Probability for physicists

Graduate Texts in Physics Simon Širca Probability for Physicists Graduate Texts in Physics Series editors Jean-Marc Di Meglio, Paris, France William T Rhodes, Boca Raton, USA Susan Scott, Acton, Australia Martin Stutzmann, Garching, Germany Andreas Wipf, Jena, Germany Graduate Texts in Physics Graduate Texts in Physics publishes core learning/teaching material for graduate- and advanced-level undergraduate courses on topics of current and emerging fields within physics, both pure and applied These textbooks serve students at the MS- or PhD-level and their instructors as comprehensive sources of principles, definitions, derivations, experiments and applications (as relevant) for their mastery and teaching, respectively International in scope and relevance, the textbooks correspond to course syllabi sufficiently to serve as required reading Their didactic style, comprehensiveness and coverage of fundamental material also make them suitable as introductions or references for scientists entering, or requiring timely knowledge of, a research field More information about this series at http://www.springer.com/series/8431 Simon Širca Probability for Physicists 123 Simon Širca Faculty of Mathematics and Physics University of Ljubljana Ljubljana Slovenia ISSN 1868-4513 Graduate Texts in Physics ISBN 978-3-319-31609-3 DOI 10.1007/978-3-319-31611-6 ISSN 1868-4521 (electronic) ISBN 978-3-319-31611-6 (eBook) Library of Congress Control Number: 2016937517 © Springer International Publishing Switzerland 2016 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland Preface University-level introductory books on probability and statistics tend to be long— too long for the attention span and immediate horizon of a typical physics student who might wish to absorb the necessary topics in a swift, direct, involving manner, relying on her existing knowledge and physics intuition rather than asking to be taken through the content at a slow and perhaps over-systematic pace In contrast, this book attempts to deliver a concise, lively, intuitive introduction to probability and statistics for undergraduate and graduate students of physics and other natural sciences Conceived primarily as a text for the second-year course on Probability in Physics at the Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, it has been designed to be as relieved of unnecessary mathematical ballast as possible, yet never to be mathematically imprecise At the same time, it is hoped to be colorful and captivating: to this end, I have strived to avoid endless, dry prototypes with tossing coins, throwing dice and births of girls and boys, and replace them wherever possible by physics-motivated examples, always in the faith that the reader is already familiar with “at least something” The book also tries to fill a few common gaps and resurrect some content that seems to be disappearing irretrievably from the modern, Bologna-style curricula Typical witnesses of such efforts are the sections on extreme-value distributions, linear regression by using singular-value decomposition, and the maximum-likelihood method The book consists of four parts In the first part (Chaps 1–6) we discuss the fundamentals of probability and probability distributions The second part (Chaps 7–10) is devoted to statistics, that is, the determination of distribution parameters based on samples Chapters 11–14 of the third part are “applied”, as they are the place to reap what has been sown in the first two parts and they invite the reader to a more concrete, computer-based engagement As such, these chapters lack the concluding exercise sections, but incorporate extended examples in the main text The fourth part consists of appendices Optional contents are denoted by asterisks H Without them, the book is tailored to a compact one-semester course; v vi Preface with them included, it can perhaps serve as a vantage point for a two-semester agenda The story-telling and the style are mine; regarding all other issues and doubts I have gladly obeyed the advice of both benevolent, though merciless reviewers, Dr Martin Horvat and Dr Gregor Šega Martin is a treasure-trove of knowledge on an incredible variety of problems in mathematical physics, and in particular of answers to these problems He does not terminate the discussions with the elusive “The solution exists!”, but rather with a fully functional, tested and documented computer code His ad hoc products saved me many hours of work Gregor has shaken my conviction that a partly loose, intuitive notation could be reader-friendly He helped to furnish the text with an appropriate measure of mathematical rigor, so that I could ultimately run with the physics hare and hunt with the mathematics hounds I am grateful to them for reading the manuscript so attentively I would also like to thank my student Mr Peter Ferjančič for leading the problem-solving classes for two years and for suggesting and solving Problem 5.6.3 I wish to express my gratitude to Professor Claus Ascheron, Senior Editor at Springer, for his effort in preparation and advancement of this book, as well as to Viradasarani Natarajan and his team for its production at Scientific Publishing Services http://pp.books.fmf.uni-lj.si Ljubljana Simon Širca Contents Part I Fundamentals of Probability and Probability Distributions Basic Terminology 1.1 Random Experiments and Events 1.2 Basic Combinatorics 1.2.1 Variations and Permutations 1.2.2 Combinations Without Repetition 1.2.3 Combinations with Repetition 1.3 Properties of Probability 1.4 Conditional Probability 1.4.1 Independent Events 1.4.2 Bayes Formula 1.5 Problems 1.5.1 Boltzmann, Bose–Einstein and Fermi–Dirac Distributions 1.5.2 Blood Types 1.5.3 Independence of Events in Particle Detection 1.5.4 Searching for the Lost Plane 1.5.5 The Monty Hall Problem H 1.5.6 Bayes Formula in Medical Diagnostics 1.5.7 One-Dimensional Random Walk H References Probability Distributions 2.1 Dirac Delta 2.1.1 Composition of the Dirac Delta with a 2.2 Heaviside Function 2.3 Discrete and Continuous Distributions 2.4 Random Variables 2.5 One-Dimensional Discrete Distributions 2.6 One-Dimensional Continuous Distributions 3 6 8 11 14 16 18 18 19 21 22 22 25 27 29 Function 31 31 33 35 36 37 37 39 vii viii Contents 2.7 Transformation of Random Variables 2.7.1 What If the Inverse of y = h(x) Is Not Unique? 2.8 Two-Dimensional Discrete Distributions 2.9 Two-Dimensional Continuous Distributions 2.10 Transformation of Variables in Two and More Dimensions 2.11 Problems 2.11.1 Black-Body Radiation 2.11.2 Energy Losses of Particles in a Planar Detector 2.11.3 Computing Marginal Probability Densities from a Joint Density 2.11.4 Independence of Random Variables in Two Dimensions 2.11.5 Transformation of Variables in Two Dimensions 2.11.6 Distribution of Maximal and Minimal Values References 41 44 45 47 50 56 56 57 58 60 61 63 64 Special Continuous Probability Distributions 3.1 Uniform Distribution 3.2 Exponential Distribution 3.2.1 Is the Decay of Unstable States Truly Exponential? 3.3 Normal (Gauss) Distribution 3.3.1 Standardized Normal Distribution 3.3.2 Measure of Peak Separation 3.4 Maxwell Distribution 3.5 Pareto Distribution 3.5.1 Estimating the Maximum x in the Sample 3.6 Cauchy Distribution 3.7 The X distribution 3.8 Student’s Distribution 3.9 F distribution 3.10 Problems 3.10.1 In-Flight Decay of Neutral Pions 3.10.2 Product of Uniformly Distributed Variables 3.10.3 Joint Distribution of Exponential Variables 3.10.4 Integral of Maxwell Distribution over Finite Range 3.10.5 Decay of Unstable States and the Hyper-exponential Distribution 3.10.6 Nuclear Decay Chains and the Hypo-exponential Distribution References 65 65 67 70 70 71 73 74 75 77 77 79 79 80 80 80 83 84 85 86 89 91 Expected Values 4.1 Expected (Average, Mean) Value 4.2 Median 4.3 Quantiles 93 93 95 96 Contents ix 4.4 Expected Values of Functions of Random Variables 4.4.1 Probability Densities in Quantum Mechanics 4.5 Variance and Effective Deviation 4.6 Complex Random Variables 4.7 Moments 4.7.1 Moments of the Cauchy Distribution 4.8 Two- and d-dimensional Generalizations 4.8.1 Multivariate Normal Distribution 4.8.2 Correlation Does Not Imply Causality 4.9 Propagation of Errors 4.9.1 Multiple Functions and Transformation of the Covariance Matrix 4.10 Problems 4.10.1 Expected Device Failure Time 4.10.2 Covariance of Continuous Random Variables 4.10.3 Conditional Expected Values of Two-Dimensional Distributions 4.10.4 Expected Values of Hyper- and Hypo-exponential Variables 4.10.5 Gaussian Noise in an Electric Circuit 4.10.6 Error Propagation in a Measurement of the Momentum Vector H References Special Discrete Probability Distributions 5.1 Binomial Distribution 5.1.1 Expected Value and Variance 5.2 Multinomial Distribution 5.3 Negative Binomial (Pascal) Distribution 5.3.1 Negative Binomial Distribution of Order k 5.4 Normal Approximation of the Binomial Distribution 5.5 Poisson Distribution 5.6 Problems 5.6.1 Detection Efficiency 5.6.2 The Newsboy Problem H 5.6.3 Time to Critical Error 5.6.4 Counting Events with an Inefficient Detector 5.6.5 Influence of Primary Ionization on Spatial Resolution H References 98 99 100 101 102 105 106 110 111 111 113 115 115 116 117 117 119 120 121 123 123 126 128 129 129 130 132 135 135 136 138 140 140 142 Stable Distributions and Random Walks 143 6.1 Convolution of Continuous Distributions 143 6.1.1 The Effect of Convolution on Distribution Moments 146 χ2.005 9.26 9.89 10.5 11.2 11.8 12.5 13.1 13.8 20.7 28.0 35.5 43.3 51.2 59.2 67.3 ν 23 24 25 26 27 28 29 30 40 50 60 70 80 90 100 10.2 10.9 11.5 12.2 12.9 13.6 14.3 15.0 22.2 29.7 37.5 45.4 53.5 61.8 70.1 χ2.01 Table D.3 (continued) 11.7 12.4 13.1 13.8 14.6 15.3 16.0 16.8 24.4 32.4 40.5 48.8 57.2 65.6 74.2 χ2.025 13.1 13.8 14.6 15.4 16.2 16.9 17.7 18.5 26.5 34.8 43.2 51.7 60.4 69.1 77.9 χ2.05 14.8 15.7 16.5 17.3 18.1 18.9 19.8 20.6 29.1 37.7 46.5 55.3 64.3 73.3 82.4 χ2.1 18.1 19.0 19.9 20.8 21.7 22.7 23.6 24.5 33.7 42.9 52.3 61.7 71.1 80.6 90.1 χ2.25 22.3 23.3 24.3 25.3 26.3 27.3 28.3 29.3 39.3 49.3 59.3 69.3 79.3 89.3 99.3 χ2.5 27.1 28.2 29.3 30.4 31.5 32.6 33.7 34.8 45.6 56.3 67.0 77.6 88.1 98.6 109 χ2.75 32.0 33.2 34.4 35.6 36.7 37.9 39.1 40.3 51.8 63.2 74.4 85.5 96.6 108 118 χ2.90 35.2 36.4 37.7 38.9 40.1 41.3 42.6 43.8 55.8 67.5 79.1 90.5 102 113 124 χ2.95 38.1 39.4 40.6 41.9 43.2 44.5 45.7 47.0 59.3 71.4 83.3 95.0 107 118 130 χ2.975 41.6 43.0 44.3 45.6 47.0 48.3 49.6 50.9 63.7 76.2 88.4 100 112 124 136 χ2.99 44.2 45.6 46.9 48.3 49.6 51.0 52.3 53.7 66.8 79.5 92.0 104 116 128 140 χ2.995 49.7 51.2 52.6 54.1 55.5 56.9 58.3 59.7 73.4 86.7 99.6 112 125 137 149 χ2.999 400 Appendix D: Tables of Distribution Quantiles t0.55 0.158 0.142 0.137 0.134 0.132 0.131 0.130 0.130 0.129 0.129 0.129 0.128 0.128 0.128 0.128 0.128 0.128 0.127 0.127 0.127 0.127 ν 10 11 12 13 14 15 16 17 18 19 20 21 0.325 0.289 0.277 0.271 0.267 0.265 0.263 0.262 0.261 0.260 0.260 0.259 0.259 0.258 0.258 0.258 0.257 0.257 0.257 0.257 0.257 t0.60 0.727 0.617 0.584 0.569 0.559 0.553 0.549 0.546 0.543 0.542 0.540 0.539 0.538 0.537 0.536 0.535 0.534 0.534 0.533 0.533 0.532 t0.70 1.000 0.816 0.765 0.741 0.727 0.718 0.711 0.706 0.703 0.700 0.697 0.695 0.694 0.692 0.691 0.690 0.689 0.688 0.688 0.687 0.686 t0.75 1.376 1.061 0.978 0.941 0.920 0.906 0.896 0.889 0.883 0.879 0.876 0.873 0.870 0.868 0.866 0.865 0.863 0.862 0.861 0.860 0.859 t0.80 3.08 1.89 1.64 1.53 1.48 1.44 1.41 1.40 1.38 1.37 1.36 1.36 1.35 1.35 1.34 1.34 1.33 1.33 1.33 1.33 1.32 t0.90 6.31 2.92 2.35 2.13 2.02 1.94 1.89 1.86 1.83 1.81 1.80 1.78 1.77 1.76 1.75 1.75 1.74 1.73 1.73 1.72 1.72 t0.95 12.7 4.30 3.18 2.78 2.57 2.45 2.36 2.31 2.26 2.23 2.20 2.18 2.16 2.14 2.13 2.12 2.11 2.10 2.09 2.09 2.08 t0.975 31.8 6.96 4.54 3.75 3.36 3.14 3.00 2.90 2.82 2.76 2.72 2.68 2.65 2.62 2.60 2.58 2.57 2.55 2.54 2.53 2.52 t0.99 63.7 9.92 5.84 4.60 4.03 3.71 3.50 3.36 3.25 3.17 3.11 3.05 3.01 2.98 2.95 2.92 2.90 2.88 2.86 2.85 2.83 t0.995 (continued) 3183 70.7 22.2 13.0 9.68 8.02 7.06 6.44 6.01 5.69 5.45 5.26 5.11 4.99 4.88 4.79 4.71 4.65 4.59 4.54 4.49 t0.999 Table D.4 Quantiles t p of the Student’s t distribution (3.22) with ν degrees of freedom for some typical (most commonly used) values of p from 0.55 to 0.999 Appendix D: Tables of Distribution Quantiles 401 t0.55 0.127 0.127 0.127 0.127 0.127 0.127 0.127 0.127 0.127 0.126 0.126 0.126 0.126 ν 22 23 24 25 26 27 28 29 30 40 60 120 ∞ Table D.4 (continued) t0.60 0.256 0.256 0.256 0.256 0.256 0.256 0.256 0.256 0.256 0.255 0.254 0.254 0.253 t0.70 0.532 0.532 0.531 0.531 0.531 0.531 0.530 0.530 0.530 0.529 0.527 0.526 0.524 t0.75 0.686 0.685 0.685 0.684 0.684 0.684 0.683 0.683 0.683 0.681 0.679 0.677 0.674 t0.80 0.858 0.858 0.857 0.856 0.856 0.855 0.855 0.854 0.854 0.851 0.848 0.845 0.842 t0.90 1.32 1.32 1.32 1.32 1.31 1.31 1.31 1.31 1.31 1.30 1.30 1.29 1.28 t0.95 1.72 1.71 1.71 1.71 1.71 1.70 1.70 1.70 1.70 1.68 1.67 1.66 1.64 2.07 2.07 2.06 2.06 2.06 2.05 2.05 2.05 2.04 2.02 2.00 1.98 1.96 t0.975 2.51 2.50 2.49 2.49 2.48 2.47 2.47 2.46 2.46 2.42 2.39 2.36 2.21 t0.99 2.82 2.81 2.80 2.79 2.78 2.77 2.76 2.76 2.75 2.70 2.66 2.62 2.58 t0.995 t0.999 4.45 4.42 4.38 4.35 4.32 4.30 4.28 4.25 4.23 4.09 3.96 3.84 3.72 402 Appendix D: Tables of Distribution Quantiles ν2 = 10 11 12 13 14 15 16 17 18 19 20 22 24 26 161 18.5 10.1 7.71 6.61 5.99 5.59 5.32 5.12 4.96 4.84 4.75 4.67 4.60 4.54 4.49 4.45 4.41 4.38 4.35 4.30 4.26 4.23 200 19.0 9.55 6.94 5.79 5.14 4.74 4.46 4.26 4.10 3.98 3.89 3.81 3.74 3.68 3.63 3.59 3.55 3.52 3.49 3.44 3.40 3.37 ν1 = 216 19.2 9.28 6.59 5.41 4.76 4.35 4.07 3.86 3.71 3.59 3.49 3.41 3.34 3.29 3.24 3.20 3.16 3.13 3.10 3.05 3.01 2.98 225 19.3 9.12 6.39 5.19 4.53 4.12 3.84 3.63 3.48 3.36 3.26 3.18 3.11 3.06 3.01 2.96 2.93 2.90 2.87 2.82 2.78 2.74 230 19.3 9.01 6.26 5.05 4.39 3.97 3.69 3.48 3.33 3.20 3.11 3.03 2.96 2.90 2.85 2.81 2.77 2.74 2.71 2.66 2.62 2.59 234 19.3 8.94 6.16 4.95 4.28 3.87 3.58 3.37 3.22 3.09 3.00 2.92 2.85 2.79 2.74 2.70 2.66 2.63 2.60 2.55 2.51 2.47 237 19.4 8.89 6.09 4.88 4.21 3.79 3.50 3.29 3.14 3.01 2.91 2.83 2.76 2.71 2.66 2.61 2.58 2.54 2.51 2.46 2.42 2.39 239 19.4 8.85 6.04 4.82 4.15 3.73 3.44 3.23 3.07 2.95 2.85 2.77 2.70 2.64 2.59 2.55 2.51 2.48 2.45 2.40 2.36 2.32 241 19.4 8.81 6.00 4.77 4.10 3.68 3.39 3.18 3.02 2.90 2.80 2.71 2.65 2.59 2.54 2.49 2.46 2.42 2.39 2.34 2.30 2.27 242 19.4 8.79 5.96 4.74 4.06 3.64 3.35 3.14 2.98 2.85 2.75 2.67 2.60 2.54 2.49 2.45 2.41 2.38 2.35 2.30 2.25 2.22 10 244 19.4 8.74 5.91 4.68 4.00 3.57 3.28 3.07 2.91 2.79 2.69 2.60 2.53 2.48 2.42 2.38 2.34 2.31 2.28 2.23 2.18 2.15 12 246 19.4 8.70 5.86 4.62 3.94 3.51 3.22 3.01 2.85 2.72 2.62 2.53 2.46 2.40 2.35 2.31 2.27 2.23 2.20 2.15 2.11 2.07 15 248 19.5 8.66 5.80 4.56 3.87 3.44 3.15 2.94 2.77 2.65 2.54 2.46 2.39 2.33 2.28 2.23 2.19 2.16 2.12 2.07 2.03 1.99 20 249 19.5 8.64 5.77 4.53 3.84 3.41 3.12 2.90 2.74 2.61 2.51 2.42 2.35 2.29 2.24 2.19 2.15 2.11 2.08 2.03 1.98 1.95 24 250 19.5 8.62 5.75 4.50 3.81 3.38 3.08 2.86 2.70 2.57 2.47 2.38 2.31 2.25 2.19 2.15 2.11 2.07 2.04 1.98 1.94 1.90 30 251 19.5 8.59 5.72 4.46 3.77 3.34 3.04 2.83 2.66 2.53 2.43 2.34 2.27 2.20 2.15 2.10 2.06 2.03 1.99 1.94 1.89 1.85 40 Table D.5 95 percentiles (F0.95 ) of the F distribution (3.23); ν1 degrees of freedom in the numerator and ν2 in the denominator 60 252 19.5 8.57 5.69 4.43 3.74 3.30 3.01 2.79 2.62 2.49 2.38 2.30 2.22 2.16 2.11 2.06 2.02 1.98 1.95 1.89 1.84 1.80 120 ∞ 254 19.5 8.53 5.63 4.37 3.67 3.23 2.93 2.71 2.54 2.40 2.30 2.21 2.13 2.07 2.01 1.96 1.92 1.88 1.84 1.78 1.73 1.69 (continued) 253 19.5 8.55 5.66 4.40 3.70 3.27 2.97 2.75 2.58 2.45 2.34 2.25 2.18 2.11 2.06 2.01 1.97 1.93 1.90 1.84 1.79 1.75 Appendix D: Tables of Distribution Quantiles 403 28 30 40 60 120 ∞ 4.20 4.17 4.08 4.00 3.92 3.84 3.34 3.32 3.23 3.15 3.07 3.00 ν1 = Table D.5 (continued) 2.95 2.92 2.84 2.76 2.68 2.41 2.71 2.69 2.61 2.53 2.45 2.11 2.56 2.53 2.45 2.37 2.29 1.92 2.45 2.42 2.34 2.25 2.18 1.79 2.36 2.33 2.25 2.17 2.09 1.70 2.29 2.27 2.18 2.10 2.02 1.62 2.24 2.21 2.12 2.04 1.96 1.56 10 2.19 2.16 2.08 1.99 1.91 1.52 12 2.12 2.09 2.00 1.92 1.83 1.44 15 2.04 2.01 1.92 1.84 1.75 1.37 20 1.96 1.93 1.84 1.75 1.66 1.28 24 1.91 1.89 1.79 1.70 1.61 1.52 30 1.87 1.84 1.74 1.65 1.55 1.46 40 1.82 1.79 1.69 1.59 1.50 1.39 60 1.77 1.74 1.64 1.53 1.43 1.32 120 1.71 1.68 1.58 1.47 1.35 1.22 ∞ 1.65 1.62 1.51 1.39 1.25 1.00 404 Appendix D: Tables of Distribution Quantiles ν2 = 10 11 12 13 14 15 16 17 18 19 20 22 24 26 4052 98.5 34.1 21.2 16.3 13.8 12.3 11.3 10.6 10.0 9.65 9.33 9.07 8.86 8.68 8.53 8.40 8.29 8.18 8.10 7.95 7.82 7.72 5000 99.0 30.8 18.0 13.3 10.9 9.55 8.65 8.02 7.56 7.21 6.93 6.70 6.51 6.36 6.23 6.11 6.01 5.93 5.85 5.72 5.61 5.53 ν1 = 5403 99.2 29.5 16.7 12.1 9.78 8.45 7.59 6.99 6.55 6.22 5.95 5.74 5.56 5.42 5.29 5.18 5.09 5.01 4.94 4.82 4.72 4.64 5625 99.3 28.7 16.0 11.4 9.15 7.85 7.01 6.42 5.99 5.67 5.41 5.21 5.04 4.89 4.77 4.67 4.58 4.50 4.43 4.31 4.22 4.14 5764 99.3 28.2 15.5 11.0 8.75 7.46 6.63 6.06 5.64 5.32 5.06 4.86 4.69 4.56 4.44 4.34 4.25 4.17 4.10 3.99 3.90 3.82 5859 99.3 27.9 15.2 10.7 8.47 7.19 6.37 5.80 5.39 5.07 4.82 4.62 4.46 4.32 4.20 4.10 4.01 3.94 3.87 3.76 3.67 3.59 5928 99.4 27.7 15.0 10.5 8.26 6.99 6.18 5.61 5.20 4.89 4.64 4.44 4.28 4.14 4.03 3.93 3.84 3.77 3.70 3.59 3.50 3.42 5981 99.4 27.5 14.8 10.3 8.10 6.84 6.03 5.47 5.06 4.74 4.50 4.30 4.14 4.00 3.89 3.79 3.71 3.63 3.56 3.45 3.36 3.29 6022 99.4 27.4 14.7 10.2 7.98 6.72 5.91 5.35 4.94 4.63 4.39 4.19 4.03 3.89 3.78 3.68 3.60 3.52 3.46 3.35 3.26 3.18 6056 99.4 27.2 14.6 10.1 7.87 6.62 5.81 5.26 4.85 4.54 4.30 4.10 3.94 3.80 3.69 3.59 3.51 3.43 3.37 3.26 3.17 3.09 10 6106 99.4 27.1 14.4 9.89 7.72 6.47 5.67 5.11 4.71 4.40 4.16 3.96 3.80 3.67 3.55 3.46 3.37 3.30 3.23 3.12 3.03 2.96 12 6157 99.4 26.9 14.2 9.72 7.56 6.31 5.52 4.96 4.56 4.25 4.01 3.82 3.66 3.52 3.41 3.31 3.23 3.15 3.09 2.98 2.89 2.81 15 6209 99.5 26.7 14.0 9.55 7.40 6.16 5.36 4.81 4.41 4.10 3.86 3.66 3.51 3.37 3.26 3.16 3.08 3.00 2.94 2.83 2.74 2.66 20 6235 99.5 26.6 13.9 9.47 7.31 6.07 5.28 4.73 4.33 4.02 3.78 3.59 3.43 3.29 3.18 3.08 3.00 2.92 2.86 2.75 2.66 2.58 24 6261 99.5 26.5 13.8 9.38 7.23 5.99 5.20 4.65 4.25 3.94 3.70 3.51 3.35 3.21 3.10 3.00 2.92 2.84 2.78 2.67 2.58 2.50 30 6287 99.5 26.4 13.8 9.29 7.14 5.91 5.12 4.57 4.17 3.86 3.62 3.43 3.27 3.13 3.02 2.92 2.84 2.76 2.69 2.58 2.49 2.42 40 Table D.6 99 percentiles (F0.99 ) of the F distribution (3.23); ν1 degrees of freedom in the numerator and ν2 in the denominator 6313 99.5 26.3 13.7 9.20 7.06 5.82 5.03 4.48 4.08 3.78 3.54 3.34 3.18 3.05 2.93 2.83 2.75 2.67 2.61 2.50 2.40 2.33 60 6366 99.5 26.1 13.5 9.02 6.88 5.65 4.86 4.31 3.91 3.60 3.36 3.17 3.00 2.87 2.75 2.65 2.57 2.49 2.42 2.31 2.21 2.13 ∞ (continued) 6339 99.5 26.2 13.6 9.11 6.97 5.74 4.95 4.40 4.00 3.69 3.45 3.25 3.09 2.96 2.84 2.75 2.66 2.58 2.52 2.40 2.31 2.23 120 Appendix D: Tables of Distribution Quantiles 405 28 30 40 60 120 ∞ 7.64 7.56 7.31 7.08 6.85 6.63 5.45 5.39 5.18 4.98 4.79 4.61 ν1 = 4.57 4.51 4.31 4.13 3.95 3.78 Table D.6 (continued) 4.07 4.02 3.83 3.65 3.48 3.32 3.75 3.70 3.51 3.34 3.17 3.02 3.53 3.47 3.29 3.12 2.96 2.80 3.36 3.30 3.12 2.95 2.79 2.64 3.23 3.17 2.99 2.82 2.66 2.51 3.12 3.07 2.89 2.72 2.56 2.41 3.03 2.98 2.80 2.63 2.47 2.32 10 2.90 2.84 2.66 2.50 2.34 2.18 12 2.75 2.70 2.52 2.35 2.19 2.04 15 2.60 2.55 2.37 2.20 2.03 1.88 20 2.52 2.47 2.29 2.12 1.95 1.79 24 2.44 2.39 2.20 2.03 1.86 1.70 30 2.35 2.30 2.11 1.94 1.76 1.59 40 2.26 2.21 2.02 1.84 1.66 1.47 60 2.17 2.11 1.92 1.73 1.53 1.32 120 2.06 2.01 1.80 1.60 1.38 1.00 ∞ 406 Appendix D: Tables of Distribution Quantiles Appendix D: Tables of Distribution Quantiles Reference S Wolfram, Wolfram Mathematica http://www.wolfram.com 407 Index A Acceptance region, 260 Airplane engines, 125 Auto-correlation (temporal), 298 Auto-regression model, 303 Average (mean) value, 93 B Ballistic diffusion, 163 Bayes formula, 16, 25 Binary communication channel, 310 Binomial distribution, 123–128, 135–138, 140 formula, 8, 124 symbol, Black-Body radiation, 56 Blood types, 19 Bombardment of London, 134, 270 Bootstrap, 276 Box diagram, 193 Box-Muller transformation, 391 Branching fraction, 88, 125 Breit-Wigner distribution, 78, 88, 393 Brownian motion, 321–323 Buffon’s experiment, 325 C Cantelli’s inequality, 101 Carry, 384 Cauchy distribution, 41, 77, 105, 393 Characteristic function, 376–382 scale of distribution, 163 Characterizing function, 315 Chebyshev inequality, 101 Coefficient of determination, 234 of linear correlation, 108, 195 Combinations, 7, Complete set of events, Complex random variable, 101 Conditional probability, 11–18 Confidence interval, 188, 235 for correlation coefficient, 196 for sample mean, 188 for sample variance, 191 level, 261 region, 191 Constraint equation, 245, 289 Constraint (Lagrange), 246, 289–303 Continuous distribution, 39, 47 Convolution, 143, 369 continuous, 143–147, 167 discrete, 147–149 Correlation and causality, 111 coefficient, 108, 195 linear (Pearson), 108, 195 non-parametric (Spearman), 196 Covariance, 107, 110 matrix, 110, 112, 114 sample, 227 Cramér-Rao inequality, 209 Cumulative function, 38, 40, 48 D Decay η meson, 125 π → 2γ, 80 constant, 67 © Springer International Publishing Switzerland 2016 S Širca, Probability for Physicists, Graduate Texts in Physics, DOI 10.1007/978-3-319-31611-6 409 410 nuclear, 133 radioactive, 67, 87–90, 128, 134 time, 67, 213, 215, 219, 276 neutron, 242 width, 88 Z0 boson, 88 Decile, 96 Detailed balance, 339 Detection efficiency, 10, 106, 135, 140 Device failure time, 115, 129, 221 Difference of events, Diffusion, 321 ballistic, 163 function, 315 log-normal, 163 normal, 163, 166 of thermal neutrons, 171 self-, 317 Dipole radiation, 390 Discrete distribution, 37, 45 Distribution binomial, 123–128, 135–138, 140, 369, 378 normal approximation, 130 Boltzmann, 18 Bose-Einstein, 18, 295 Breit-Wigner, 78, 88, 393 candidate, 339 Cauchy, 41, 77, 105, 374, 393 χ2 , 79 continuous, 39, 47 cumulative, 40, 48 discrete, 37, 45 equilibrium, 310, 339 exponential, 67–70, 84 F, 80 Fermi-Dirac, 18, 296 Fréchet, 158 function, 38, 40, 48 continuous, 40 discrete, 38 empirical, 271 Gumbel, 158 hyper-exponential, 86, 117 hypo-exponential, 89, 117 isotropic (R3 , Rd ), 389 isotropic (hyperplane), 389 Lévy, 154 log-normal, 381 Lorentz, 78 maximum-entropy continuous, 297 discrete, 289–297 Index Maxwell, 74, 85, 105 Maxwell-Boltzmann, 292 mode, 104 multinomial, 128 multivariate normal, 110, 300, 392 negative binomial, 129 order k, 129, 138 normal, 70–74, 109–111, 375, 377, 391 normal, multivariate, 110, 300, 392 of blood types, 19 of energy losses in detector, 57 of extreme rainfall, 159, 222–224 of extreme values, 63–64, 156–161, 172, 222–224 of maximal values, 63–64, 156–161 of minimal values, 63–64, 156–161 of sample median, 194 Pareto, 75–77, 97 Pascal, 129 Poisson, 132–135, 140–142, 369, 371, 372, 377 stable, 153–155 Student’s (t), 79 target, 339 uniform, 65, 80–84 in circle, 388 in sphere, 388 over directions (R3 , Rd ), 389 Voigt, 169 Weibull, 158 Dosage of active ingredient, 201 Drawing, 37, 337, 383 Drift function, 315 E Effective deviation, 100 Efficiency of detector, 10, 106, 135, 140 of estimator, 210, 211 of vaccine, 133 Electron mobility in semiconductor, 255 El Niño, 254 Energy free, 294 internal, 294 ENSO (El Niño Southern Oscillation), 254 Entropy information (Shannon), 283 of continuous distribution, 286 of discrete distribution, 286 relative, 287 thermodynamic, 294–297 Index Equation Chapman–Kolmogorov, 308, 312, 314 constraint, 245 Fokker–Planck, 316, 317, 319 Langevin, 321 likelihood, 204 Montroll-Weiss, 166 Yule-Walker, 302 Error function (erf), 71 Estimate, 178 of spectrum (max-entropy), 298 Estimator, 178 biased, 178 consistent, 178 efficient, 210 unbiased, 178, 198, 199, 210 Eugene Onegin, 307 Event certain, complementary, compound, elementary, exclusive, impossible, incompatible, independent, 14–16, 21, 47, 48 universal, Exam grades, 279 Expected value, 93–115 Experiment Buffon, 325 random, Exponential distribution, 67–70, 84 Extinction probability, 353 time, 349 Extreme value distribution, 156–161 F Fat tails, 155, 163 Fisher transformation, 196 Flu medicine, 279 Formula Bayes, 16, 25 binomial, 8, 124 Planck, 56 product, 11 Sokhotsky–Plemelj, 34 total probability, 17 Free energy, 294 Function characteristic, 376–382 411 characterizing, 315 (cumulative) distribution, 38, 40 diffusion, 315 Dirac (δ), 31, 364 distribution, 38, 40 drift, 315 erf, 71 Heaviside (step), 35 importance, 334 likelihood, 203 moment-generating, 374–376 partition, 290 probability-generating, 367–374 propagator moment, 314 step (Heaviside), 35 transition, 384 FWHM, 73 G General linear regression, 248 Generating function and convolutions, 369 probability, 367–374 Generator of random numbers linear, 384–386 non-linear, 386–387 Global release of CO2 , 251 H HXRBS, 76 Hypercube, 327 Hypersphere, 327, 338 Hypothesis, 16, 259 alternative, 260 null, 259 I Importance function, 334 Importance sampling, 333 Independent variables, 47, 48 Inequality Cantelli’s, 101 Chebyshev, 101 Cramér-Rao, 209 Jensen’s, 99 Information, 12, 283 entropy (Shannon), 283 of sample, 209 Internal energy, 294 Inter-quartile range (IQR), 96, 193 412 J Jacobi matrix, 51 Jensen’s inequality, 99 Joint probability density, 47 K Kullback-Leibler distance, 287 Kurtosis, 104 L Lagrange multiplier, 246, 289–303 LCG, 384 Least median of squares (LMS), 249 Lévy flights, 163 Likelihood, 203 interval, 212–214 region, 217–219 Linear congruential generator, 384 correlation, 108, 195 regression, 228–250 error in both coordinates, 240 Log-normal diffusion, 163 LOLA, 76 Lorentz distribution, 78 M MAD, MADN, 193 Magnetic dipoles, 294 Magnetization in superconductor, 257 Marginal probability density, 48 Markov chain, 308–313 at long times, 309 ergodic, 309 irreducible, 309 Monte Carlo, 339–345 reversible, 339 process, 307–323 propagator, 314 density function, 314 Matrix covariance, 110, 112, 114, 227 Jacobi, 51 Markov (stochastic), 308 stochastic (Markov), 308 Toeplitz, 299 Maximum-entropy distribution, 289–298 spectral analysis (MESA), 298 Maxwell distribution, 74, 85, 105 Index Mean, 93 sample, 179 Measurable set, space, 363 Measure, 363 Dirac, 364 Median, 95 absolute deviation (MAD), 193 of squares of residuals, 249 sample, 193 MESA (Maximum-entropy spectral analysis), 298 Meteorites, 133 Method maximum likelihood, 203–224 MCMC, 339–345 MESA, 303 Monte Carlo, 325–345 Markov-chain (MCMC), 339–345 numerical integration, 328–338 variance reduction, 333–338 of least squares, 227–257 rejection, 392 transformation (inverse), 390 Metropolis-Hastings algorithm, 341–345 Mixing (MCMC), 343 Mode, 104 Model nested, 268 of births, 347 of births and deaths, 351–356 of deaths, 348 of weather, 311 Moment, 102–106 generating function, 374–376 indeterminate, 381 Multinomial distribution, 128 symbol, 7, 128 N Neutron decay time, 242 Nimbus satellite, 192 Noise in electric circuit, 101, 119 Non-Parametric correlation, 196 Normal diffusion, 163, 166 distribution, 70–74, 109–111, 375, 377, 391 multivariate, 110, 300, 392 system of equations, 229 Nuclear decay chain, 89 Index O Order of generator, 384 Outcome of experiment, Outliers, 192, 193 P Pareto distribution, 75–77, 97 principle (80/20), 97 Partition function, 290 Percentile, 96, 397 Permutations, Phase sum, 290 Photo-disintegration of He, 263 Planck formula, 56 Pochammer symbol, Polymer molecule, 169 Population, 177 dynamics, 347–358 Power of test, 261 set, 364 spectral density (PSD), 302 tails, 155, 163 Primary ionization, 140 Principle maximum likelihood, 204 of indifference, 288 of insufficient reason, 288 of maximum entropy, 288 Pareto (80/20), 97 Probability acceptance (MCMC), 341 conditional, 11–18 density, 39 in quantum mechanics, 99 joint, 47 marginal, 48 Markov propagator, 314 function, 37 generating function, 367–374 measure, 364 of event, of extinction, 353 posterior, 17 prior, 17 transition (single-step), 308 Problem Monty Hall, 22 newsboy, 136 of indeterminate moments, 381 Process 413 Markov, 307–323 continuous-time, 313–323 discrete-time, 308–313 propagator, 314 memoryless, 307 Ornstein–Uhlenbeck, 318 random, 162 Wiener, 317 Product formula, 11 of events, Propagation of errors, 111–115, 120 Propagator density function, 314 moment function, 314 of Markov process, 314 Pseudo-random numbers, 383–393 Q Q–Q plot, 199 Quantile, 96, 397 Quantum defects in atoms, 256 Quartile, 96 Quasi-random sequence, 337 R Rabbits and foxes, 357–358 Radioactive decay, 213 Random experiment, process, 162 variable, 37 complex, 101 continuous, 39 discrete, 37 realization, 37 vector, 101 walk, 27, 162–167 continuous-time, 165 discrete-time, 162 Rank, 196 correlation coefficient, 197 Realization of random variable, 37 Regression by singular-value decomposition, 248 fitting a constant, 240 fitting a polynomial, 230–236 fitting a straight line, 237–240 for binned data, 242–245 general linear, 248 linear, 228–250 414 error in both coordinates, 240 with constraints, 245–247 non-linear, 250–257 robust, 249 with orthogonal polynomials, 236 Rejection region, 260 Residual, 232 Return period, 159 value, 159 Risk level, 189 Robust regression, 249 statistics, 192–194 ROC curve, 260, 263 S Sample, 177 correlation, 195 covariance matrix, 227 distribution, 178 of sums and differences, 184 of variance ratios, 186 of variances, 185 mean, 179 rank, 197 space, statistic, 178 variance, 179 Scintillator (light yield), 86 Searching for the lost plane, 22 Seed (random number generator), 384 Seismic velocity, 287 Self-diffusion, 317 Sensitivity of test, 25, 261 Sequence quasi-random, 337 serially uniform, 383 Sobol, 337 unbiased, 383 uncorrelated, 383 uniform, 383 Set measurable, 363 power, 364 Sharks and ice cream, 111 σ-algebra, 363 Significance of test, 260 Single-step transition probability, 308 Skewness, 103 Sokhotsky–Plemelj formula, 34 Southern oscillation, 254 Index Spatial resolution of detector, 140 Specificity of test, 25, 261 Spectral analysis (maximum-entropy), 298 line shape, 73, 78 line width, 168 Stable distributions, 153–155 Standard deviation, 71, 100 State periodic, 309 reproducible, 309 Statistic, 178 Statistical significance, 260 tests, 259–276 Stochastic variable, 37 Student’s distribution, 79 Sub-diffusion, 166 Sum of events, Super-diffusion, 163, 166 Symbol binomial, multinomial, 7, 128 Pochammer, T Temperature anomaly, 233 Tensile strength of glass fibers, 224 Test Anderson-Darling, 276 χ2 (Pearson’s), 269–271, 278 comparing binned data, 271 comparing sample means, 265, 266 comparing sample variances, 267 confidence level, 261 F, 267–269 Kolmogorov–Smirnov, 271–275 of sample mean, 264 of sample variance, 265 parametric, 263–269 power, 261 risk level, 189 sensitivity, 25, 261 specificity, 25, 261 statistical, 259–276 statistical significance, 260 t (Student’s), 264 Theorem Berry-Esséen, 151 central limit, 149–152 generalized, 155 Fisher-Tippett-Gnedenko, 158 Index Gauss–Markov, 229 Laplace’s limit, 130, 379 Perron-Frobenius, 310 Szeg˝o, 301 Thermal expansion of copper, 255 Thinning, 140 “Three-σ” rule, 193 Three-level system, 293 Time decay, 67, 213, 215, 219, 276 of extinction, 349 to failure, 138 Total probability formula, 17 Transformation Box-Muller, 391 Fisher, 196 415 of random variable n-dim, 50–56 1-dim, 41–45 Transition function, 384 V Variance, 100 lower bound, 209 minimal, 209 sample, 179 Variations, W Width of spectral line, 168 ... references for scientists entering, or requiring timely knowledge of, a research field More information about this series at http://www.springer.com/series/8431 Simon Širca Probability for Physicists. .. B occurs nAB times, therefore P(A|B) = lim n→∞ P(AB) nAB /n = nB /n P(B) The conditional probability for A given B (P(B) = 0) is therefore computed by dividing the probability of the simultaneous... grateful to them for reading the manuscript so attentively I would also like to thank my student Mr Peter Ferjančič for leading the problem-solving classes for two years and for suggesting and