Continuous Probability Distributions in Concurrent Games

In many real-world applications, the phenomena of interest are continuous in nature and modeled through continuous probability distributions, but their observed values are actually discrete and hence it would be more reasonable and convenient to choose an appropriate (multivariate) discrete distribution generated from the underlying continuous model preserving one or more important features. In this paper, two methods are discussed for deriving a bivariate discrete probability distribution from a continuous one by retaining some specific features of the original stochastic model, namely (1) the joint density function, or (2) the joint survival function. Examples of applications are presented, which involve two types of bivariate exponential distributions, in order to illustrate how the discretization procedures work and show whether and to which extent they alter the dependence structure of the original model. We also prove that some bivariate discrete distributions that were recently proposed in the literature can be actually regarded as discrete counterparts of well-known continuous models. A numerical study is presented in order to illustrate how the procedures are practically implemented and to present inferential aspects. Two real datasets, considering correlated discrete recurrence times (the former) and counts (the latter) are eventually fitted using two discrete analogues of a bivariate exponential distribution.

Higher-order probabilistic programming languages allow programmers to write sophisticated models in machine learning and statistics in a succinct and structured way, but step outside the standard measure-theoretic formalization of probability theory. Programs may use both higher-order functions and continuous distributions, or even define a probability distribution on functions. But standard probability theory does not handle higher-order functions well: the category of measurable spaces is not cartesian closed. Here we introduce quasi-Borel spaces. We show that these spaces: form a new formalization of probability theory replacing measurable spaces; form a cartesian closed category and so support higher-order functions; form a well-pointed category and so support good proof principles for equational reasoning; and support continuous probability distributions. We demonstrate the use of quasi-Borel spaces for higher-order functions and probability by: showing that a well-known construction of probability theory involving random functions gains a cleaner expression; and generalizing de Finetti's theorem, that is a crucial theorem in probability theory, to quasi-Borel spaces.

We propose an original classification of several discrete and continuous probability distributions. We establish links between these distributions, in particular the little known relationship between the negative hypergeometric distribution and the beta distribution. These relations allow us to propose a structure of relations which is summarized in graphic form.Our classification emphasises the analogy between certain discrete and continuous distributions. This analogy makes it possible to establish relations between the theory of point processes and the theory of survey sampling. It also makes it possible to envisage the use of link functions that are little used in generalised regression.

The survival function method is a well-known alternative procedure for calculating the expectation of a nonnegative random variable with a continuous probability distribution. This paper proves a generalization of this method that can be used to calculate the expectation of a continuously differentiable function of a nonnegative random variable with a continuous, discrete or mixed probability distribution. A similar result is used in actuarial probability manuals, but proofs for the general case along with required assumptions are often not provided. Moreover, generalizations of the survival function method are typically not mentioned in probability textbooks. One of the main contributions of this paper is that it explores some technical conditions that guarantee that the method can be applied. This paper also provides an alternate proof of the generalized survival function method along with a different set of conditions under which the method can be proved to hold. Finally, examples of how the method can be applied to calculate expectations and moment-generating functions and to derive an integral identity are given.

As probabilistic computations play an increasing role in solving various problems, researchers have designed probabilistic languages that treat probability distributions as primitive datatypes. Most probabilistic languages, however, focus only on discrete distributions and have limited expressive power. In this paper, we present a probabilistic language, called λο, which uniformly supports all kinds of probability distributions -- discrete distributions, continuous distributions, and even those belonging to neither group. Its mathematical basis is sampling functions, i.e. , mappings from the unit interval (0.0,1.0] to probability domains.We also briefly describe the implementation of λο as an extension of Objective CAML and demonstrate its practicality with three applications in robotics: robot localization, people tracking, and robotic mapping. All experiments have been carried out with real robots.

This paper introduces a new class of continuous probability distributions that are flexible enough to represent a wide range of uncertainties such as those that commonly arise in business, technology, and science. In many such cases, the nature of the uncertainty is more naturally characterized by quantiles than by parameters of familiar continuous probability distributions. In the practice of decision analysis, it is common to fit a hand-drawn curve to quantile outputs from probability elicitations on a continuous uncertain quantity and to then discretize the curve. The resulting discrete probability distribution is an approximation that cuts off the distribution's tails and eliminates intermediate values. Quantile-parameterized distributions address this problem by using quantiles themselves to parameterize a continuous probability distribution. We define quantile-parameterized distributions, illustrate their flexibility and range of applicability, and conclude with practical considerations when parameterizing distributions using inconsistent quantile assessments.

Continuous probability distribution was one of the mathematics concept that students had difficulty.This study analyzed the definition and introduction of the continuous probability distribution under the 2009-revised curriculum and 2015-revised curriculum.In this study, the following subjects were studied.Firstly, definitions of continuous probability variable in ' Probability and Statistics' textbook developed under the 2009-revised curriculum and 2015-revised curriculum were analyzed.Secondly, introductions of continuous probability distribution in ' Probability and Statistics' textbook developed under the 2009-revised curriculum and 2015-revised curriculum were analyzed.The results of this study were as follows.First, 8 textbooks under the 2009-revised curriculum defined the continuous probability variable as probability variable with all the real values within a range or an interval.And 1 textbook under the 2009-revised curriculum defined the continuous probability variable as probability variable when the set of its value is uncountable.But all textbooks under the 2015-revised curriculum defined the continuous probability variable as probability variable with all the real values within a range.Second, 4 textbooks under the 2009-revised curriculum and 4 textbooks under 2015-revised curriculum introduced a continuous random distribution using an uniformly distribution.And 5 textbooks under the 2009-revised curriculum and 5 textbooks under the 2015-revised curriculum introduced a continuous random distribution using a relative frequency density.

Frequency-dependent acoustical loss due to a multitude of physical mechanisms is commonly modeled by multiple relaxations. For discrete relaxation distributions, such models correspond with causal wave equations of integer-order temporal derivatives. It has also been shown that certain continuous distributions may give causal wave equations with fractional-order temporal derivatives. This paper demonstrates analytically that if the wave-frequency ω satisfies ΩL≪ω ≪ΩH, a continuous relaxation distribution populating only Ω∈[ΩL,ΩH] gives the same effective wave equation as for a fully populated distribution. This insight sparks the main contribution: the elaboration of a method to determine discrete relaxation parameters intended for mimicking a desired attenuation behavior for band-limited waves. In particular, power-law attenuation is discussed as motivated by its prevalence in complex media, e.g., biological tissue. A Mittag-Leffler function related distribution of relaxation mechanisms has previously been shown to be related to the fractional Zener wave equation of three power-law attenuation regimes. Because these regimes correspond to power-law regimes in the relaxation distribution, the idea is to sample the distribution's compressibility contributions evenly in logarithmic frequency while appropriately taking the stepsize into account. This work thence claims to provide a model-based approach to determination of discrete relaxation parameters intended to adequately model attenuation power-laws.

In many management science or economic applications, it is common to represent the key uncertain inputs as continuous random variables. However, when analytic techniques fail to provide a closed-form solution to a problem or when one needs to reduce the computational load, it is often necessary to resort to some problem-specific approximation technique or approximate each given continuous probability distribution by a discrete distribution. Many discretization methods have been proposed so far; in this work, we revise the most popular techniques, highlighting their strengths and weaknesses, and empirically investigate their performance through a comparative study applied to a well-known engineering problem, formulated as a stress–strength model, with the aim of weighting up their feasibility and accuracy in recovering the value of the reliability parameter, also with reference to the number of discrete points. The results overall reward a recently introduced method as the best performer, which derives the discrete approximation as the numerical solution of a constrained non-linear optimization, preserving the first two moments of the original distribution. This method provides more accurate results than an ad-hoc first-order approximation technique. However, it is the most computationally demanding as well and the computation time can get even larger than that required by Monte Carlo approximation if the number of discrete points exceeds a certain threshold.

This system description paper introduces the probabilistic programming language Hakaru10, for expressing, and performing inference on (general) graphical models. The language supports discrete and continuous distributions, mixture distributions and conditioning. Hakaru10 is a DSL embedded in Haskell and supports Monte-Carlo Markov Chain (MCMC) inference.

Chapter 4 - Examples of Continuous Probability Distributions

: This paper presents a method for training connectional networks that adhere to the principles of graded, random, adaptive, and interactive propagation of information (GRAIN). While our analysis has bee motivated by our desire to find a learning algorithm that would work in this environment, we have succeeded in implementing a model that encompasses a large class of previous connectionist algorithms under the same theoretical principles and that expands the scope of problems they can learn. Stimulations show examples where GRAIN networks successfully approximate both discrete and continuous probability distributions, demonstrating that their scope extends beyond what can be learned by backpropagation networks or standard Boltzmann machines.

This chapter is aimed at introducing and explaining some basic concepts of statistics and probability in order to aid the reader in understanding some of the more advanced concepts presented in the main text of the book. The main topics that are discussed are set theory, permutations and combinations, discrete and continuous probability distributions, descriptive statistics, and bivariate distributions.

At first in Chap. 3 are described such important notions as characteristic functions of a probability distribution defined as Fourier transform of a probability density function and constants characterizing random variables: an average value, a median value describing the middle of a distribution, a variation ratio closely connected with a standard deviation. Other characteristic constants characterizing random variables are: an asymmetry coefficient and an excess coefficient. Other parameters describing shape of a probability density functions called quantiles are also defined. A number of useful probability distributions are presented, among them are discrete probability distributions: a binomial distribution, a multinomial distribution, a Poisson distribution, and continuous probability distributions: a rectangular distribution, an exponential distribution, a Weibull distribution and a Student distribution are described. Finally are given some introductory remarks concerning multidimensional probability distributions, developed further in Chaps. 5 and 7 for analyzing various practical problems.

In socio-economics sciences, researchers need to model behaviour of ordinal variables in simultaneous equation system involving multiple endogenous ordinal variables. In existing literature the tedious solution of such likelihood function is misunderstood by applied researchers see O’Brien and Homer (Qual Quant 21(4):349–360, 1987). This paper is an attempt to convert ordinal variables into continuous format for the common use in standard simultaneous regression models. The proposed method involves generating random numbers from continuous probability distributions (say, uniform and/or truncated normal distributions) within a given discrete probability distribution. The new method can be used in ordered-logit and probit models by avoiding existing restrictive assumptions. However, the proposed method can be used under certain restrictions.