Package umontreal.ssj.probdist

Probability Distributions. More...

## Classes | |

class | AndersonDarlingDist |

Extends the class ContinuousDistribution for the Anderson–Darling distribution (see [5], [139], [146], [192] ). More... | |

class | AndersonDarlingDistQuick |

Extends the class AndersonDarlingDist for the Anderson–Darling distribution (see [5], [139], [192] ). More... | |

class | BernoulliDist |

Extends the class DiscreteDistributionInt for the Bernoulli distribution [107] with parameter \(p\), where \(0\le p\le1\). More... | |

class | BetaDist |

Extends the class ContinuousDistribution for the beta distribution [89] (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 [107] (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 [88] (page 299) with location parameter \(\alpha\) and scale parameter \(\beta> 0\). More... | |

class | ChiDist |

Extends the class ContinuousDistribution for the chi distribution [88] (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 chi-square distribution with \(n\) degrees of freedom, where \(n\) is a positive integer [88] (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 chi-square distribution with \(\nu\) degrees of freedom and noncentrality parameter \(\lambda\), where \(\nu> 0\) and \(\lambda> 0\) [89] (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 Cramér-von Mises distribution (see [48], [191], [192] ). 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 [88] (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 [18] 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 [89] (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 [88] (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 [107] (page 322) with parameter \(p\), where \(0 < p < 1\). More... | |

class | GumbelDist |

Extends the class ContinuousDistribution for the Gumbel distribution [89] (page 2), with location parameter \(\delta\) and scale parameter \(\beta\neq0\). More... | |

class | HalfNormalDist |

Extends the class ContinuousDistribution for the half-normal 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 [60] (page 101) with \(k\) elements chosen among \(l\), \(m\) being of one type, and \(l-m\) 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 \(\lambda_i = (n+1-i)h\). 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 [39] . More... | |

class | InverseGammaDist |

Extends the class ContinuousDistribution for the inverse gamma distribution with shape parameter \(\alpha> 0\) and scale parameter \(\beta> 0\). 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 [90], [107], [56] 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 [90], [88] ). More... | |

class | JohnsonSUDist |

Extends the class ContinuousDistribution for the Johnson \(S_U\) distribution (see [107] (page 316)). More... | |

class | JohnsonSystem |

This class contains common parameters and methods for the Johnson system of distributions [90], [88] with shape parameters \(\gamma\) and \(\delta> 0\), location parameter \(\xi\), and scale parameter \(\lambda>0\). More... | |

class | KolmogorovSmirnovDist |

Extends the class ContinuousDistribution for the Kolmogorov-Smirnov distribution with parameter \(n\) [48] . 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 [36], [48], [25] ). More... | |

class | LaplaceDist |

Extends the class ContinuousDistribution for the Laplace distribution (see, e.g., [89] (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., [89] (page 115)). More... | |

class | LoglogisticDist |

Extends the class ContinuousDistribution for the Log-Logistic distribution with shape parameter \(\alpha> 0\) and scale parameter \(\beta> 0\). More... | |

class | LognormalDist |

Extends the class ContinuousDistribution for the lognormal distribution [88] . 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 [107] (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., [88] (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 \(\beta> 0\) [88] (page 574). More... | |

class | PascalDist |

The Pascal distribution is a special case of the negative binomial distribution [107] (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 piecewise-linear approximation of the empirical distribution function, based on the observations \(X_{(1)},…,X_{(n)}\) (sorted by increasing order), and defined as follows (e.g., [107] (page 318)). More... | |

class | PoissonDist |

Extends the class DiscreteDistributionInt for the Poisson distribution [107] (page 325) with mean \(\lambda\ge0\). More... | |

class | PowerDist |

Extends the class ContinuousDistribution for the power distribution [50] (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 [50] with location parameter \(a\), and scale parameter \(\beta> 0\). More... | |

class | StudentDist |

Extends the class ContinuousDistribution for the Student \(t\)-distribution [89] (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 [89] (page 297) and [107] (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 [89] (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 [37], [203] ). More... | |

class | WatsonUDist |

Extends the class ContinuousDistribution for the Watson U distribution (see [48], [191], [192] ). More... | |

class | WeibullDist |

This class extends the class ContinuousDistribution for the Weibull distribution [88] (page 628) with shape parameter \(\alpha> 0\), location parameter \(\delta\), and scale parameter \(\lambda> 0\). More... | |

Probability Distributions.

This package contains a set of Java classes providing methods to compute mass, density, distribution, complementary distribution, and inverse distribution functions for some discrete and continuous probability distributions. It also provides 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. It is possible to plot the density or the cumulative probabilities of a distribution function either on screen, or in a LaTeX file, but for this, one has to use the package umontreal::ssj::charts.

We recall that the *distribution function* of a *continuous* random variable \(X\) with *density* \(f\) over the real line is

\[ F(x) = P[X\le x] = \int_{-\infty}^x f(s)ds \tag{FDist} \]

while that of a *discrete* random variable \(X\) with *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 \(p(x_i) = P[X = x_i]\). For a discrete distribution over the set of integers, one has

\[ F (x) = P[X\le x] = \sum_{s=-\infty}^x p(s), \tag{FDistDiscInt} \]

where \(p(s) = P[X=s]\).

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 \(\bar{F}(x) = 1 - F (x-1)\) for discrete distributions over the integers. This definition is *non-standard* for the discrete case: we have \(\bar{F} (x) = P[X\ge x]\) instead of \(\bar{F} (x) = P[X > x] = 1-F(x)\). We find it more convenient especially for computing \(p\)-values in goodness-of-fit tests.

The *inverse 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 often 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 (for the Poisson distribution, for example), 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 one-time setup cost, for speed. In addition to the non-static 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 non-negative 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.

In fact, only the non-negligible 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 \(1-F (x)\) 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 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.

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