Local Average Probabilities of Randomistic Variables (original) (raw)

Discretization of Probability Distributions: Random, Deterministic and Randomistic Sampling

ForsChem Research Reports, 2019

Sampling procedures are commonly used to extract a finite number of elements from a particular probability distribution. This discretization of the probability distribution is usually performed using pseudo-random number generators. This type of discretization, known as random sampling, requires suitable functions for transforming standard uniform random numbers into random numbers following any arbitrary probability distribution. While random sampling resembles the natural behavior of experimentation, individual samples do not necessarily preserve all the properties of the original probability distribution. Those properties include the cumulative probability and the moments of the distribution. The match between the cumulative probability observed in a sample and that of the original distribution can be determined using the random goodness-of-fit criterion. Random samples seldom achieve a 100% fit to the original distribution. Deterministic sampling methods, on the other hand, always present a 100% random goodness-of-fit, but their values are always the same, depending on the size of the sample. One particular case of deterministic sampling is optimal sampling, which ensure goodness-of-fit but also allows preserving the moments of the original distribution. Finally, randomistic sampling combines the fitness of deterministic sampling with the changing behavior of random samples, resulting in an interesting alternative for representing random variables, particularly in applications involving Monte Carlo methods, where the sample is expected to represent the properties of the full distribution.

INTRODUCTION TO PROBABILITY AND PROBABILITY DISTRIBUTIONS

The book is written with the realization that concepts of probability and probability distributions – even though they often appear deceptively simple – are in fact difficult to comprehend. Every basic concept and method is therefore explained in full, in a language that is easily understood. A prominent feature of the book is the inclusion of many examples. Each example is carefully selected to illustrate the application of a particular statistical technique and or interpretation of results. Another feature is that each chapter has an extensive collection of exercises. Many of these exercises are from published sources, including past examination questions from King Saud University in Saudi Arabia, and Methodist University College Ghana. Answers to all the exercises are given at the end of the book.

A practical overview on probability distributions

Journal of thoracic disease, 2015

Aim of this paper is a general definition of probability, of its main mathematical features and the features it presents under particular circumstances. The behavior of probability is linked to the features of the phenomenon we would predict. This link can be defined probability distribution. Given the characteristics of phenomena (that we can also define variables), there are defined probability distribution. For categorical (or discrete) variables, the probability can be described by a binomial or Poisson distribution in the majority of cases. For continuous variables, the probability can be described by the most important distribution in statistics, the normal distribution. Distributions of probability are briefly described together with some examples for their possible application.

Numerical Determination of the Probability Density of Functions of Randomistic Variables

ForsChem Research Reports, 2022

The change of variable theorem is a useful equation for analytically determining the resulting probability density of an arbitrary function of one or more independent randomistic variables. The term randomistic may represent purely deterministic variables, purely random variables, or a combination of both. The change of variable theorem requires an inverse of the original function where one of the independent variables is explicitly solved in terms of all other variables. Unfortunately, this is not always the case, and therefore the analytical change of variable theorem cannot be used in those situations. In addition, when two or more independent variables are involved, the analytical change of variable theorem requires solving one or more definite integrals, and there are situations where the integrals cannot be expressed as known analytical functions. In this report, a numerical version of the change of variable theorem is presented for obtaining the probability density of a function, when the analytical change of variable theorem does not succeed or cannot be used. The proposed method is implemented in R language, and its use is illustrated with several examples. Most examples considered are comparative, where the exact analytical solution is known, in order to validate the performance of the method. Similitude percentages above 95% were obtained. Of course, both the accuracy and computational demand of the numerical method will strongly depend on the step sizes and tolerance considered. Additional examples are included to show that numerical solutions are possible even when the analytical method fails.