The Britannica Guide to Statistics and Probability

by Gregersen, Erik

• ISBN13: 9781615301188
• ISBN10: 1615301186
• Edition: 1st
• Format: Hardcover
• Copyright: 2010-08-15
• Publisher: Britannica Educational Pub

Summary

In an effort to better forecast the future for gains, and combat the potential losses of uncertainty, numerous disciplines have come to rely on the power of statistics and probability. By observing patterns and repeated behaviors, mathematicians have devised calculations to significantly reduce human potential for error. This volume introduces the historical and mathematical basis of statistics and probability as well as their application to everyday situations. Readers will also meet the prominent thinkers who advanced the field and established a numerical basis for prediction.

Table of Contents

Introduction	p. 12
History of Statistics and Probability	p. 21
Early Probability	p. 21
Games of Chance	p. 21
Risks, Expectations, and Fair Contracts	p. 24
Probability as the Logic of Uncertainty	p. 26
The Probability of Causes	p. 30
The Rise of Statistics	p. 33
Political Arithmetic	p. 33
Social Numbers	p. 34
A New Kind of Regularity	p. 37
Statistical Physics	p. 38
The Spread of Statistical Mathematics	p. 39
Statistical Theories in the Sciences	p. 41
Biometry	p. 42
Samples and Experiments	p. 45
The Modern Role of Statistics	p. 46
Probability Theory	p. 48
Experiments, Sample Space, Events, and Equally Likely Probabilities	p. 50
Applications of Simple Probability Experiments	p. 50
The Principle of Additivity	p. 53
Multinomial Probability	p. 54
The Birthday Problem	p. 57
Conditional Probability	p. 59
Applications of Conditional Probability	p. 60
Independence	p. 63
Bayes's Theorem	p. 64
Random Variables, Distributions, Expectation, and Variance	p. 65
Random Variables	p. 65
Probability Distribution	p. 66
Expected Value	p. 68
Variance	p. 70
An Alternative Interpretation of Probability	p. 72
The Law of Large Numbers, the Central Limit Theorem, and the Poisson Approximation	p. 76
The Law of Large Numbers	p. 76
The Central Limit Theorem	p. 78
The Poisson Approximation	p. 80
Infinite Sample Spaces and Axiomatic Probability	p. 82
Infinite Sample Spaces	p. 82
The Strong Law of Large Numbers	p. 84
Measure Theory	p. 85
Probability Density Functions	p. 89
Conditional Expectation and Least Squares Prediction	p. 92
The Poisson Process and the Brownian Motion Process	p. 94
The Poisson Process	p. 94
Brownian Motion Process	p. 95
Stochastic Processes	p. 102
Stationary Processes	p. 102
Markovian Processes	p. 104
The Ehrenfest Model of Diffusion	p. 105
The Symmetric Random Walk	p. 107
Queuing Models	p. 108
Insurance Risk Theory	p. 111
Martingale Theory	p. 112
Statistics	p. 115
Descriptive Statistics	p. 116
Tabular Methods	p. 117
Graphical Methods	p. 118
Numerical Measures	p. 119
Outliers	p. 120
Exploratory Data Analysis	p. 121
Probability	p. 122
Events and Their Probabilities	p. 123
Random Variables and Probability Distributions	p. 123
Special Probability Distributions	p. 125
The Binomial Distribution	p. 125
The Poisson Distribution	p. 126
The Normal Distribution	p. 127
Estimation	p. 127
Sampling and Sampling Distributions	p. 128
Estimation of a Population Mean	p. 129
Estimation of Other Parameters	p. 131
Estimation Procedures for Two Populations	p. 131
Hypothesis Testing	p. 132
Bayesian Methods	p. 135
Experimental Design	p. 136
Analysis of Variance and Significance Testing	p. 138
Regression and Correlation Analysis	p. 138
Regression Model	p. 138
Least Squares Method	p. 139
Analysis of Variance and Goodness of Fit	p. 141
Significance Testing	p. 142
Residual Analysis	p. 142
Model Building	p. 143
Correlation	p. 144
Time Series and Forecasting	p. 145
Nonparametric Methods	p. 146
Statistical Quality Control	p. 148
Acceptance Sampling	p. 148
Statistical Process Control	p. 149
Sample Survey Methods	p. 150
Decision Analysis	p. 152
Game Theory	p. 154
Classification of Games	p. 156
One-Person Games	p. 158
Two-Person Constant-Sum Games	p. 159
Games of Perfect Information	p. 159
Games of Imperfect Information	p. 160
Mixed Strategies and the Minimax Theorem	p. 162
Utility Theory	p. 165
Two-Person Variable-Sum Games	p. 166
Cooperative Versus Noncooperative Games	p. 168
The Nash Solution	p. 170
The Prisoners' Dilemma	p. 171
Theory of Moves	p. 174
Biological Applications	p. 176
N-Person Games	p. 178
Sequential and Simultaneous Truels	p. 179
Power in Voting: The Paradox of the Chair's Position	p. 183
The von Neumann-Morgenstern Theory	p. 188
The Banzhaf Value in Voting Games	p. 192
Combinations	p. 197
History	p. 198
Early Developments Combinatorics During the 20th Century	p. 200
Problems of Enumeration	p. 202
Permutations and Combinations	p. 202
Binomial Coefficients	p. 202
Multinomial Coefficients	p. 203
Recurrence Relations and Generating Functions	p. 204
Partitions	p. 205
The Ferrer's Diagram	p. 206
The Principle of Inclusion and Exclusion: Derangements	p. 209
Polya's Theorem	p. 211
The Möbius Inversion Theorem	p. 211
Special Problems	p. 212
The Ising Problem	p. 213
Self-Avoiding Random Walk	p. 213
Problems of Choice	p. 213
Systems of Distinct Representatives	p. 213
Ramsey's Numbers	p. 214
Design Theory	p. 215
BIB (Balanced Incomplete Block) Designs	p. 215
Pbib (Partially Balanced Incomplete Block) Designs	p. 218
Latin Squares and the Packing Problem	p. 220
Orthogonal Latin Squares	p. 220
Orthogonal Arrays and the Packing Problem	p. 222
Graph Theory	p. 224
Definitions	p. 224
Enumeration of Graphs	p. 225
Characterization Problems of Graph Theory	p. 226
Applications of Graph Theory	p. 228
Planar Graphs	p. 228
The Four-Colour Map Problem	p. 229
Eulerian Cycles and the Königsberg Bridge Problem	p. 231
Directed Graphs	p. 232
Combinatorial Geometry	p. 232
Some Historically Important Topics of Combinatorial Geometry	p. 234
Packing and Covering	p. 234
Polytopes	p. 236
Incidence Problems	p. 238
Helly's Theorem	p. 238
Methods of Combinatorial Geometry	p. 240
Exhausting the Possibilities	p. 240
Use of Extremal Properties	p. 241
Use of Transformations Between Different Spaces and Applications of Helly's Theorem	p. 243
Biographies	p. 244
Special Topics	p. 291
Bayes's Theorem	p. 291
Binomial Distribution	p. 293
Central Limit Theorem	p. 295
Chebyshev's Inequality	p. 296
Decision Theory	p. 297
Distribution Function	p. 298
Error	p. 298
Estimation	p. 299
Indifference	p. 300
Inference	p. 301
Interval Estimation	p. 301
Law of Large Numbers	p. 302
Least Squares Approximation	p. 303
Markov Process	p. 305
Mean	p. 305
Normal Distribution	p. 308
Permutations and Combinations	p. 310
Point Estimation	p. 313
Poisson Distribution	p. 314
Queuing Theory	p. 315
Random Walk	p. 316
Sampling	p. 316
Standard Deviation	p. 318
Stochastic Process	p. 318
Student's T-Test	p. 319
Glossary	p. 321
Bibliography	p. 324
Index	p. 326
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