Chapter

A review of random variable theory

Daniel T. Gillespie and Effrosyni Seitaridou

in Simple Brownian Diffusion

Published in print October 2012 | ISBN: 9780199664504
Published online January 2013 | e-ISBN: 9780191748516 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199664504.003.0002
A review of random variable theory

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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, selective introduction to random variable theory, presenting some definitions and results that will be used repeatedly throughout the rest of the book. The presentation features utilitarian definitions of probability and random variables, and derivations of the mathematical rules they obey. Topics covered include: the probability density function and the cumulative distribution function; the uniform, exponential, normal, Cauchy, and binomial random variables; moments, means, variances, standard deviations, covariances, and correlations; the Dirac delta function and sure random variables; multivariate random variables; statistical independence; the random variable transformation theorem and its many useful consequences, such as the central limit theorem and linear combination theorems for normal random variables; the bivariate normal random variable; and the computer generation of random numbers by the inversion method. The presentation assumes fluency in standard calculus, but it does not require a knowledge of more advanced topics in mathematics.

Keywords: probability; random variable; probability density function; mean, variance; covariance; correlation; Gaussian; Cauchy; multivariate; bivariate normal; inversion generating method

Chapter.  9894 words.  Illustrated.

Subjects: Physics

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