// <copyright file="InverseGaussian.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) 2009-2019 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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// 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 IStation.Numerics.Random;
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using IStation.Numerics.Statistics;
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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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namespace IStation.Numerics.Distributions
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{
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public class InverseGaussian : IContinuousDistribution
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{
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System.Random _random;
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/// <summary>
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/// Gets the mean (μ) of the distribution. Range: μ > 0.
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/// </summary>
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public double Mu { get; }
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/// <summary>
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/// Gets the shape (λ) of the distribution. Range: λ > 0.
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/// </summary>
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public double Lambda { get; }
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/// <summary>
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/// Initializes a new instance of the InverseGaussian class.
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/// </summary>
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/// <param name="mu">The mean (μ) of the distribution. Range: μ > 0.</param>
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/// <param name="lambda">The shape (λ) of the distribution. Range: λ > 0.</param>
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/// <param name="randomSource">The random number generator which is used to draw random samples.</param>
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public InverseGaussian(double mu, double lambda, System.Random randomSource = null)
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{
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if (!IsValidParameterSet(mu, lambda))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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_random = randomSource ?? SystemRandomSource.Default;
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Mu = mu;
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Lambda = lambda;
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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 $"InverseGaussian(μ = {Mu}, λ = {Lambda})";
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}
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/// <summary>
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/// Tests whether the provided values are valid parameters for this distribution.
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/// </summary>
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/// <param name="mu">The mean (μ) of the distribution. Range: μ > 0.</param>
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/// <param name="lambda">The shape (λ) of the distribution. Range: λ > 0.</param>
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public static bool IsValidParameterSet(double mu, double lambda)
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{
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var allFinite = mu.IsFinite() && lambda.IsFinite();
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return allFinite && mu > 0.0 && lambda > 0.0;
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}
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/// <summary>
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/// Gets the random number generator which is used to draw random samples.
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/// </summary>
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public System.Random RandomSource
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{
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get => _random;
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set => _random = value ?? SystemRandomSource.Default;
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}
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/// <summary>
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/// Gets the mean of the Inverse Gaussian distribution.
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/// </summary>
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public double Mean => Mu;
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/// <summary>
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/// Gets the variance of the Inverse Gaussian distribution.
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/// </summary>
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public double Variance => Math.Pow(Mu, 3) / Lambda;
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/// <summary>
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/// Gets the standard deviation of the Inverse Gaussian 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 median of the Inverse Gaussian distribution.
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/// No closed form analytical expression exists, so this value is approximated numerically and can throw an exception.
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/// </summary>
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public double Median => InvCDF(0.5);
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/// <summary>
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/// Gets the minimum of the Inverse Gaussian 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 Inverse Gaussian distribution.
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/// </summary>
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public double Maximum => double.PositiveInfinity;
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/// <summary>
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/// Gets the skewness of the Inverse Gaussian distribution.
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/// </summary>
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public double Skewness => 3 * Math.Sqrt(Mu / Lambda);
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/// <summary>
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/// Gets the kurtosis of the Inverse Gaussian distribution.
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/// </summary>
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public double Kurtosis => 15 * Mu / Lambda;
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/// <summary>
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/// Gets the mode of the Inverse Gaussian distribution.
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/// </summary>
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public double Mode => Mu * (Math.Sqrt(1 + (9 * Mu * Mu) / (4 * Lambda * Lambda)) - (3 * Mu) / (2 * Lambda));
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/// <summary>
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/// Gets the entropy of the Inverse Gaussian distribution (currently not supported).
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/// </summary>
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public double Entropy => throw new NotSupportedException();
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/// <summary>
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/// Generates a sample from the inverse Gaussian 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, Mu, Lambda);
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}
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/// <summary>
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/// Fills an array with samples generated from the distribution.
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/// </summary>
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/// <param name="values">The array to fill with the samples.</param>
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public void Samples(double[] values)
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{
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SamplesUnchecked(_random, values, Mu, Lambda);
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}
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/// <summary>
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/// Generates a sequence of samples from the inverse Gaussian 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, Mu, Lambda);
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}
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/// <summary>
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/// Generates a sample from the inverse Gaussian 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="mu">The mean (μ) of the distribution. Range: μ > 0.</param>
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/// <param name="lambda">The shape (λ) of the distribution. Range: λ > 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 mu, double lambda)
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{
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if (!IsValidParameterSet(mu, lambda))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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return SampleUnchecked(rnd, mu, lambda);
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}
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/// <summary>
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/// Fills an array with samples generated from the distribution.
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/// </summary>
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/// <param name="rnd">The random number generator to use.</param>
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/// <param name="values">The array to fill with the samples.</param>
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/// <param name="mu">The mean (μ) of the distribution. Range: μ > 0.</param>
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/// <param name="lambda">The shape (λ) of the distribution. Range: λ > 0.</param>
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public static void Samples(System.Random rnd, double[] values, double mu, double lambda)
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{
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if (!IsValidParameterSet(mu, lambda))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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SamplesUnchecked(rnd, values, mu, lambda);
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}
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/// <summary>
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/// Generates a sequence of samples from the Burr 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="mu">The mean (μ) of the distribution. Range: μ > 0.</param>
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/// <param name="lambda">The shape (λ) of the distribution. Range: λ > 0.</param>
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/// <returns>a sequence of samples from the distribution.</returns>
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public static IEnumerable<double> Samples(System.Random rnd, double mu, double lambda)
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{
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if (!IsValidParameterSet(mu, lambda))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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return SamplesUnchecked(rnd, mu, lambda);
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}
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internal static double SampleUnchecked(System.Random rnd, double mu, double lambda)
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{
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double v = IStation.Numerics.Distributions.Normal.Sample(rnd, 0, 1);
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double test = rnd.NextDouble();
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return InverseGaussianSampleImpl(mu, lambda, v, test);
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}
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internal static void SamplesUnchecked(System.Random rnd, double[] values, double mu, double lambda)
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{
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if (values.Length == 0)
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{
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return;
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}
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double[] v = new double[values.Length];
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IStation.Numerics.Distributions.Normal.Samples(rnd, v, 0, 1);
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double[] test = rnd.NextDoubles(values.Length);
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for (var j = 0; j < values.Length; ++j)
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{
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values[j] = InverseGaussianSampleImpl(mu, lambda, v[j], test[j]);
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}
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}
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internal static IEnumerable<double> SamplesUnchecked(System.Random rnd, double mu, double lambda)
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{
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while (true)
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{
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yield return SampleUnchecked(rnd, mu, lambda);
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}
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}
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internal static double InverseGaussianSampleImpl(double mu, double lambda, double normalSample, double uniformSample)
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{
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double y = normalSample * normalSample;
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double x = mu + (mu * mu * y) / (2 * lambda) - (mu / (2 * lambda)) * Math.Sqrt(4 * mu * lambda * y + mu * mu * y * y);
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if (uniformSample <= mu / (mu + x))
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return x;
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else
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return mu * mu / x;
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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="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 DensityImpl(Mu, Lambda, 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 DensityLnImpl(Mu, Lambda, 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 CumulativeDistributionImpl(Mu, Lambda, x);
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}
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/// <summary>
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/// Computes the inverse cumulative distribution (CDF) of the distribution at p, i.e. solving for P(X ≤ x) = p.
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/// </summary>
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/// <param name="p">The location at which to compute the inverse cumulative distribution function.</param>
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/// <returns>the inverse cumulative distribution at location <paramref name="p"/>.</returns>
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public double InvCDF(double p)
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{
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Func<double, double> equationToSolve = (x) => CumulativeDistribution(x) - p;
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if (RootFinding.NewtonRaphson.TryFindRoot(equationToSolve, Density, Mode, 0, double.PositiveInfinity, 1e-8, 100, out double quantile))
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return quantile;
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else
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throw new NonConvergenceException("Numerical estimation of the statistic has failed. The used solver did not succeed in finding a root.");
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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="mu">The mean (μ) of the distribution. Range: μ > 0.</param>
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/// <param name="lambda">The shape (λ) of the distribution. Range: λ > 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 mu, double lambda, double x)
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{
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if (!IsValidParameterSet(mu, lambda))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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return DensityImpl(mu, lambda, 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="mu">The mean (μ) of the distribution. Range: μ > 0.</param>
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/// <param name="lambda">The shape (λ) of the distribution. Range: λ > 0.</param>
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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="DensityLn"/>
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public static double PDFLn(double mu, double lambda, double x)
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{
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if (!IsValidParameterSet(mu, lambda))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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return DensityLnImpl(mu, lambda, 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="mu">The mean (μ) of the distribution. Range: μ > 0.</param>
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/// <param name="lambda">The shape (λ) of the distribution. Range: λ > 0.</param>
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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="CumulativeDistribution"/>
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public static double CDF(double mu, double lambda, double x)
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{
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if (!IsValidParameterSet(mu, lambda))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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return CumulativeDistributionImpl(mu, lambda, x);
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}
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/// <summary>
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/// Computes the inverse cumulative distribution (CDF) of the distribution at p, i.e. solving for P(X ≤ x) = p.
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/// </summary>
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/// <param name="mu">The mean (μ) of the distribution. Range: μ > 0.</param>
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/// <param name="lambda">The shape (λ) of the distribution. Range: λ > 0.</param>
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/// <param name="p">The location at which to compute the inverse cumulative distribution function.</param>
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/// <returns>the inverse cumulative distribution at location <paramref name="p"/>.</returns>
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/// <seealso cref="CumulativeDistribution"/>
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public static double ICDF(double mu, double lambda, double p)
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{
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if (!IsValidParameterSet(mu, lambda))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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var igDist = new InverseGaussian(mu, lambda);
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return igDist.InvCDF(p);
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}
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/// <summary>
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/// Estimates the Inverse Gaussian parameters from sample data with maximum-likelihood.
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/// </summary>
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/// <param name="samples">The samples to estimate the distribution parameters from.</param>
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/// <param name="randomSource">The random number generator which is used to draw random samples. Optional, can be null.</param>
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/// <returns>An Inverse Gaussian distribution.</returns>
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public static InverseGaussian Estimate(IEnumerable<double> samples, System.Random randomSource = null)
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{
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var sampleVec = samples.ToArray();
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var muHat = sampleVec.Mean();
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var lambdahat = 1 / (1 / samples.HarmonicMean() - 1 / muHat);
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return new InverseGaussian(muHat, lambdahat, randomSource);
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}
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internal static double DensityImpl(double mu, double lambda, double x)
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{
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return Math.Sqrt(lambda / (2 * Math.PI * Math.Pow(x, 3))) * Math.Exp(-((lambda * Math.Pow(x - mu, 2)) / (2 * mu * mu * x)));
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}
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internal static double DensityLnImpl(double mu, double lambda, double x)
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{
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return Math.Log(Math.Sqrt(lambda / (2 * Math.PI * Math.Pow(x, 3)))) - ((lambda * Math.Pow(x - mu, 2)) / (2 * mu * mu * x));
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}
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internal static double CumulativeDistributionImpl(double mu, double lambda, double x)
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{
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return Normal.CDF(0, 1, Math.Sqrt(lambda / x) * (x / mu - 1)) + Math.Exp(2 * lambda / mu) * Normal.CDF(0, 1, -Math.Sqrt(lambda / x) * (x / mu + 1));
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}
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}
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}
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