Probability and Statistics

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.

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.

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.

3 - A Primer on Financial Time Series Analysis

One chapter cannot possibly review what one learned in one or two pre-requisite courses in statistics. This is an econometrics book, and it is imperative that the student have taken at least one solid course in statistics. The concepts of a random variable, whether discrete or continuous, and the associated probability function or probability density function (p.d.f) are assumed known. Similarly, the reader should know the following statistical terms: Cumulative distribution function, marginal, conditional and joint p.d.f ’s. The reader should be comfortable with computing mathematical expectations, and familiar with the concepts of independence, Bayes Theorem and several continuous and discrete probability distributions. These distributions include: the Bernoulli, Binomial, Poisson, Geometric, Uniform, Normal, Gamma, Chi-squared (χ2), Exponential, Beta, t and F distributions.

This study explores the integration of the GeoGebra application into statistical learning and its effectiveness in supporting problem-solving processes. GeoGebra serves not only as a computational tool but also as a pedagogical medium that fosters conceptual understanding, visualization, and interactive demonstrations. Through its dynamic and visual features, GeoGebra enables lecturers and students to engage in deeper discussions of statistical topics, thereby strengthening the comprehension of fundamental concepts. Several interactive applets were designed to facilitate learning on topics such as data processing, discrete and continuous probability distributions, and hypothesis testing. Examples include an applet for the Z-test of a single mean and a visualization of the area under the F-distribution curve. The slider function in GeoGebra provides flexibility for dynamically adjusting parameters that are directly linked to graphical representations, helping students to intuitively grasp abstract statistical ideas. By transforming abstract statistical concepts into interactive visual forms, GeoGebra contributes to a more meaningful learning experience. Moreover, the integration of GeoGebra in the classroom opens new opportunities for action research focused on enhancing students’ conceptual understanding and developing innovative, technology-based learning strategies. The findings underscore that GeoGebra can be effectively utilized as a bridge between theoretical and applied statistics, promoting active learning and deeper engagement among students in higher education settings.

: 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.

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.

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.

The time to extinction (TTE) is very important topic in some fields of studies; ecology, economics, corporate competition, bacterial sciences and epidemiology. The aim of the present paper is to investigate about the empirical parametric and empirical nonparametric probability distribution of the time to extinction for two related stochastic models; Rosenzweig and Macarthur model and May model. First, we assume the amplitude of r is a random variable with a continuous uniform probability distribution on a closed interval [r 1, r 2]. Second, we assume the number of consecutive years’ during which the amplitude of r remains constant is a random variable with a discrete geometric probability distribution with parameter p which is supported on a positive integer ℤ+ and p ∈ (0,1).

A Strategy for the Integration of Production Planning and Scheduling in Refineries under Uncertainty

Continuous Probability Distributions in Concurrent Games

Permutation tests are based on all possible arrangements of observed data sets. Consequently, such tests yield exact probability values obtained from discrete probability distributions. An exact nondirectional method to combine independent probability values that obey discrete probability distributions is introduced. The exact method is the discrete analog to Fisher's classical method for combining probability values from independent continuous probability distributions. If the combination of probability values includes even one probability value that obeys a sparse discrete probability distribution, then Fisher's classical method may be grossly inadequate.

There is a growing movement to introduce elements of statistics and probability into the secondary and even the elementary school curriculum, as part of basic literacy in mathematics. Although many articles in the education literature recommend how to teach statistics better, there is little published research on how students actually learn statistics concepts. The experience of psychologists, educators, and statisticians alike is that a large proportion of students, even in college, do not understand many of the basic statistical concepts they have studied. Inadequacies in prerequisite mathematics skills and abstract reasoning are part of the problem. In addition, research in cognitive science demonstrates the prevalence of some intuitive ways of thinking that interfere with the learning of correct statistical reasoning. The literature reviewed in this paper indicates a need for collaborative, cross-disciplinary research on how students come to think correctly about probability and statistics.