Between Certainty and Uncertainty

Ludomir M. Laudański Between Certainty and Uncertainty Intelligent Systems Reference Library, Volume 31 Editors-in-Chief Prof. Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul. Newelska 6 01-447 Warsaw Poland E-mail: kacprzyk@ibspan.waw.pl Prof. Lakhmi C. Jain University of South Australia Adelaide Mawson Lakes Campus South Australia 5095 Australia E-mail: Lakhmi.jain@unisa.edu.au Further volumes of this series can be found on our homepage: springer.com Vol. 11. Samuli Niiranen and Andre Ribeiro (Eds.) Information Processing and Biological Systems, 2011 ISBN 978-3-642-19620-1 Vol. 12. Florin Gorunescu Data Mining, 2011 ISBN 978-3-642-19720-8 Vol. 13. Witold Pedrycz and Shyi-Ming Chen (Eds.) Granular Computing and Intelligent Systems, 2011 ISBN 978-3-642-19819-9 Vol. 14. George A. Anastassiou and Oktay Duman Towards Intelligent Modeling: Statistical Approximation Theory, 2011 ISBN 978-3-642-19825-0 Vol. 15. 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Data Mining: Foundations and Intelligent Paradigms, 2011 ISBN 978-3-642-23150-6 Vol. 26. Tauseef Gulrez and Aboul Ella Hassanien (Eds.) Advances in Robotics and Virtual Reality, 2011 ISBN 978-3-642-23362-3 Vol. 27. Cristina Urdiales Collaborative Assistive Robot for Mobility Enhancement (CARMEN), 2011 ISBN 978-3-642-24901-3 Vol. 28. Tatiana Valentine Guy, Miroslav Kárný and David H. Wolpert (Eds.) Decision Making with Imperfect Decision Makers, 2012 ISBN 978-3-642-24646-3 Vol. 29. Roumen Kountchev and Kazumi Nakamatsu (Eds.) Advances in Reasoning-Based Image Processing Intelligent Systems, 2012 ISBN 978-3-642-24692-0 Vol. 20. Lech Polkowski Approximate Reasoning by Parts, 2011 ISBN 978-3-642-22278-8 Vol. 30. Marina V. Sokolova and Antonio Fernández-Caballero Decision Making in Complex Systems, 2012 ISBN 978-3-642-25543-4 Vol. 21. Igor Chikalov Average Time Complexity of Decision Trees, 2011 ISBN 978-3-642-22660-1 Vol. 31. Ludomir M. Laudański Between Certainty and Uncertainty, 2013 ISBN 978-3-642-25696-7 Ludomir M. Laudański Between Certainty and Uncertainty Statistics and Probability in Five Units with Notes on Historical Origins and Illustrative Numerical Examples 123 Author Ludomir M. Laudański Rzeszow Technical University Rzeszow Poland ISSN 1868-4394 e-ISSN 1868-4408 ISBN 978-3-642-25696-7 e-ISBN 978-3-642-25697-4 DOI 10.1007/978-3-642-25697-4 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2012930478 c Springer-Verlag Berlin Heidelberg 2013  This work is subject to copyright. 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Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Acknowledgements Bertrand Russell (1872–1970), who left about 40 books for the coming generations, in a preface to one of them confessed that they were completely ready in his mind before he sat down to write them. But this is not the case even with a single one of my books whose number does not came to one third of Russell's books. Most frequently the preparatory procedure of my book resembles a mosaic where the general concept becomes the opening stage of the design process. Here I must mention the pleasure stemming from a suitable selection of the pieces used to make successive parts of the mosaic. On the other hand, my books have been predominantly the result and product of my lectures and of the stimulating processes of my students' reception of them. As Czesław Miłosz once said after giving a lecture: "Again I was lucky enough to draw a rabbit from the hat". Therefore my first words of acknowledgement are addressed to my students. On a more personal note – I commence by mentioning my best American colleague and friend, the late Professor Frederick O. Smetana (1928-2011), whose assistance over several decades was an important stimulus in all I did – despite thousands of kilometres dividing us – he lived in Raleigh and I in Rzeszow (even the first letters of our home towns are the same). I also have profited, which a careful reader might note in the book, from the assistance of Professor Antoni Smoluk, now emeritus of Wrocław University of Economics and long-time Head of the Mathematics Department. Next I would like to thank Professor Stephan M. Stigler of Chicago University, an internationally recognized figure, the author of histories and stories on statistics and probability. His help with respect to literature for my book was exceptional: I can confess that each and every of my calls for help, even during his summer holidays, was answered and appropriate assistance was provided. Lecturing for two weeks at Yasar University, Ismir, Turkey, I had the pleasure to met Professor Giuseppe Burgio of La Sapienza, Rome, and our further contacts became fruitfully stimulating for this book. I have also received friendly support from Professor Anthony W. F. Edwards, Cambridge. The true Godfather of this book was Dr. Thomas Ditzinger from Springer Verlag, Heidelberg; I would also like to thank Helger Schaepe from the same place, whose friendly help I greatly appreciate. I received assistance from a very distant place, Chennai, India and here I must mention Ms. Suguna Ramalignan, the leader of the team which patiently and thoroughly took the book through its final printing stage. Let me also express my gratitude to professor Daniel Simson who gave his permission to reproduce a drawing by Leon JeĞmanowicz. My daughter Maria Klara Laudanska did the final proof reading of the book. To all those persons, named and unnamed, I would like once more to express my deep appreciation, gratitude and sincere thanks. Rzeszow May 2012 Ludomir M. LaudaĔski Contents Polish Probabilists .................................................................................................1 Prologue and Logistics – per se ............................................................................3 Dice Players ............................................................................................................5 BOOK ONE THEORY Chapter 1: Descriptive Statistics ..........................................................................9 1.1 1.2 1.3 A Dialogue ..............................................................................................9 Defining the Subject ..............................................................................11 Descriptive Statistics Dimension One ...................................................13 1.3.1 Mean Value – Definition and Significance ...............................13 1.3.2 Variance, and Variability ..........................................................16 1.3.3 Linear Transformations .............................................................19 1.3.4 Z-Score Statistics ......................................................................25 1.4 Famous and Admired ............................................................................26 Literature References ......................................................................................35 Chapter 2: Grouped Data. Introduction to General Statistics ........................37 2.1 Grouping Due to Attributes ...................................................................38 2.2 Inner Appendix ......................................................................................45 2.3 Grouping of Variables ...........................................................................50 2.4 Direct Method to Derive Averages ........................................................53 2.5 Coded Method .......................................................................................56 2.6 Discussing Two Special Cases ..............................................................58 2.7 Percentiles for the Grouped Data ...........................................................62 References .......................................................................................................64 Chapter 3: Regression vrs. Correlation .............................................................67 3.1 Linear Regression – The Idea ................................................................67 3.2 Regression Lines ...................................................................................69 3.3 Arithmetical Appendix without Comments ...........................................75 3.4 Correlation – Descriptive Statistics .......................................................76 3.5 Correlation – Grouped Data ..................................................................80 3.6 The Great Table of Correlation..............................................................80 References .......................................................................................................85 VIII Contents Chapter 4: Binomial Distribution ......................................................................87 4.1 Tracing the Origin .................................................................................87 Abuthnot ...........................................................................................88 Pascal ................................................................................................90 Stifel ..................................................................................................93 Bayes .................................................................................................96 4.2 Close Acquaintance ...............................................................................99 4.3 Three Problems of S. Pepys [22], p.400-401 .........................................99 4.4 Weldon’s Dice Data.............................................................................102 4.5 Two Shores – Two Tails ......................................................................103 4.6 Jacob Bernoulli’ Weak Law of Large Numbers ..................................109 4.7 Following Abraham de Moivre ...........................................................110 4.8 Beyond the Binomial Distribution .......................................................113 4.9 Derivation of the Poisson Distribution ................................................116 4.10 Notes on the Multinomial and Negative Binomial Distributions ........121 References .....................................................................................................125 Chapter 5: Normal Distribution Binomial Heritage .......................................129 5.1 Normal Statistics, Preliminaries ..........................................................129 5.2 Four Properties of the Normal Distribution .........................................135 5.3 Making Use of the Statistical Tables of the Normal Distribution........135 5.4 Two Proofs ..........................................................................................137 5.5 The Central Limit Theorem – An Intuitive Approach .........................138 5.6 Distribution of Sample Means .............................................................140 5.7 Properties of the Distribution of Sample Means ..................................141 5.8 To Initiate the Monte Carlo Simulation ...............................................144 5.9 De Moivre–Laplace Limit Theorems ..................................................149 5.10 Remarks on the Binomials Convergence ............................................154 References .....................................................................................................157 Les Gross Poissons .............................................................................................159 BOOK TWO EXERCISES Unit 1: Descriptive Statistics .............................................................................165 Problem 1.1 (see: [1], Prob. 4.10, p.59).....................................................165 Problem 1.2 (see: [1], Prob. 2.8, p.23).......................................................165 Problem 1.3 (see [1], Prob. 2.20, p.25) ......................................................166 Problem 1.4 ...................................................................................................168 Problem 1.5 (Follows Prob.4.25 [1]) .........................................................170 Problem 1.6 (see [1], Prob.2.29) ................................................................174 Problem 1.7 (see [1], Prob.2.26) ................................................................175 Problem 1.8 (see: [1], Prob.2.23)...............................................................176 Problem 1.9 (see [1], Prob.3.9) ..................................................................177 Short Note .....................................................................................................178 Contents IX Problem 1.10 (see: [1], Prob.3.7) ................................................................179 Problem 1.11 (see: [1], Prob.3.21) ..............................................................180 Problem 1.12 (see: [1], Prob.3.13, p.42) .....................................................180 Problem 1.13 (see [1], 3.28, p.45) ...............................................................181 Problem 1.14 (see [6], 2.21) ........................................................................181 Problem 1.15 (see: [1], Prob.3.27) ..............................................................185 References .....................................................................................................187 Unit 2: Grouped Data ........................................................................................189 Problem 2.1 (see [1], p.15) ........................................................................189 Problem 2.2 (see [1], p.16) ........................................................................190 Problem 2.3 (see: [1]), p.16) ......................................................................191 Problem 2.4 (see [2], Prob.5.12) - The Continuous Case ..........................192 Problem 2.5 (see [2], Prob.5.10) – The Discrete Case ...............................194 Problem 2.6 (see: [4], pp.32-33) ................................................................196 Problem 2.7 ...................................................................................................199 Problem 2.8 (see [2], Prob.5.21) ................................................................205 Problem 2.9 (see [2], Review I, Prob.1.13) ...............................................206 Problem 2.10 (see [1], p.96) ........................................................................208 Problem 2.11 (Example of the Final Examination Problem) ......................211 Problem 2.12 (see [1], p.104) ......................................................................213 Problem 2.13 (Another Example of the Final Examination Problem) ........213 References .....................................................................................................215 Unit 3: Regression vs. Correlation ...................................................................217 Problem 3.1 (see [3]) .................................................................................217 Problem 3.2 ...................................................................................................221 Problem 3.3 [6] ..........................................................................................223 Problem 3.4 ...................................................................................................226 Problem 3.5 ...................................................................................................226 Problem 3.6 ...................................................................................................226 Problem 3.7 ...................................................................................................227 Problem 3.8 ...................................................................................................228 Problem 3.9 ...................................................................................................231 Problem 3.10 .................................................................................................235 Problem.3.11 (see [4], Problem.16.11) .......................................................237 Problem 3.12 .................................................................................................238 Problem Extra One ........................................................................................241 References .....................................................................................................242 Unit 4: Binomial Distribution ...........................................................................245 Problem 4.1 (see [12], Problem 3.1, p.256, p.453) ....................................245 Problem 4.2 (see [12], Problem 2.12, p.251) .............................................248 Problem 4.3 ...................................................................................................249 Problem 4.4 ...................................................................................................252 X Contents Problem 4.5 ...................................................................................................252 Problem 4.6 ...................................................................................................253 Problem 4.7 (see: [13], p.125) ...................................................................256 Problem 4.8 ...................................................................................................259 Problem 4.9 ...................................................................................................261 Problem 4.10 (see [2], Problem 9.15, p.178) ..............................................262 Problem 4.11 (see [13], p.128) ....................................................................263 Problem 4.12 (see [5], p.165, p.63) .............................................................264 Problem 4.13 (Source: Internet) ..................................................................265 Problem 4.14 .................................................................................................266 Problem 4.15 (see [14]) ...............................................................................268 Problem 4.16 (see [19]) ...............................................................................270 Problem 4.17 .................................................................................................272 References .....................................................................................................273 Unit 5: Normal Distribution. Binomial Heritage ............................................275 Problem.5.1 (see [5], Problem 7.16, p.129) ...............................................276 Problem 5.2 (see [5], Problem 7.20) ..........................................................276 Problem 5.3 (see 7.22 in [5], Modified) ....................................................279 Problem 5.4 ([1], 7.28 – No Answer) ........................................................280 Problem 5.5 ([1], 7.27 – Answers Enclosed) .............................................281 Problem 5.6 ([1], 7.29, with a Single Answer; Modified) .........................282 Problem 5.7 (Following Problem 4.5) .......................................................283 Problem 5.8 (Following Problem 4.7) .......................................................285 Problem 5.9 (Following Problem 4.8) .......................................................285 Problem 5.10 (Following Problem 4.10)...................................................286 Problem 5.11 (see [5], Problem 7.25) .........................................................286 Problem 5.12 (Weinberg [1] 10.5 p. 196) ...................................................288 Problem 5.13 Weinberg [1], 10.7, p.196) .................................................289 Problem 5.14 (Weinberg [1], 10.9, s.196) ...................................................291 Problem 5.15 (Weinberg [1], 10.13, p.197) ................................................293 Problem 5.16 (Weinberg [1], 10.15, p.197) ................................................295 Problem 5.17 (Weinberg [1] 10.23, p.198) .................................................297 Problem 5.18 (Weinberg [1], 8.8, p.157, No Answers)...............................298 Problem 5.19 (Weinberg [1], 8.10, p.157 – No Answers)...........................300 Problem 5.20 ..............................................................................................301 References .....................................................................................................302 Error Function ...................................................................................................303 References ..........................................................................................................305 Index ...................................................................................................................311