of the distribution. Range: xm > 0.
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/// </summary>
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public double Scale => _scale;
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/// <summary>
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/// Gets the shape (α) of the distribution. Range: α > 0.
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/// </summary>
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public double Shape => _shape;
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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
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{
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get
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{
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if (_shape <= 1)
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{
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throw new NotSupportedException();
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}
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return _shape*_scale/(_shape - 1.0);
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}
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}
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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
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{
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get
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{
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if (_shape <= 2.0)
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{
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return double.PositiveInfinity;
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}
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return _scale*_scale*_shape/((_shape - 1.0)*(_shape - 1.0)*(_shape - 2.0));
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}
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}
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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 => (_scale*Math.Sqrt(_shape))/(Math.Abs(_shape - 1.0)*Math.Sqrt(_shape - 2.0));
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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(_shape/_scale) - (1.0/_shape) - 1.0;
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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 => (2.0*(_shape + 1.0)/(_shape - 3.0))*Math.Sqrt((_shape - 2.0)/_shape);
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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 => _scale;
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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 => _scale*Math.Pow(2.0, 1.0/_shape);
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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 => _scale;
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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 _shape*Math.Pow(_scale, _shape)/Math.Pow(x, _shape + 1.0);
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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="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(_shape) + _shape*Math.Log(_scale) - (_shape + 1.0)*Math.Log(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 1.0 - Math.Pow(_scale/x, _shape);
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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 _scale*Math.Pow(1.0 - p, -1.0/_shape);
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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, _scale, _shape);
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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, _scale, _shape);
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}
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/// <summary>
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/// Generates a sequence of samples from the Pareto 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, _scale, _shape);
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}
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static double SampleUnchecked(System.Random rnd, double scale, double shape)
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{
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return scale*Math.Pow(rnd.NextDouble(), -1.0/shape);
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}
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static IEnumerable<double> SamplesUnchecked(System.Random rnd, double scale, double shape)
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{
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var power = -1.0/shape;
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return rnd.NextDoubleSequence().Select(x => scale*Math.Pow(x, power));
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}
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static void SamplesUnchecked(System.Random rnd, double[] values, double scale, double shape)
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{
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var power = -1.0/shape;
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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] = scale*Math.Pow(values[i], power);
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}
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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="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
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/// <param name="shape">The shape (α) of the distribution. Range: α > 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 scale, double shape, double x)
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{
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if (scale <= 0.0 || shape <= 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 shape*Math.Pow(scale, shape)/Math.Pow(x, shape + 1.0);
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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="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
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/// <param name="shape">The shape (α) of the distribution. Range: α > 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 scale, double shape, double x)
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{
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if (scale <= 0.0 || shape <= 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(shape) + shape*Math.Log(scale) - (shape + 1.0)*Math.Log(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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/// <param name="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
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/// <param name="shape">The shape (α) of the distribution. Range: α > 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 scale, double shape, double x)
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{
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if (scale <= 0.0 || shape <= 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 - Math.Pow(scale/x, shape);
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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="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
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/// <param name="shape">The shape (α) of the distribution. Range: α > 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 scale, double shape, double p)
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{
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if (scale <= 0.0 || shape <= 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 scale*Math.Pow(1.0 - p, -1.0/shape);
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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="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
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/// <param name="shape">The shape (α) of the distribution. Range: α > 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 scale, double shape)
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{
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if (scale <= 0.0 || shape <= 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 scale*Math.Pow(rnd.NextDouble(), -1.0/shape);
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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="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
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/// <param name="shape">The shape (α) of the distribution. Range: α > 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 scale, double shape)
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{
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if (scale <= 0.0 || shape <= 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, scale, shape);
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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="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
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/// <param name="shape">The shape (α) of the distribution. Range: α > 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 scale, double shape)
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{
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if (scale <= 0.0 || shape <= 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, scale, shape);
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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="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
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/// <param name="shape">The shape (α) of the distribution. Range: α > 0.</param>
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/// <returns>a sample from the distribution.</returns>
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public static double Sample(double scale, double shape)
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{
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if (scale <= 0.0 || shape <= 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, scale, shape);
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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="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
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/// <param name="shape">The shape (α) of the distribution. Range: α > 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 scale, double shape)
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{
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if (scale <= 0.0 || shape <= 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, scale, shape);
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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="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
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/// <param name="shape">The shape (α) of the distribution. Range: α > 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 scale, double shape)
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{
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if (scale <= 0.0 || shape <= 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, scale, shape);
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}
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}
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}
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