Elementary Applications of Probability Theory

Henry C. Tuckwell

• Publisher: Chapman and Hall/CRC
• Year: 2018
• eBook ISBN: 9781351452953
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

This book provides a clear and straightforward introduction to applications of probability theory with examples given in the biological sciences and engineering. The first chapter contains a summary of basic probability theory. Chapters two to five deal with random variables and their applications. Topics covered include geometric probability, estimation of animal and plant populations, reliability theory and computer simulation. Chapter six contains a lucid account of the convergence of sequences of random variables, with emphasis on the central limit theorem and the weak law of numbers. The next four chapters introduce random processes, including random walks and Markov chains illustrated by examples in population genetics and population growth. This edition also includes two chapters which introduce, in a manifestly readable fashion, the topic of stochastic differential equations and their applications.

1

A review of basic probability theory

This is a book about the applications of probability. It is hoped to convey that this subject is both a fascinating and important one. The examples are drawn mainly from the biological sciences but some originate in the engineering, physical, social and statistical sciences. Furthermore, the techniques are not limited to any one area.

The reader is assumed to be familiar with the elements of probability or to be studying it concomitantly. In this chapter we will briefly review some of this basic material. This will establish notation and provide a convenient reference place for some formulas and theorems which are needed later at various points.

1.1 PROBABILITY AND RANDOM VARIABLES

When an experiment is performed whose outcome is uncertain, the collection of possible elementary outcomes is called a sample space, often denoted by Ω. Points in Ω, denoted in the discrete case by ω_i, i = 1, 2, … have an associated probability P{ω_i}. This enables the probability of any subset A of Ω, called an event, to be ascertained by finding the total probability associated with all the points in the given subset:

$P\left\{A\right\}={\displaystyle \sum _{\omega ,ϵA}P\left\{{\omega }_i\right\}}$

We always have

$0⩽P\left\{A\right\}⩽1,$

and in particular P{Ω} = 1 and P{∅} = 0, where ∅ is the empty set relative to Ω.

A random variable is a real-valued function defined on the elements of a sample space. Roughly speaking it is an observable which takes on numerical values with certain...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface
• Preface to the first edition
• 1 A review of basic probability theory
• 2 Geometric probability
• 3 Some applications of the hypergeometric and Poisson distributions
• 4 Reliability theory
• 5 Simulation and random numbers
• 6 Convergence of sequences of random variables: the central limit theorem and the laws of large numbers
• 7 Simple random walks
• 8 Population genetics and Markov chains
• 9 Population growth I: birth and death processes
• 10 Population growth II: branching processes
• 11 Stochastic processes and an introduction to stochastic differential equations
• 12 Diffusion processes, stochastic differential equations and applications
• Appendix Table of critical values of the χ2 distribution
• Index