Statistics. David W. Scott

      Statistics

      A Concise Mathematical Introduction for Students, Scientists, and Engineers

       David W. Scott

       Rice University

       Houston, Texas

      This edition first published 2020

      © 2020 John Wiley & Sons Ltd

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      Paperback ISBN: 9781119675846

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      Cover Image: Courtesy of David W. Scott

       To my parents, John and Nancy Scott

Preface

      My aim in writing this book is to provide a self‐contained, one‐semester probability and statistics introduction that covers core material without ballooning into a huge tome. Since statistics requires an understanding of distributions and relationships (for example, predicting

from

), some introductory knowledge of multivariate calculus and linear algebra will be assumed. Examples will use the

language, but they can easily be modified to other systems such as Matlab. Mathematica will be used for symbolic computations. JMP can be used to perform statistical tests in a unified manner.

      The course divides naturally into three sections: (1) classical probability; (2) distribution functions, density functions, and random variables; and (3) statistical inference and hypothesis testing.

      In selecting material to include, I have favored models that follow directly from simple, intuitive assumptions. I have also favored statistical topics that are widely used. In this era of data science, I have occasionally selected new topics that are relevant and easily understood. For example, robustness is relevant because bad data or outliers can adversely affect classical methodology.

      Students who have taken AP Statistics will have an advantage in that they will have seen a large number of cookbook statistical procedures and tests. We will cover only a selection, as the mathematical foundations (or outline thereof) will be of equal interest here. Often we will sacrifice mathematical rigor in favor of an engineering‐level understanding without apology. Motivated students will naturally follow this course with more mathematically rigorous courses in statistics, probability, and stochastic processes. Reading about other statistical tests and methods should be straightforward after mastering the material covered here.

      I have included a handful of problems and case studies, to keep things simple. There will be a live course website with numerous sample problems and exams. Instructors with special interests can easily insert their own examples and problems in appropriate sections.

      The URL for the additional course material is

      http://www.stat.rice.edu/∼scottdw/wiley-dws-2020/

      The directory contains problems, sample exams, and the pdf file all-figs.pdf, which displays all 57 figures, including 45 color diagrams. The author may be reached at [email protected]

I wish to thank James R. Thompson, who introduced me to the beauty of model building and statistical thinking. He served as one of my thesis advisers, directing me into the joys of nonparametric modeling. He was in turn highly influenced by his thesis adviser, John W. Tukey, one of the most important statisticians of the 20th century. Tukey's contributions ranged from the fast Fourier transform to the body of graphical work introduced in his monograph Exploratory Data Analysis. Their ideas appear throughout this book.

       David W. Scott

       Houston, Texas

       September, 2019

1 Data Analysis and Understanding

      The field of statistics has a rich history that has become tightly integrated into the emerging field of data sciences. Collaboration with computer scientists, numerical analysts, and decision makers characterizes the field. The role of statistics and statisticians is to find actionable information in a noisy collection of data. Every field of academic endeavor encounters this problem: from the electrical engineer trying to find a signal in a noisy channel to an English professor trying to determine the authorship of a contested newly discovered manuscript.

      There are two basic tasks for the statistician. First is to characterize the distribution of possible outcomes using a batch of representative data. An actuary may be asked to find a dollar loss for car accidents that is not exceeded 99.999% of the time. An economist may be asked to provide useful summaries of a collection of income data. The histogram is our primary tool here, an idea that did not appear until the 17th century; see Graunt (1662), who analyzed death