umontreal.iro.lecuyer.probdistmulti (Java Libraries for Stochastic Simulation)

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Package umontreal.iro.lecuyer.probdistmulti

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Description

Class Summary
BiNormalDist	Extends the class `ContinuousDistribution2Dim` for the bivariate normal distribution.
BiNormalDonnellyDist	Extends the class `BiNormalDist` for the bivariate normal distribution using a translation of Donnelly's FORTRAN code.
BiNormalGenzDist	Extends the class `BiNormalDist` for the bivariate normal distribution using Genz's algorithm as described in.
BiStudentDist	Extends the class `ContinuousDistribution2Dim` for the standard bivariate Student's t distribution.
ContinuousDistribution2Dim	Classes implementing 2-dimensional continuous distributions should inherit from this class.
ContinuousDistributionMulti	Classes implementing continuous multi-dimensional distributions should inherit from this class.
DirichletDist	Implements the abstract class `ContinuousDistributionMulti` for the Dirichlet distribution with parameters (α₁,...,α_d), α_i > 0.
DiscreteDistributionIntMulti	Classes implementing multi-dimensional discrete distributions over the integers should inherit from this class.
MultinomialDist	Implements the abstract class `DiscreteDistributionIntMulti` for the multinomial distribution with parameters n and (p₁, ...,p_d).
MultiNormalDist	Implements the abstract class `ContinuousDistributionMulti` for the multinormal distribution with mean vector μ and covariance matrix Σ.
NegativeMultinomialDist	Implements the class `DiscreteDistributionIntMulti` for the negative multinomial distribution with parameters n > 0 and ( p₁,…, p_d) such that all 0 < p_i < 1 and ∑_i=1^dp_i < 1.

Package umontreal.iro.lecuyer.probdistmulti Description

This package contains Java classes providing methods to compute mass, density, distribution and complementary distribution functions for some multi-dimensional discrete and continuous probability distributions. It does not generate random numbers for multivariate distributions; for that, see the package umontreal.iro.lecuyerrandvarmulti.

Distributions

We recall that the distribution function of a continuous random vector X = {x₁, x₂,…, x_d} with density f (x₁, x₂,…, x_d) over the d-dimensional space R^d is

F(x₁, x₂,…, x_d)	=	P[X₁≤x₁, X₂≤x₂,…, X_d≤x_d]
	=	$\displaystyle \htint_{{-\infty}}^{{x_1}}$ $\displaystyle \htint_{{-\infty}}^{{x_2}}$ ^... $\displaystyle \htint_{{-\infty}}^{{x_d}}$ f (s₁, s₂,…, s_d) ds₁ds₂…ds_d

while that of a discrete random vector X with mass function {p₁, p₂,…, p_d} over a fixed set of real numbers is

F(x₁, x₂,…, x_d)	=	P[X₁≤x₁, X₂≤x₂,…, X_d≤x_d]
	=	$\displaystyle \htsum_{{i_1\le x_1}}^{}$ $\displaystyle \htsum_{{i_2\le x_2}}^{}$ ^... $\displaystyle \htsum_{{i_d\le x_d}}^{}$ p(x₁, x₂,…, x_d),

where p(x₁, x₂,…, x_d) = P[X₁ = x₁, X₂ = x₂,…, X_d = x_d]. For a discrete distribution over the set of integers, one has

F(x₁, x₂,…, x_d)	=	P[X₁≤x₁, X₂≤x₂,…, X_d≤x_d]
	=	$\displaystyle \htsum_{{s_1=-\infty}}^{{x_1}}$ $\displaystyle \htsum_{{s_2=-\infty}}^{{x_2}}$ ^... $\displaystyle \htsum_{{s_d=-\infty}}^{{x_d}}$ p(s₁, s₂,…, s_d),