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1 UNITEXT for Physics Series editors Paolo Biscari, Milano, Italy Michele Cini, Roma, Italy Attilio Ferrari, Torino, Italy Stefano Forte, Milano, Italy Morten Hjorth-Jensen, Oslo, Norway Nicola Manini, Milano, Italy Guido Montagna, Pavia, Italy Oreste Nicrosini, Pavia, Italy Luca Peliti, Napoli, Italy Alberto Rotondi, Pavia, Italy

2 UNITEXT for Physics series, formerly UNITEXT Collana di Fisica e Astronomia, publishes textbooks and monographs in Physics and Astronomy, mainly in English language, characterized of a didactic style and comprehensiveness. The books published in UNITEXT for Physics series are addressed to graduate and advanced graduate students, but also to scientists and researchers as important resources for their education, knowledge and teaching. More information about this series at

3 Carlos Maña Probability and Statistics for Particle Physics 123

4 Carlos Maña Departamento de Investigación Básica Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas Madrid Spain ISSN ISSN (electronic) UNITEXT for Physics ISBN ISBN (ebook) DOI / Library of Congress Control Number: Springer International Publishing AG 2017 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. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

5 Contents 1 Probability The Elements of Probability: ðx; B; lþ Events and Sample Space: ðxþ r-algebras ðb X Þ and Measurable Spaces ðx; B X Þ Set Functions and Measure Space: ðx; B X ; lþ Random Quantities Conditional Probability and Bayes Theorem Statistically Independent Events Theorem of Total Probability Bayes Theorem Distribution Function Discrete and Continuous Distribution Functions Distributions in More Dimensions Stochastic Characteristics Mathematical Expectation Moments of a Distribution The Error Propagation Expression Integral Transforms The Fourier Transform The Mellin Transform Ordered Samples Limit Theorems and Convergence Chebyshev s Theorem Convergence in Probability Almost Sure Convergence Convergence in Distribution Convergence in L p Norm Uniform Convergence v

6 vi Contents Appendices References Bayesian Inference Elements of Parametric Inference Exchangeable Sequences Predictive Inference Sufficient Statistics Exponential Family Prior Functions Principle of Insufficient Reason Parameters of Position and Scale Covariance Under Reparameterizations Invariance Under a Group of Transformations Conjugated Distributions Probability Matching Priors Reference Analysis Hierarchical Structures Priors for Discrete Parameters Constrains on Parameters and Priors Decision Problems Hypothesis Testing Point Estimation Credible Regions Bayesian (B) Versus Classical (F ) Philosophy Some Worked Examples Regression Characterization of a Possible Source of Events Anisotropies of Cosmic Rays References Monte Carlo Methods Pseudo-Random Sequences Basic Algorithms Inverse Transform Acceptance-Rejection (Hit-Miss; J. Von Neumann 1951) Importance Sampling Decomposition of the Probability Density Everything at Work The Compton Scattering An Incoming Flux of Particles Markov Chain Monte Carlo Sampling from Conditionals and Gibbs Sampling

7 Contents vii 3.5 Evaluation of Definite Integrals References Information Theory Quantification of Information Expected Information and Entropy Conditional and Mutual Information Generalization for Absolute Continuous Random Quantities Kullback Leibler Discrepancy and Fisher s Matrix Fisher s Matrix Asymptotic Behaviour of the Likelihood Function Some Properties of Information Geometry and Information References

8 Introduction They say that understanding ought to work by the rules of right reason. These rules are, or ought to be, contained in Logic; but the actual science of logic is conversant at present only with things either certain, impossible, or entirely doubtful, none of which (fortunately) we have to reason on. Therefore the true logic of this world is the calculus of Probabilities, which takes account of the magnitude of the probability which is, or ought to be, in a reasonable man s mind. J.C. Maxwell These notes, based on a one-semester course on probability and statistics given in the former Doctoral Program of the Department of Theoretical Physics at the Universidad Complutense in Madrid, are a more elaborated version of a series of lectures given at different places to advanced graduate and Ph.D. students. Although they certainly have to be tailored for undergraduate students, they contain a humble overview of the basic concepts and ideas one should have in mind before getting involved in data analysis and I believe they will be a useful reference for both students and researchers. I feel, maybe wrongly, that there is a recent tendency in a subset of the particle physics community to consider statistics as a collection of prescriptions written in some holy references that are used blindly with the only arguments that either everybody does it that way or that it has always been done this way. In the lectures, I have tried to demystify the how-to recipes not because they are not useful but because, on the one hand, they are applicable under some conditions that tend to be forgotten and, on the other, because if the concepts are clear so will be the way to proceed ( at least formally ) for the problems that come across in particle physics. At the end, the quote from Laplace given at the beginning of the first lecture is what it is all about. There is a countable set of books on probability and statistics and a sizable subset of them are very good, out of which I would recommend the following ones (a personal choice function). Chapter 1 deals with probability and this is just a measure, a finite nonnegative measure, so it will be very useful to read some sections of Measure Theory (2006; Springer) by V.I. Bogachev, in particular Chaps. 1 and 2 of the first volume. However, for those students who are not yet ix

9 x Introduction familiar with measure theory, there is an appendix to this chapter with a short digression on some basic concepts. A large fraction of the material presented in this lecture can be found in more depth, together with other interesting subjects, in the book Probability: A Graduate Course (2013; Springer Texts in Statistics) by A. Gut. Chapter 2 is about statistical inference, Bayesian inference in fact, and a must for this topic is the Bayesian Theory (1994; John Wiley & Sons) by J.M. Bernardo and A.F.M. Smith that contains also an enlightening discussion about the Bayesian and frequentist approaches in the Appendix B. It is beyond question that in any worthwhile course on statistics the ubiquitous frequentist methodology has to be taught as well and there are excellent references on the subject. Students are encouraged to look, for instance, at Statistical Methods in Experimental Physics (2006; World Scientific) by F. James, Statistics for Nuclear and Particle Physicists (1989; Cambridge University Press) by L. Lyons, or Statistical Data Analysis (1997; Oxford Science Pub.) by G. Cowan. Last, Chap. 3 is devoted to Monte Carlo simulation, an essential tool in statistics and particle physics, and Chap. 4 to information theory, and, like for the first chapters, both have interesting references given along the text. Time is short, my strength is limited,, Kafka dixit, so many interesting subjects that deserve a whole lecture by themselves are left aside. To mention some: an historical development of probability and statistics, Bayesian networks, generalized distributions (a different approach to probability distributions), decision theory (games theory), and Markov chains for which we shall state only the relevant properties without further explanation. I am grateful to Drs. J. Berdugo, J. Casaus, C. Delgado, and J. Rodriguez for their suggestions and a careful reading of the text and much indebted to Dr. Hisako Niko. Were not for her interest, this notes would still be in the drawer. My gratitude goes also to Mieke van der Fluit for her assistance with the edition.