SSJ
3.3.1
Stochastic Simulation in Java

Probability distributions. More...
Classes  
class  AndersonDarlingDist 
Extends the class ContinuousDistribution for the Anderson–Darling distribution (see [6], [165], [173], [225] ). More...  
class  AndersonDarlingDistQuick 
Extends the class AndersonDarlingDist for the Anderson–Darling distribution (see [6], [165], [225] ). More...  
class  BernoulliDist 
Extends the class DiscreteDistributionInt for the Bernoulli distribution [118] with parameter \(p\), where \(0\le p\le1\). More...  
class  BetaDist 
Extends the class ContinuousDistribution for the beta distribution [100] (page 210) with shape parameters \(\alpha> 0\) and \(\beta> 0\), over the interval \([a,b]\), where \(a < b\). More...  
class  BetaSymmetricalDist 
Specializes the class BetaDist to the case of a symmetrical beta distribution over the interval \([0,1]\), with shape parameters \(\alpha= \beta\). More...  
class  BinomialDist 
Extends the class DiscreteDistributionInt for the binomial distribution [118] (page 321) with parameters \(n\) and \(p\), where \(n\) is a positive integer and \(0\le p\le1\). More...  
class  CauchyDist 
Extends the class ContinuousDistribution for the Cauchy distribution [99] (page 299) with location parameter \(\alpha\) and scale parameter \(\beta> 0\). More...  
class  ChiDist 
Extends the class ContinuousDistribution for the chi distribution [99] (page 417) with shape parameter \(\nu > 0\), where the number of degrees of freedom \(\nu\) is a positive integer. More...  
class  ChiSquareDist 
Extends the class ContinuousDistribution for the chisquare distribution with \(n\) degrees of freedom, where \(n\) is a positive integer [99] (page 416). More...  
class  ChiSquareDistQuick 
Provides a variant of ChiSquareDist with faster but less accurate methods. More...  
class  ChiSquareNoncentralDist 
Extends the class ContinuousDistribution for the noncentral chisquare distribution with \(\nu\) degrees of freedom and noncentrality parameter \(\lambda\), where \(\nu> 0\) and \(\lambda> 0\) [100] (page 436). More...  
class  ConstantDist 
Represents a constant discrete distribution taking a single real value with probability 1. More...  
class  ConstantIntDist 
Represents a constant discrete distribution taking a single integer value with probability 1. More...  
class  ContinuousDistribution 
Classes implementing continuous distributions should inherit from this base class. More...  
class  CramerVonMisesDist 
Extends the class ContinuousDistribution for the Cramérvon Mises distribution (see [55], [224], [225] ). More...  
class  DiscreteDistribution 
This class implements discrete distributions over a finite set of real numbers (also over integers as a particular case). More...  
class  DiscreteDistributionInt 
Classes implementing discrete distributions over the integers should inherit from this class. More...  
interface  Distribution 
This interface should be implemented by all classes supporting discrete and continuous distributions. More...  
class  DistributionFactory 
This class implements a string API for the package probdist . More...  
class  EmpiricalDist 
Extends DiscreteDistribution to an empirical distribution function, based on the observations \(X_{(1)},…,X_{(n)}\) (sorted by increasing order). More...  
class  ErlangDist 
Extends the class GammaDist for the special case of the Erlang distribution with shape parameter \(k > 0\) and scale parameter \(\lambda> 0\). More...  
class  ExponentialDist 
Extends the class ContinuousDistribution for the exponential distribution [99] (page 494) with mean \(1/\lambda\) where \(\lambda> 0\). More...  
class  ExponentialDistFromMean 
Extends the ExponentialDist class with a constructor accepting as argument the mean \(1/\lambda\) instead of the rate \(\lambda\). More...  
class  ExtremeValueDist 
This class has been replaced by GumbelDist . More...  
class  FatigueLifeDist 
Extends the class ContinuousDistribution for the fatigue life distribution [20] with location parameter \(\mu\), scale parameter \(\beta\) and shape parameter \(\gamma\). More...  
class  FisherFDist 
Extends the class ContinuousDistribution for the Fisher F distribution with \(n_1\) and \(n_2\) degrees of freedom, where \(n_1\) and \(n_2\) are positive integers. More...  
class  FoldedNormalDist 
Extends the class ContinuousDistribution for the folded normal distribution with parameters \(\mu\ge0\) and \(\sigma> 0\). More...  
class  FrechetDist 
Extends the class ContinuousDistribution for the Fréchet distribution [100] (page 3), with location parameter \(\delta\), scale parameter \(\beta> 0\), and shape parameter \(\alpha> 0\), where we use the notation \(z = (x\delta)/\beta\). More...  
class  GammaDist 
Extends the class ContinuousDistribution for the gamma distribution [99] (page 337) with shape parameter \(\alpha> 0\) and scale parameter \(\lambda> 0\). More...  
class  GammaDistFromMoments 
Extends the GammaDist distribution with constructors accepting the mean \(\mu\) and variance \(\sigma^2\) as arguments instead of a shape parameter \(\alpha\) and a scale parameter \(\lambda\). More...  
class  GeometricDist 
Extends the class DiscreteDistributionInt for the geometric distribution [118] (page 322) with parameter \(p\), where \(0 < p < 1\). More...  
class  GumbelDist 
Extends the class ContinuousDistribution for the Gumbel distribution [100] (page 2), with location parameter \(\delta\) and scale parameter \(\beta\neq0\). More...  
class  HalfNormalDist 
Extends the class ContinuousDistribution for the halfnormal distribution with parameters \(\mu\) and \(\sigma> 0\). More...  
class  HyperbolicSecantDist 
Extends the class ContinuousDistribution for the hyperbolic secant distribution with location parameter \(\mu\) and scale parameter \(\sigma> 0\). More...  
class  HypergeometricDist 
Extends the class DiscreteDistributionInt for the hypergeometric distribution [66] (page 101) with \(k\) elements chosen among \(l\), \(m\) being of one type, and \(lm\) of the other. More...  
class  HypoExponentialDist 
This class implements the hypoexponential distribution, also called the generalized Erlang distribution. More...  
class  HypoExponentialDistEqual 
This class implements the hypoexponential distribution for the case of equidistant \(\lambda_i = (n+1i)h\). More...  
class  HypoExponentialDistQuick 
This class is a subclass of HypoExponentialDist and also implements the hypoexponential distribution. More...  
class  InverseDistFromDensity 
Implements a method for computing the inverse of an arbitrary continuous distribution function when only the probability density is known [43] . More...  
class  InverseGammaDist 
Extends the class ContinuousDistribution for the inverse gamma distribution with shape parameter \(\alpha> 0\) and scale parameter \(\beta> 0\), also known as the Pearson type V distribution. More...  
class  InverseGaussianDist 
Extends the class ContinuousDistribution for the inverse Gaussian distribution with location parameter \(\mu> 0\) and scale parameter \(\lambda> 0\). More...  
class  JohnsonSBDist 
Extends the class ContinuousDistribution for the Johnson \(S_B\) distribution [101], [118], [63] with shape parameters \(\gamma\) and \(\delta> 0\), location parameter \(\xi\), and scale parameter \(\lambda>0\). More...  
class  JohnsonSLDist 
Extends the class ContinuousDistribution for the Johnson \(S_L\) distribution (see [101], [99] ). More...  
class  JohnsonSUDist 
Extends the class ContinuousDistribution for the Johnson \(S_U\) distribution (see [118] (page 316)). More...  
class  JohnsonSystem 
This class contains common parameters and methods for the Johnson system of distributions [101], [99] with shape parameters \(\gamma\) and \(\delta> 0\), location parameter \(\xi\), and scale parameter \(\lambda>0\). More...  
class  KolmogorovSmirnovDist 
Extends the class ContinuousDistribution for the KolmogorovSmirnov distribution with parameter \(n\) [55] . More...  
class  KolmogorovSmirnovDistQuick 
Extends the class KolmogorovSmirnovDist for the Kolmogorov–Smirnov distribution. More...  
class  KolmogorovSmirnovPlusDist 
Extends the class ContinuousDistribution for the Kolmogorov–Smirnov+ distribution (see [40], [55], [27] ). More...  
class  LaplaceDist 
Extends the class ContinuousDistribution for the Laplace distribution (see, e.g., [100] (page 165)). More...  
class  LogarithmicDist 
Extends the class DiscreteDistributionInt for the logarithmic distribution. More...  
class  LogisticDist 
Extends the class ContinuousDistribution for the logistic distribution (e.g., [100] (page 115)). More...  
class  LoglogisticDist 
Extends the class ContinuousDistribution for the LogLogistic distribution with shape parameter \(\alpha> 0\) and scale parameter \(\beta> 0\). More...  
class  LognormalDist 
Extends the class ContinuousDistribution for the lognormal distribution [99] . More...  
class  LognormalDistFromMoments 
Extends the LognormalDist class with a constructor accepting the mean \(m\) and the variance \(v\) of the distribution as arguments. More...  
class  NakagamiDist 
Extends the class ContinuousDistribution for the Nakagami distribution with location parameter \(a\), scale parameter \(\lambda> 0\) and shape parameter \(c > 0\). More...  
class  NegativeBinomialDist 
Extends the class DiscreteDistributionInt for the negative binomial distribution [118] (page 324) with real parameters \(n\) and \(p\), where \(n > 0\) and \(0\le p\le1\). More...  
class  NormalDist 
Extends the class ContinuousDistribution for the normal distribution (e.g., [99] (page 80)). More...  
class  NormalDistQuick 
A variant of the class NormalDist (for the normal distribution with mean \(\mu\) and variance \(\sigma^2\)). More...  
class  NormalInverseGaussianDist 
Extends the class ContinuousDistribution for the normal inverse gaussian distribution with location parameter \(\mu\), scale parameter \(\delta> 0\), tail heavyness \(\alpha> 0\), and asymmetry parameter \(\beta\) such that \(0 \le\beta < \alpha\). More...  
class  ParetoDist 
Extends the class ContinuousDistribution for a distribution from the Pareto family, with shape parameter \(\alpha> 0\) and location parameter \(\beta> 0\) [99] (page 574). More...  
class  PascalDist 
The Pascal distribution is a special case of the negative binomial distribution [118] (page 324) with parameters \(n\) and \(p\), where \(n\) is a positive integer and \(0\le p\le1\). More...  
class  Pearson5Dist 
THIS CLASS HAS BEEN RENAMED InverseGammaDist . More...  
class  Pearson6Dist 
Extends the class ContinuousDistribution for the Pearson type VI distribution with shape parameters \(\alpha_1 > 0\) and \(\alpha_2 > 0\), and scale parameter \(\beta> 0\). More...  
class  PiecewiseLinearEmpiricalDist 
Extends the class ContinuousDistribution for a piecewiselinear approximation of the empirical distribution function, based on the observations \(X_{(1)},…,X_{(n)}\) (sorted by increasing order), and defined as follows (e.g., [118] (page 318)). More...  
class  PoissonDist 
Extends the class DiscreteDistributionInt for the Poisson distribution [118] (page 325) with mean \(\lambda\ge0\). More...  
class  PowerDist 
Extends the class ContinuousDistribution for the power distribution [57] (page 161) with shape parameter \(c > 0\), over the interval \([a,b]\), where \(a < b\). More...  
class  RayleighDist 
This class extends the class ContinuousDistribution for the Rayleigh distribution [57] with location parameter \(a\), and scale parameter \(\beta> 0\). More...  
class  StudentDist 
Extends the class ContinuousDistribution for the Student \(t\)distribution [100] (page 362) with \(n\) degrees of freedom, where \(n\) is a positive integer. More...  
class  StudentDistQuick 
Extends the class StudentDist for the Student \(t\)distribution. More...  
class  TriangularDist 
Extends the class ContinuousDistribution for the triangular distribution (see [100] (page 297) and [118] (page 317)) with domain \([a,b]\) and mode (or shape parameter) \(m\), where \(a\le m\le b\). More...  
class  TruncatedDist 
This container class takes an arbitrary continuous distribution and truncates it to an interval \([a,b]\), where \(a\) and \(b\) can be finite or infinite. More...  
class  UniformDist 
Extends the class ContinuousDistribution for the uniform distribution [100] (page 276) over the interval \([a,b]\). More...  
class  UniformIntDist 
Extends the class DiscreteDistributionInt for the discrete uniform distribution over the range \([i,j]\). More...  
class  WatsonGDist 
Extends the class ContinuousDistribution for the Watson \(G\) distribution (see [41], [238] ). More...  
class  WatsonUDist 
Extends the class ContinuousDistribution for the Watson U distribution (see [55], [224], [225] ). More...  
class  WeibullDist 
This class extends the class ContinuousDistribution for the Weibull distribution [99] (page 628) with shape parameter \(\alpha> 0\), location parameter \(\delta\), and scale parameter \(\lambda> 0\). More...  
Probability distributions.
This package contains Java classes that implement discrete and continuous univariate probability distributions. These classes provide methods to evaluate the probability density (or mass) function (pdf), the cumulative distribution function (cdf), the complementary cdf, the inverse cdf, some moments, etc. It also offers methods to estimate the parameters of some distributions from empirical data. It does not generate random variates; for that, see the package umontreal.ssj.randvar. One can plot the density or the cdf of a distribution either on screen, or in a LaTeX file, by using the package umontreal.ssj.charts.
The cumulative distribution function (cdf) of a continuous random variable \(X\) with probability density function (pdf) \(f\) over the real line is
\[ F(x) = P[X\le x] = \int_{\infty}^x f(s)ds \tag{FDist} \]
while the cdf of a discrete random variable \(X\) with probability mass function \(p\) over a fixed set of real numbers \(x_0 < x_1 < x_2 < \cdots\) is
\[ F(x) = P[X\le x] = \sum_{x_i\le x} p(x_i), \tag{FDistDisc} \]
where \(p(x_i) = P[X = x_i]\). Sometimes, the mass function is just called the pdf as well. It is the density with respect to a discrete measure.
We define \(\bar{F}\), the complementary distribution function of \(X\), by
\[ \bar{F} (x) = P[X\ge x]. \]
With this definition of \(\bar{F}\), one has \(\bar{F}(x) = 1  F (x)\) for continuous distributions and \(\bar{F}(x) = 1  F (x1)\) for discrete distributions over the integers. This definition is nonstandard for the discrete case: we have \(\bar{F} (x) = P[X\ge x]\) instead of \(\bar{F} (x) = P[X > x] = 1F(x)\). We find it more convenient especially for computing \(p\)values in goodnessoffit tests.
The inverse cumulative distribution function is defined as
\[ F^{1}(u) = \inf\{x\in\mathbb{R}: F (x)\ge u\}, \tag{inverseF} \]
for \(0\le u\le1\). This function \(F^{1}\) is used among other things to generate the random variable \(X\) by inversion, by passing a \(U (0,1)\) random variate as the value of \(u\).
The package probdist
offers two types of tools for computing \(p\), \(f\), \(F\), \(\bar{F}\), and \(F^{1}\): static methods, for which no object needs to be created, and methods associated with distribution objects. Standard distributions are implemented each in their own class. Constructing an object from one of these classes can be convenient if \(F\), \(\bar{F}\), etc., has to be evaluated several times for the same distribution. In certain cases, creating the distribution object would precompute tables that would speed up significantly all subsequent method calls for computing \(F\), \(\bar{F}\), etc. This trades memory, plus a onetime setup cost, for speed. In addition to the nonstatic methods, the distribution classes also provide static methods that do not require the creation of an object.
The distribution classes extend one of the (abstract) classes umontreal.ssj.probdist.DiscreteDistribution and umontreal.ssj.probdist.ContinuousDistribution (which both implement the interface umontreal.ssj.probdist.Distribution ) for discrete and continuous distributions over the real numbers, or umontreal.ssj.probdist.DiscreteDistributionInt, for discrete distributions over the nonnegative integers.
For example, the class umontreal.ssj.probdist.PoissonDist extends umontreal.ssj.probdist.DiscreteDistributionInt. Calling a static method from this class will compute the corresponding probability from scratch. Constructing a umontreal.ssj.probdist.PoissonDist object, on the other hand, will precompute tables that contain the probability terms and the distribution function for a given parameter \(\lambda\) (the mean of the Poisson distribution). These tables will then be used whenever a method is called for the corresponding object. This second approach is recommended if some of \(F\), \(\bar{F}\), etc., has to be computed several times for the same parameter \(\lambda\). As a rule of thumb, creating objects and using their methods is faster than just using static methods as soon as two or three calls are made, unless the parameters are large.
Only the nonnegligible probability terms (those that exceed the threshold umontreal.ssj.probdist.DiscreteDistributionInt.EPSILON ) are stored in the tables. For \(F\) and \(\bar{F}\), a single table actually contains \(F (x)\) for \(F (x) \le1/2\) and \(1F (x)\) for \(F (x) > 1/2\). When the distribution parameters are so large that the tables would take too much space, these are not created and the methods automatically call their static equivalents instead of using tables.
Objects that implement the interface umontreal.ssj.probdist.Distribution (and sometimes the abstract class umontreal.ssj.probdist.ContinuousDistribution ) are required by some methods in package umontreal.ssj.randvar and also in classes umontreal.ssj.gof.GofStat and umontreal.ssj.gof.GofFormat of package umontreal.ssj.gof.
Some of the classes in probdist
also provide methods that compute parameter estimations of the corresponding distribution from a set of empirical observations, in most cases based on the maximum likelihood method.