## Representing Response Times as Random Variables

*R. Duncan Luce*

### in Response Times: Their Role in Inferring Elementary Mental Organization

- Published in print:
- 1991
- Published Online:
- January 2008
- ISBN:
- 9780195070019
- eISBN:
- 9780199869879
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195070019.003.0001
- Subject:
- Psychology, Cognitive Models and Architectures

This chapter begins with a discussion of the study of response times. Response times are treated as observations of a random variable. The mathematics of stochastic processes is then used to ... More

## Probability and Statistical Distributions

*Željko Ivezi, Andrew J. Connolly, Jacob T. VanderPlas, Alexander Gray, Željko Ivezi, Andrew J. Connolly, Jacob T. VanderPlas, and Alexander Gray*

### in Statistics, Data Mining, and Machine Learning in Astronomy: A Practical Python Guide for the Analysis of Survey Data

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691151687
- eISBN:
- 9781400848911
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691151687.003.0003
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

This chapter reviews notation and basic concepts in probability and statistics. The coverage of various topics cannot be complete, and it is aimed at concepts needed to understand material covered in ... More

## Introduction to Probability and Random Variables

*M. Vidyasagar*

### in Hidden Markov Processes: Theory and Applications to Biology

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691133157
- eISBN:
- 9781400850518
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691133157.003.0001
- Subject:
- Mathematics, Probability / Statistics

This chapter provides an introduction to probability and random variables. Probability theory is an attempt to formalize the notion of uncertainty in the outcome of an experiment. For instance, ... More

## Going, Going …

*Ken Binmore*

### in Playing for Real: Game Theory

- Published in print:
- 2007
- Published Online:
- May 2007
- ISBN:
- 9780195300574
- eISBN:
- 9780199783748
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195300574.003.0021
- Subject:
- Economics and Finance, Microeconomics

The most successful area of game theory is auction design. This chapter offers a short review of the subject. Various traditional auctions are described. After showing that most of these generate the ... More

## Introduction to Information Theory

*M. Vidyasagar*

### in Hidden Markov Processes: Theory and Applications to Biology

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691133157
- eISBN:
- 9781400850518
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691133157.003.0002
- Subject:
- Mathematics, Probability / Statistics

This chapter provides an introduction to some elementary aspects of information theory, including entropy in its various forms. Entropy refers to the level of uncertainty associated with a random ... More

## Fourier Transforms in Probability, Random Variables and Stochastic Processes

*Robert J Marks II*

### in Handbook of Fourier Analysis & Its Applications

- Published in print:
- 2009
- Published Online:
- November 2020
- ISBN:
- 9780195335927
- eISBN:
- 9780197562567
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195335927.003.0009
- Subject:
- Computer Science, Mathematical Theory of Computation

In this Chapter, we present application of Fourier analysis to probability, random variables and stochastic processes [1089, 1097, 1387, 1329]. Arandom variable, X, is the assignment of a number to ... More

## A review of random variable theory

*Daniel T. Gillespie and Effrosyni Seitaridou*

### in Simple Brownian Diffusion: An Introduction to the Standard Theoretical Models

- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199664504
- eISBN:
- 9780191748516
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199664504.003.0002
- Subject:
- Physics, Soft Matter / Biological Physics

In extending the analysis of diffusion beyond the traditional approach described in Chapter 1, the chapter here finds it necessary to work with random variables. This chapter gives a self-contained, ... More

## Classical Probability Theory and Stochastic Processes

*Heinz-Peter Breuer and Francesco Petruccione*

### in The Theory of Open Quantum Systems

- Published in print:
- 2007
- Published Online:
- February 2010
- ISBN:
- 9780199213900
- eISBN:
- 9780191706349
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199213900.003.01
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter contains a survey of classical probability theory and stochastic processes. It starts with a description of the fundamental concepts of probability space and Kolmogorov axioms. These ... More

## Entropy and Mutual Information

*Partha P. Mitra and Hemant Bokil*

### in Observed Brain Dynamics

- Published in print:
- 2007
- Published Online:
- May 2009
- ISBN:
- 9780195178081
- eISBN:
- 9780199864829
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195178081.003.0014
- Subject:
- Neuroscience, Techniques, Molecular and Cellular Systems

In the neuroscience literature, significant use has been made of entropy as a measure of variability, and mutual information as a measure of association. Part of the attraction of these measures ... More

## Taking Chances

*Ken Binmore*

### in Playing for Real: Game Theory

- Published in print:
- 2007
- Published Online:
- May 2007
- ISBN:
- 9780195300574
- eISBN:
- 9780199783748
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195300574.003.0003
- Subject:
- Economics and Finance, Microeconomics

This chapter shows how risk can be introduced into the rules of a game by admitting chance moves. The Monty Hall problem is offered as an example. Elementary probability theory is reviewed, and ... More

## Simulating molecular evolution

*ZIHENG YANG*

### in Computational Molecular Evolution

- Published in print:
- 2006
- Published Online:
- April 2010
- ISBN:
- 9780198567028
- eISBN:
- 9780191728280
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567028.003.0009
- Subject:
- Biology, Evolutionary Biology / Genetics

This chapter discusses basic techniques of computer simulation. Topics covered include random number generator, generation of continuous random variables, generation of discrete random variables, and ... More

## Stochastic Processes and Stochastic Calculus

*Claus Munk*

### in Fixed Income Modelling

- Published in print:
- 2011
- Published Online:
- September 2011
- ISBN:
- 9780199575084
- eISBN:
- 9780191728648
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199575084.003.0003
- Subject:
- Economics and Finance, Financial Economics

The price of an asset at a future point in time will typically be unknown, i.e. a random variable. In order to describe the uncertain evolution in the price of the asset over time, we need a ... More

## PROBABILITY THEORY

*Robert M. Mazo*

### in Brownian Motion: Fluctuations, Dynamics, and Applications

- Published in print:
- 2008
- Published Online:
- January 2010
- ISBN:
- 9780199556441
- eISBN:
- 9780191705625
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199556441.003.0002
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter begins with a review of elementary probability theory, conditional probability, and statistical independence. It explains the notion of a random variable and its distribution function or ... More

## Real-valued Random Processes

*Arno Berger and Theodore P. Hill*

### in An Introduction to Benford's Law

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691163062
- eISBN:
- 9781400866588
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691163062.003.0008
- Subject:
- Mathematics, Probability / Statistics

Benford's law arises naturally in a variety of stochastic settings, including products of independent random variables, mixtures of random samples from different distributions, and iterations of ... More

## Properties of random variables

*M. D. Edge*

### in Statistical Thinking from Scratch: A Primer for Scientists

- Published in print:
- 2019
- Published Online:
- October 2019
- ISBN:
- 9780198827627
- eISBN:
- 9780191866463
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198827627.003.0006
- Subject:
- Biology, Biomathematics / Statistics and Data Analysis / Complexity Studies

In this chapter, the behavior of random variables is summarized using the concepts of expectation, variance, and covariance. The expectation is a measurement of the location of a random variable’s ... More

## Random Variables

*James Davidson*

### in Stochastic Limit Theory: An Introduction for Econometricians

- Published in print:
- 1994
- Published Online:
- November 2003
- ISBN:
- 9780198774037
- eISBN:
- 9780191596117
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198774036.003.0008
- Subject:
- Economics and Finance, Econometrics

Specializing the concepts of Ch. 7 to the case of real variables, this chapter introduces distribution functions, discrete and continuous distributions, and describes examples such as the binomial, ... More

## Degree of Belief is Expected Truth Value

*Nicholas J. J. Smith*

### in Cuts and Clouds: Vagueness, its Nature, & its Logic

- Published in print:
- 2010
- Published Online:
- May 2010
- ISBN:
- 9780199570386
- eISBN:
- 9780191722134
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199570386.003.0029
- Subject:
- Philosophy, Logic/Philosophy of Mathematics, Metaphysics/Epistemology

A number of authors have noted that vagueness engenders degrees of belief, but that these degrees of belief do not behave like subjective probabilities. So should we countenance two different kinds ... More

## Basic probability theory

*Thomas P. Trappenberg*

### in Fundamentals of Machine Learning

- Published in print:
- 2019
- Published Online:
- January 2020
- ISBN:
- 9780198828044
- eISBN:
- 9780191883873
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198828044.003.0006
- Subject:
- Neuroscience, Behavioral Neuroscience

The discussion provides a refresher of probability theory, in particular with respect to the formulations that build the theoretical language of modern machine learning. Probability theory is the ... More

## Langevin treatment of the Fokker–Planck process

*Melvin Lax, Wei Cai, and Min Xu*

### in Random Processes in Physics and Finance

- Published in print:
- 2006
- Published Online:
- January 2010
- ISBN:
- 9780198567769
- eISBN:
- 9780191718359
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567769.003.0010
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter discusses the Langevin treatment of the Fokker–Planck process and diffusion. The form of Langevin equation used is different from the stochastic differential equation using Ito's ... More

## Probability and random variables

*M. D. Edge*

### in Statistical Thinking from Scratch: A Primer for Scientists

- Published in print:
- 2019
- Published Online:
- October 2019
- ISBN:
- 9780198827627
- eISBN:
- 9780191866463
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198827627.003.0005
- Subject:
- Biology, Biomathematics / Statistics and Data Analysis / Complexity Studies

This chapter considers the rules of probability. Probabilities are non-negative, they sum to one, and the probability that either of two mutually exclusive events occurs is the sum of the probability ... More