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SSJ: Package umontreal.ssj.probdist

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 \(\bar{F}(x) = 1 - F (x-1)\) 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 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 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.

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 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.

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