Probably not : future prediction using probability and statistical inference PDF
Preview Probably not : future prediction using probability and statistical inference
Probably Not Probably Not Future Prediction Using Probability and Statistical Inference Lawrence N. Dworsky Second Edition This second edition first published 2019 © 2019 John Wiley & Sons, Inc. All rights reserved. Edition History John Wiley & Sons, Inc. (1e, 2008) All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions. The right of Lawrence N. Dworsky to be identified as the author of this work has been asserted in accordance with law. 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Title: Probably not : future prediction using probability and statistical inference / Lawrence N. Dworsky. Description: Second edition. | Hoboken, NJ : Wiley, 2019. | Includes index. | Identifiers: LCCN 2019019378 (print) | LCCN 2019021721 (ebook) | ISBN 9781119518136 (Adobe PDF) | ISBN 9781119518129 (ePub) | ISBN 9781119518105 (pbk.) Subjects: LCSH: Prediction theory. | Probabilities–Problems, exercises, etc. | Mathematical statistics–Problems, exercises, etc. Classification: LCC QA279.2 (ebook) | LCC QA279.2 .D96 2020 (print) | DDC 519.2/87–dc23 LC record available at https://lccn.loc.gov/2019019378 Cover design by Wiley Cover image: © Ralf Hiemisch/Getty Images Set in 10/12pt Warnock by SPi Global, Pondicherry, India Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 To all the people who realize, or want to learn, that Probability and Statistics isn’t just a mathematics discipline, it’s a way to see and better understand the world. vii Contents Acknowledgments xi About the Companion Website xiii Introduction 1 1 An Introduction to Probability 5 Predicting the Future 5 Rule Making 7 Random Events and Probability 9 The Lottery {Very Improbable Events and Very Large Data Sets} 15 Coin Flipping {Fair Games, Looking Backward for Insight} 17 The Coin Flip Strategy That Can’t Lose 24 The Prize Behind the Door {Looking Backward for Insight, Again} 25 The Checker Board {Dealing With Only Part of the Data Set} 27 Comments 31 Problems 32 2 Probability Distribution Functions and Some Math Basics 35 The Probability Distribution Function 35 Averages and Weighted Averages 38 Expected Values (Again) 41 The Basic Coin Flip Game 43 PDF Symmetry 43 Standard Deviation 46 Cumulative Distribution Function 55 The Confidence Interval 57 Final Points 58 Rehash and Histograms 59 Problems 66 3 Building a Bell 71 Problems 87 viii Contents 4 Random Walks 89 The One‐Dimensional Random Walk 89 Some Subsequent Calculations 93 Diffusion 95 Problems 99 5 Life Insurance 103 Introduction 103 Life Insurance 103 Insurance as Gambling 104 Life Tables 107 Birth Rates and Population Stability 112 Life Tables, Again 113 Premiums 115 Social Security – Sooner or Later? 120 Problems 125 6 The Binomial Theorem 129 Introduction 129 The Binomial Probability Formula 130 Permutations and Combinations 132 Large Number Approximations 134 The Poisson Distribution 136 Disease Clusters 140 Clusters 140 Problems 142 7 Pseudorandom Numbers and Monte Carlo Simulations 145 Random Numbers and Simulations 145 Pseudorandom Numbers 145 The Middle Square PRNG 146 The Linear Congruential PRNG 148 A Normal Distribution Generator 150 An Arbitrary Distribution Generator 151 Monte Carlo Simulations 153 A League of Our Own 156 Discussion 159 Notes 160 8 Some Gambling Games in Detail 161 The Basic Coin Flip Game 161 The “Ultimate Winning Strategy” 166 Parimutuel Betting 169 The Gantt Chart and a Hint of Another Approach 172 Problems 174 Contents ix 9 Scheduling and Waiting 177 Introduction 177 Scheduling Appointments in the Doctor’s Office 177 Lunch with a Friend 180 Waiting for a Bus 182 Problems 185 10 Combined and Conditional Probabilities 187 Introduction 187 Functional Notation (Again) 187 Conditional Probability 189 Medical Test Results 192 The Shared Birthday Problem 195 Problems 197 11 Bayesian Statistics 199 Bayes Theorem 199 Multiple Possibilities 202 Will Monty Hall Ever Go Away? 207 Philosophy 209 The Prosecutor’s Fallacy 210 Continuous Functions 211 Credible Intervals 214 Gantt Charts (Again) 215 Problems 217 12 Estimation Problems 221 The Number of Locomotives Problem 221 Number of Locomotives, Improved Estimate 222 Decision Making 224 The Lighthouse Problem 227 The Likelihood Function 229 The Lighthouse Problem II 232 13 Two Paradoxes 233 Introduction 233 Parrondo’s Paradox 233 Another Parrondo Game 236 The Parrondo Ratchet 239 Simpson’s Paradox 240 Problems 244 14 Benford’s Law 247 Introduction 247 History 247 x Contents The 1/x Distribution 249 Surface Area of Countries of the World 252 Goodness of Fit Measure 253 Smith’s Analysis 255 Problems 259 15 Networks, Infectious Diseases, and Chain Letters 261 Introduction 261 Degrees of Separation 261 Propagation Along the Networks 265 Some Other Networks 270 Neighborhood Chains 271 Chain Letters 273 Comments 276 16 Introduction to Frequentist Statistical Inference 277 Introduction 277 Sampling 277 Sample Distributions and Standard Deviations 280 Estimating Population Average from a Sample 282 The Student‐T Distribution 285 Did a Sample Come from a Given Population? 289 A Little Reconciliation 289 Correlation and Causality 291 Correlation Coefficient 293 Regression Lines 294 Regression to the Mean 295 Problems 298 17 Statistical Mechanics and Thermodynamics 303 Introduction 303 Statistical Mechanics 304 (Concepts of) Thermodynamics 306 18 Chaos and Quanta 311 Introduction 311 Chaos 311 Probability in Quantum Mechanics 319 Appendix 323 Index 329 xi Acknowledgments Ten years ago my wife, Suzanna, patiently encouraged me to write this book during the many months it took me to actually get started. She listened to my ideas, read and commented on drafts and perhaps most importantly, suggested the title. This year she once again supplied the encouragement and support for me to repeat the process. The folks at John Wiley were very encouraging and helpful at the start and continue to be so. I am grateful to all of you. —Lawrence N. Dworsky xiii About the Companion Website This book is accompanied by a companion website: www.wiley.com/go/probablynot2e The website includes the following materials for instructors: ● Solutions for all the problems in the chapters.
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