Elementary Applications of Probability Theory: With an introduction to stochastic differential equations

307 Pages·1995·5.468 MB·English

by  Henry C. Tuckwell (auth.)

Elementary Applications of
Probability Theory
CHAPMAN & HALL STATISTICS TEXTBOOK SERIES
Editors:
Dr Chris Chatfield Professor Jim V. Zidek
Reader in Statistics Department of Statistics
School of Mathematical Sciences University of British Columbia, Canada
University of Bath, UK
OTHER TITLES IN THE SERIES INCLUDE
Practical Statistics for Medical Research An Introduction to Generalized Linear
D.G. Altman Models
A.J. Dobson
Interpreting Data
A.J.B. Anderson Multivariate Analysis of Variance and
Repeated Measures
Statistical Methods for SPC and TQM
D.J. Hand and C.C. Taylor
D. Bissell
The Theory of Linear Models
Statistics in Research and Development
B. Jorgensen
Second edition
R. Caulcutt Statistical Theory
Fourth edition
The Analysis of Time Series
B. Lindgren
Fourth edition
C. Chatfield Essential Statistics
Second edition
Problem Solving - A Statistician's Guide
D.G. Rees
C. Chatfield
Decision Analysis: A Bayesian Approach
Statistics for Technology
J.Q. Smith
Third edition
C. Chatfield Applied Nonparametric Statistical
Methods
Introduction to Multivariate Analysis
Second edition
C. Chatfield and A.J. Collins
P. Sprent
Modelling Binary Data
Elementary Applications of Probability
D. Collett
Theory
H.C. Tuckwell
Model6ng Survival Data in Medical
Research
Statistical Process Control: Theory and
D. Collett
Practice
Third edition
Applied Statistics
G.B. Wetherill and D.W. Brown
D.R. Cox and E.J. Snell
Statistics in Engineering
Statistical Analysis of Reliability Data
A practical approach
M.J. Crowder, A.C. Kimber, T.J.
A.V. Metcalfe
Sweeting and R.L. Smith
Full information on the complete range of Chapman & Hall statistics books is available
from the publishers.
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
SPRINGER-SCIENCE+BUSINESS MEDIA. B.V.
First edition 1988
Second edition 1995
© 1988, 1995 Henry C. Tuckwell
Originally published by Chapman & Hall in 1995
Typeset in 10/12 pt Times by Thomson Press (India) Ltd, New Delhi
ISBN 978-0-412-57620-1 ISBN 978-1-4899-3290-7 (eBook)
DOI 10.1007/978-1-4899-3290-7
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To Silvia Dori
Contents
Preface xi
Preface to the first edition xiii
1 A review of basic probability theory 1
1.1 Probability and random variables 1
1.2 Mean and variance 5
1.3 Conditional probability and independence 5
1.4 Law of total probability 7
1.5 Change of variables 8
1.6 Two-dimensional random variables 10
1.7 Hypothesis testing-the x2 goodness of fit test 11
1.8 Notation 13
References 14
2 Geometric probability 16
2.1 Buffon's needle problem 16
2.2 The distance between two random points on a line segment 19
2.3 The distance between two points dropped randomly in a
circle 21
2.4 Sum of two random variables 25
References 27
Exercises 27
3 Some applications of the hypergeometric and Poisson
distributions 30
3.1 The hypergeometric distribution 30
3.2 Estimating a population from capture-recapture data 33
3.3 The Poisson distribution 37
3.4 Homogeneous Poisson point process in one dimension 38
3.5 Occurrence of Poisson processes in Nature 41
3.6 Poisson point processes in two dimensions 44
3.7 Compound Poisson random variables 48
3.8 The delta function 50
viii Contents
3.9 An application in neurobiology 52
References 56
Exercises 57
4 Reliability theory 61
4.1 Failure time distributions 61
4.2 Reliability function and failure rate function 63
4.3 The spare parts problem 69
4.4 Complex systems 70
4.5 Se.ries and parallel systems 72
4.6 Combinations and other structures 75
Further reading 77
References 77
Exercises 77
5 Simulation and random numbers 81
5.1 The need for simulation 81
5.2 The usefulness of a random sample from a uniform
distribution 83
5.3 Generation of uniform (0, 1) random numbers 86
5.4 Generation of random numbers from a normal distribution 88
5.5 Statistical tests for random numbers 90
5.6 Testing for independence 92
References 96
Exercises 96
6 Convergence of sequences of random variables: the central limit
theorem and the laws of large numbers 98
6.1 Characteristic functions 98
6.2 Examples 101
6.3 Convergence in distribution 104
6.4 The central limit theorem 107
6.5 The Poisson approximation to the binomial distribution 110
6.6 Convergence in probability 111
6.7 Chebyshev's inequality 113
6.8 The weak law of large numbers 115
References 119
Exercises 119
7 Simple random walks 123
7.1 Random processes - definitions and classifications 123
7.2 Unrestricted simple random walk 126
7.3 Random walk with absorbing states 131
Contents 1x
7.4 The probabilities of absorption at 0 132
7.5 Absorption at c > 0 137
7.6 The case c = oo 138
7.7 How long will absorption take? 139
7.8 Smoothing the random walk - the Wiener process and Brownian
motion 142
References 145
Exercises 145
8 Population genetics and Markov chains 148
8.1 Genes and their frequencies in populations 148
8.2 The Hardy-Weinberg principle 150
8.3 Random mating in finite populations: a Markov chain model 153
8.4 General description of Markov chains 154
8.5 Temporally homogeneous Markov chains 155
8.6 Random genetic drift 158
8.7 Markov chains with absorbing states 160
8.8 Absorption probabilities 162
8.9 The mean time to absorption 167
8.10 Mutation 171
8.11 Stationary distributions 173
8.12 Approach to a stationary distribution as n-+ oo 174
References 178
Exercises 17 9
9 Population growth 1: birth and death processes 183
9.1 Introduction 183
9.2 Simple Poisson processes 185
9.3 Markov chains in continuous time 187
9.4 The Yule process 188
9.5 Mean and variance for the Yule process 192
9.6 A simple death process 194
9.7 Simple birth and death process 196
9.8 Mean and variance for the birth and death process 199
References 201
Exercises 201
10 Population growth II: branching processes 204
10.1 Cell division 204
10.2 The Galton-Watson branching process 205
10.3 Mean and variance for the Galton-Watson process 207
10.4 Probability generating functions of sums of random
variables 209
Contents
X
10.5 The probability of extinction 212
References 216
Exercises 217
11 Stochastic processes and an introduction to stochastic
differential equations 219
11.1 Deterministic and stochastic differential equations 219
11.2 The Wiener process (Brownian motion) 222
11.3 White noise 226
11.4 The simplest stochastic differential equations - the Wiener
process with drift 228
11.5 Transition probabilities and the Chapman-Kolmogorov
equation 231
References 234
Exercises 234
12 Diffusion processes, stochastic differential equations and
applications 237
12.1 Diffusion processes and the Kolmogorov (or Fokker-Planck)
equations 237
12.2 Stationary distributions 242
12.3 The Wiener process with drift 244
12.4 The Ornstein-Uhlenbeck process 256
12.5 Stochastic integrals and stochastic differential equations 260
12.6 Modelling with stochastic differential equations 269
12.7 Applications 270
References 280
Exercises 282
Appendix Table of critical values of the z2-distribution 285
Index 286