The Britannica Guide to Statistics and Probability (Math Explained)

335 Pages·2011·5.422 MB·English

by  Erik Gregersen

Published in 2011 by Britannica Educational Publishing
(a trademark of Encyclopædia Britannica, Inc.)
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First Edition
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Library of Congress Cataloging-in-Publication Data
The Britannica guide to statistics and probability / edited by Erik Gregersen.—1st ed.
p. cm.—(Math explained)
“In association with Britannica Educational Publishing, Rosen Educational Services.”
Includes bibliographical references and index.
ISBN 978-1-61530-228-4 (eBook)
1. Probabilities—Popular works. 2. Mathematical statistics—Popular works. I. Gregersen,
Erik.
QA273.15.B75 2011
519.2—dc22
2010002546
Cover © www.istockphoto.com/Duncan Walker; pp. 12, 20, 21, 48, 115, 154, 197, 244, 291, 321,
324, 326 Shutterstock.com.
C
ontents
Introduction 12
Chapter 1: History of Statistics
and Probability 21
Early Probability 21
Games of Chance 21
Risks, Expectations, and
Fair Contracts 24 29
Probability as the Logic
of Uncertainty 26
The Probability of Causes 30
The Rise of Statistics 33
Political Arithmetic 33
Social Numbers 34
A New Kind of Regularity 37
Statistical Physics 38
The Spread of Statistical Mathematics 39
Statistical Theories in the Sciences 41
Biometry 42
Samples and Experiments 45
43
The Modern Role of Statistics 46
Chapter 2: Probability Theory 48
Experiments, Sample Space, Events,
and Equally Likely Probabilities 50
Applications of Simple
Probability Experiments 50
The Principle of Additivity 53
Multinomial Probability 54
The Birthday Problem 57
Conditional Probability 59
Applications of Conditional
49
Probability 60
Independence 63
Bayes’s Theorem 64
Random Variables, Distributions,
Expectation, and Variance 65
Random Variables 65
Probability Distribution 66
Expected Value 68
Variance 70
An Alternative Interpretation of
Probability 72
The Law of Large Numbers, the
Central Limit Theorem, and
the Poisson Approximation 76
The Law of Large Numbers 76
The Central Limit Theorem 78
The Poisson Approximation 80 96
Infinite Sample Spaces and
Axiomatic Probability 82
Infinite Sample Spaces 82
The Strong Law of Large Numbers 84
Measure Theory 85
Probability Density Functions 89
Conditional Expectation and Least
Squares Prediction 92
The Poisson Process and the
Brownian Motion Process 94
The Poisson Process 94
Brownian Motion Process 95
Stochastic Processes 102
Stationary Processes 102
Markovian Processes 104
The Ehrenfest Model
of Diffusion 105
The Symmetric Random Walk 107
Queuing Models 108
Insurance Risk Theory 111 109
Martingale Theory 112
Chapter 3: Statistics 115
Descriptive Statistics 116
Tabular Methods 117
Graphical Methods 118
Numerical Measures 119
Outliers 120
Exploratory Data Analysis 121
Probability 122
Events and Their Probabilities 123
Random Variables and Probability
Distributions 123
118
Special Probability Distributions 125
The Binomial Distribution 125
The Poisson Distribution 126
The Normal Distribution 127
Estimation 127
Sampling and Sampling
Distributions 128
Estimation of a Population Mean 129
Estimation of Other Parameters 131
Estimation Procedures for Two
Populations 131
Hypothesis Testing 132
Bayesian Methods 135 122
Experimental Design 136
Analysis of Variance and
Significance Testing 138
Regression and Correlation Analysis 138
Regression Model 138
Least Squares Method 139
Analysis of Variance and
Goodness of Fit 141
Significance Testing 142
Residual Analysis 142
Model Building 143
Correlation 144
Time Series and Forecasting 145 140
Nonparametric Methods 146
Statistical Quality Control 148
Acceptance Sampling 148
Statistical Process Control 149
Sample Survey Methods 150
Decision Analysis 152
Chapter 4: Game Theory 154
Classification of Games 156
One-Person Games 158
Two-Person Constant-Sum Games 159
Games of Perfect Information 159
155
Games of Imperfect
Information 160
Mixed Strategies and the 159
Minimax Theorem 162
Utility Theory 165
Two-Person Variable-Sum Games 166
Cooperative Versus
Noncooperative Games 168
The Nash Solution 170
The Prisoners’ Dilemma 171
Theory of Moves 174
Biological Applications 176
N-Person Games 178
Sequential and
Simultaneous Truels 179
198
Power in Voting: The Paradox of
the Chair’s Position 183
The von Neumann–Morgenstern
Theory 188
The Banzhaf Value in
Voting Games 192
Chapter 5: Combinations 197
History 198
Early Developments 198
Combinatorics During
the 20th Century 200
Problems of Enumeration 202
Permutations and Combinations 202
Binomial Coefficients 202
Multinomial Coefficients 203
Recurrence Relations and
Generating Functions 204
Partitions 205
The Ferrer’s Diagram 206
The Principle of Inclusion and
Exclusion: Derangements 209
Polya’s Theorem 211
The Möbius Inversion Theorem 211
Special Problems 212
The Ising Problem 213
Self-Avoiding Random Walk 213
Problems of Choice 213
Systems of Distinct Representatives 213 207
Ramsey’s Numbers 214
Design Theory 215
BIB (Balanced Incomplete
Block) Designs 215
Pbib (Partially Balanced
Incomplete Block) Designs 218
Latin Squares and the Packing Problem 220
Orthogonal Latin Squares 220
Orthogonal Arrays and
the Packing Problem 222
Graph Theory 224
Definitions 224
Enumeration of Graphs 225
Characterization Problems
of Graph Theory 226
Applications of Graph Theory 228
Planar Graphs 228
The Four-Colour Map Problem 229
Eulerian Cycles and the
231
Königsberg Bridge Problem 231
Directed Graphs 232