Elementary Applications of Probability Theory

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1 Elementary Applications of Probability Theory With an introduction to stochastic differential equations Second edition Henry C. Tuckwell Senior Research Fellow Stochastic Analysis Group of the Centre for Mathematics and its Applications Australian National University Australia CHAPMAN & HALL London Glasgow Weinheim New York Tokyo Melbourne Madras

2 Preface Preface to the first edition xi xiii 1 A review of basic probability theory Probability and random variables Mean and variance Conditional probability and independence Law of total probability Change of variables Two-dimensional random variables Hypothesis testing - the x 2 goodness of fit test Notation Geometric probability Buffon's needle problem The distance between two random points on a line segment The distance between two points dropped randomly in a circle Sum of two random variables Some applications of the hypergeometric and Poisson distributions The hypergeometric distribution Estimating a population from capture-recapture data The Poisson distribution Homogeneous Poisson point process in one dimension Occurrence of Poisson processes in Nature Poisson point processes in two dimensions Compound Poisson random variables The delta function 50

3 viii 3.9 An application in neurobiology Reliability theory Failure time distributions Reliability function and failure rate function The spare parts problem Complex systems Series and parallel systems Combinations and other structures 75 Further reading Simulation and random numbers The need for simulation The usefulness of a random sample from a uniform distribution Generation of uniform (0, 1) random numbers Generation of random numbers from a normal distribution Statistical tests for random numbers Testing for independence Convergence of sequences of random variables: the central limit theorem and the laws of large numbers Characteristic functions Examples Convergence in distribution The central limit theorem The Poisson approximation to the binomial distribution Convergence in probability Chebyshev's inequality The weak law of large numbers Simple random walks Random processes - definitions and classifications Unrestricted simple random walk Random walk with absorbing states 131

4 The probabilities of absorption at 0 Absorption at c>0 The case c = oo How long will absorption take? Smoothing the random walk - the Wiener process and Brownian motion ix Population genetics and Markov chains Genes and their frequencies in populations The Hardy-Weinberg principle Random mating in finite populations: a Markov chain model General description of Markov chains Temporally homogeneous Markov chains Random genetic drift Markov chains with absorbing states Absorption probabilities The mean time to absorption Mutation Stationary distributions Approach to a stationary distribution as n -»oo Population growth I: birth and death processes Introduction Simple Poisson processes Markov chains in continuous time The Yule process Mean and variance for the Yule process A simple death process Simple birth and death process Mean and variance for the birth and death process Population growth II: branching processes Cell division The Galton-Watson branching process Mean and variance for the Galton-Watson process Probability generating functions of sums of random variables 209

5 x 10.5 The probability of extinction Stochastic processes and an introduction to stochastic differential equations Deterministic and stochastic differential equations The Wiener process (Brownian motion) White noise The simplest stochastic differential equations - the Wiener process with drift Transition probabilities and the Chapman-Kolmogorov equation Diffusion processes, stochastic differential equations and applications Diffusion processes and the Kolmogorov (or Fokker-Planck) equations Stationary distributions 242 "12.3 The Wiener process with drift The Ornstein-Uhlenbeck process Stochastic integrals and stochastic differential equations Modelling with stochastic differential equations Applications Appendix Table of critical values of the ^-distribution 285 Index 286

COPYRIGHTED MATERIAL CONTENTS. Preface Preface to the First Edition

COPYRIGHTED MATERIAL CONTENTS. Preface Preface to the First Edition Preface Preface to the First Edition xi xiii 1 Basic Probability Theory 1 1.1 Introduction 1 1.2 Sample Spaces and Events 3 1.3 The Axioms of Probability 7 1.4 Finite Sample Spaces and Combinatorics 15