Statistics for Scientists and Engineers E-Book

Ramalingam Shanmugam

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 808

Veröffentlichungsjahr: 2015

Beschreibung

This book provides the theoretical framework needed to build, analyze and interpret various statistical models. It helps readers choose the correct model, distinguish among various choices that best captures the data, or solve the problem at hand. This is an introductory textbook on probability and statistics. The authors explain theoretical concepts in a step-by-step manner and provide practical examples. The introductory chapter in this book presents the basic concepts. Next, the authors discuss the measures of location, popular measures of spread, and measures of skewness and kurtosis. Probability theory, discrete distributions, and important continuous distributions that are often encountered in practical applications are analyzed. Mathematical Expectation is covered, along with Generating Functions and Functions of Random Variables. It discusses joint distributions, and novel methods to find the mean deviation of discrete and continuous statistical distributions. * Provides insight on coding complex algorithms using the 'loop unrolling technique' * Covers illuminating discussions on Poisson limit theorem, central limit theorem, mean deviation generating functions, CDF generating function and extensive summary tables * Contains extensive exercises at the end of each chapter and examples from interdisciplinary fields Statistics for Scientists and Engineers is a great resource for students in engineering, physical sciences, and management, and also practicing engineers who require skill sets to model practical problems in a statistical setting.

Leseprobe

Table of Contents

Cover

Title Page

Copyright

Preface

Audience

Purpose

Approach

Main Features

Mathematics Level

Coverage

About the Companion Website

Chapter 1: Descriptive Statistics

1.1 Introduction

1.2 Statistics as A Scientific Discipline

1.3 The NOIR Scale

1.4 Population Versus Sample

1.5 Combination Notation

1.6 Summation Notation

1.7 Product Notation

1.8 Rising and Falling Factorials

1.9 Moments and Cumulants

1.10 Data Transformations

1.11 Data Discretization

1.12 Categorization of Data Discretization

1.13 Testing for Normality

1.14 Summary

Exercises

Chapter 2: Measures of Location

2.1 Meaning of Location Measure

2.2 Measures of Central Tendency

2.3 Arithmetic Mean

2.4 Median

2.5 Quartiles and Percentiles

2.6 Mode

2.7 Geometric Mean

2.8 Harmonic Mean

2.9 Which Measure to Use?

2.10 Summary

Exercises

Chapter 3: Measures of Spread

3.1 Need For a Spread Measure

3.2 Range

3.3 Inter-Quartile Range (IQR)

3.4 The Concept of Degrees of Freedom

3.5 Averaged Absolute Deviation (AAD)

3.6 Variance and Standard Deviation

3.7 Coefficient of Variation

3.8 Gini Coefficient

3.9 Summary

Exercises

Chapter 4: Skewness and Kurtosis

4.1 Meaning of Skewness

4.2 Categorization of Skewness Measures

4.3 Measures of Skewness

4.4 Concept of Kurtosis

4.5 Measures of Kurtosis

4.6 Summary

Exercises

Chapter 5: Probability

5.1 Introduction

5.2 Probability

5.3 Different Ways to Express Probability

5.4 Sample Space

5.5 Mathematical Background

5.6 Events

5.7 Event Algebra

5.8 Basic Counting Principles

5.9 Permutations and Combinations

5.10 Principle of Inclusion and Exclusion (PIE)

5.11 Recurrence Relations

5.12 Urn Models

5.13 Partitions

5.14 Axiomatic Approach

5.15 The Classical Approach

5.16 Frequency Approach

5.17 Bayes Theorem

5.18 Summary

Exercises

Chapter 6: Discrete Distributions

6.1 Discrete Random Variables

6.2 Binomial Theorem

6.3 Mean Deviation of Discrete Distributions

6.4 Bernoulli Distribution

6.5 Binomial Distribution

6.6 Discrete Uniform Distribution

6.7 Geometric Distribution

6.8 Negative Binomial Distribution

6.9 Poisson Distribution

6.10 Hypergeometric Distribution

6.11 Negative Hypergeometric Distribution

6.12 Beta Binomial Distribution

6.13 Logarithmic Series Distribution

6.14 Multinomial Distribution

6.15 Summary

Exercises

Chapter 7: Continuous Distributions

7.1 Introduction

7.2 Mean Deviation of Continuous Distributions

7.3 Continuous Uniform Distribution

7.4 Exponential Distribution

7.5 Beta Distribution

7.6 The Incomplete Beta Function

7.7 General Beta Distribution

7.8 Arc-Sine Distribution

7.9 Gamma Distribution

7.10 Cosine Distribution

7.11 The Normal Distribution

7.12 Cauchy Distribution

7.13 Inverse Gaussian Distribution

7.14 Lognormal Distribution

7.15 Pareto Distribution

7.16 Double Exponential Distribution

7.17 Central Distribution

7.18 Student'S

T

Distribution

7.19 Snedecor's

F

Distribution

7.20 Fisher's

Z

Distribution

7.21 Weibull Distribution

7.22 Rayleigh Distribution

7.23 Chi-Distribution

7.24 Maxwell Distribution

7.25 Summary

Exercises

Chapter 8: Mathematical Expectation

8.1 Meaning of Expectation

8.2 Random Variable

8.3 Expectation of Functions of Random Variables

8.4 Conditional Expectations

8.5 Inverse Moments

8.6 Incomplete Moments

8.7 Distances as Expected Values

8.8 Summary

Exercises

Chapter 9: Generating Functions

9.1 Types of Generating Functions

9.2 Probability Generating Functions (PGF)

9.3 Generating Functions for CDF (GFCDF)

9.4 Generating Functions for Mean Deviation (GFMD)

9.5 Moment Generating Functions (MGF)

9.6 Characteristic Functions (ChF)

9.7 Cumulant Generating Functions (CGF)

9.8 Factorial Moment Generating Functions (FMGF)

9.9 Conditional Moment Generating Functions (CMGF)

9.10 Convergence of Generating Functions

9.11 Summary

Exercises

Chapter 10: Functions of Random Variables

10.1 Functions of Random Variables

10.2 Distribution of Translations

10.3 Distribution of Constant Multiples

10.4 Method of Distribution Functions (MoDF)

10.5 Change of Variable Technique

10.6 Distribution of Squares

10.7 Distribution of Square-Roots

10.8 Distribution of Reciprocals

10.9 Distribution of Minimum and Maximum

10.10 Distribution of Trigonometric Functions

10.11 Distribution of Transcendental Functions

10.12 Transformations of Normal Variates

10.13 Summary

Exercises

Chapter 11: Joint Distributions

11.1 Joint and Conditional Distributions

11.2 Jacobian of Transformations

11.3 Polar Transformations

11.4 Summary

Exercises

References

Index

End User License Agreement

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Guide

Cover

Table of Contents

preface

Begin Reading

List of Illustrations

Chapter 1: Descriptive Statistics

Figure 1.1 The NOIR scale of measurement.

Chapter 2: Measures of Location

Figure 2.1 Location measures.

Figure 2.2 Quantiles.

Chapter 3: Measures of Spread

Figure 3.1 Dispersion low and high.

Figure 3.2 Dispersion measures. (AAD = Average Absolute Deviation from the mean, GMD = Gini's Mean Difference, CV = Coefficient of Variation, IQR = Inter-Quartile Range, Abs = Absolute, Rel = Relative.)

Chapter 5: Probability

Figure 5.1

Figure 5.2 Venn diagram for .

Figure 5.3 Venn diagram for .

Figure 5.4 Coin-tossing sample space.

Chapter 6: Discrete Distributions

Figure 6.1 BINO(10,0.1) and BINO(10,0.9).

Figure 6.2 Geometric distribution .

Figure 6.3 Normal approximation.

Figure 6.4 Geometric distribution .

Figure 6.5 Three poisson distributions.

Figure 6.6 Two negative binomials.

Figure 6.7 Limiting behavior of binomial distributions.

Chapter 7: Continuous Distributions

Figure 7.1 CDF of CUNI(

a

,

b

).

Figure 7.2 Three exponential distributions.

Figure 7.3 Three beta distributions.

Figure 7.4 BETA(4,4) and BETA(6,5).

Figure 7.5 Standard arc-sine distribution.

Figure 7.6 Normal versus Cauchy distributions.

Figure 7.7 versus Cauchy CDF.

Figure 7.8 Three gamma distributions.

Figure 7.9 Inverse Gaussian distributions.

Figure 7.10 Lognormal distributions.

Figure 7.11 Laplace distributions.

Figure 7.12 Chi-square distribution.

Figure 7.13 Fisher's

Z

distribution.

Figure 7.14 Incomplete beta function.

Figure 7.15 Rayleigh distributions.

Figure 7.16 Maxwell distributions.

Chapter 8: Mathematical Expectation

Figure 8.1 Continuous density function takes any value in a range

Figure 8.2 Discrete distribution function is a step function.

Figure 8.3 Parallel circuit.

Figure 8.4 Continuous density function.

Chapter 11: Joint Distributions

Figure 11.1 Region of integration (i).

Figure 11.2 Region of integration (ii).

Figure 11.3 Region of integration for

X

/

Y

.

Figure 11.4 Region of integration

List of Tables

Chapter 1: Descriptive Statistics

Table 1.1 Computation of Entropies

Table 1.2 Normality Testing Using Graphical and Analytical Methods

Chapter 2: Measures of Location

Table 2.1 Weighted Mean Example: Calories Burned on Threadmill

Table 2.2 Hexavalent Chromium Levels

Table 2.3 Recursive Computation of Geometric Mean

Table 2.4 Recursive Computation of Harmonic Mean

Table 2.5 Median of Grouped Data

Chapter 3: Measures of Spread

Table 3.1 Categorization of Dispersion Measures

Table 3.2 Recursive Calculation of Variance using Theorem 3.1

Chapter 4: Skewness and Kurtosis

Table 4.1 Asymmetry of Some Statistical Distributions

Table 4.2 BMI of 30 Patients—Unsorted

Table 4.3 BMI Frequency Distribution

Table 4.4 Range of Skewness Measures

Chapter 5: Probability

Table 5.1 Some Common Symbols in Probability

Table 5.2 Some Equally Likely Experiments and Their Probabilities

Table 5.3 Roots of Quadratic Equation

Table 5.4 Some Permutation Formulas

Table 5.5 Divisibility of Integers by 3, 5, or 7

Table 5.6 Urns and Balls Without Restrictions

Table 5.7 Soup and Meal Combination

Table 5.8 Frequency Distribution of BMI Values

Table 5.9 Child's Blood-type from Those of Parents'

Table 5.10 Blood Type Frequency Data

Table 5.11 Customers to a Store

Table 5.12 Cancer Incidence among Smokers and Nonsmokers

Table 5.13 Impurity in Minerals

Chapter 6: Discrete Distributions

Table 6.1 Mean Deviation of Binomial Distribution Using Our Power Method (equation 6.8) for

np

a Half-integer

Table 6.2 Properties of Bernoulli Distribution

Table 6.3 Binomial Probabilities Example

Table 6.4 Properties of Binomial Distribution

Table 6.5 Properties of Discrete Uniform Distribution

Table 6.6 Distribution of for

n

Even

Table 6.7 Mean Deviation of Binomial Distribution Using Equation (6.6)

Table 6.8 Variance of Discrete Distributions

Table 6.9 Properties of Geometric Distribution

Table 6.10 Properties of Negative Binomial Distribution

Table 6.11 Properties of Poisson Distribution

Table 6.12 Mean Deviation of Poisson Distribution Using Our Power Method (equation (6.8)) for Integer

Table 6.13 Properties of Hypergeometric Distribution

Table 6.14 Properties of Beta Binomial Distribution

Table 6.15 Properties of Logarithmic Distribution

Table 6.16 Properties of Multinomial Distribution

Table 6.17 Properties of Photo-electric Surface

Chapter 7: Continuous Distributions

Table 7.1 Summary Table of Expressions for MD

Table 7.2 Properties of Continuous Uniform Distribution

Table 7.3 Properties of Exponential Distribution ()

Table 7.4 Summary Table of Additivity Property

Table 7.5 Properties of Beta-I Distribution

Table 7.6 Mean Deviation of Beta-I using Equations (7.31) and (7.32)

Table 7.7 Properties of Beta-II Distribution

Table 7.8 Properties of Arc-Sine Distribution

Table 7.9 Properties of Gamma Distribution ()

Table 7.10 Properties of Cosine Distribution

Table 7.11 Properties of Normal Distribution()

Table 7.13 Properties of Cauchy Distribution ()

Table 7.14 Properties of IGD

Table 7.12 Area Under Normal Variates

Table 7.15 Properties of Lognormal Distribution

Table 7.16 Properties of Pareto Distribution ()

Table 7.17 Properties of Double Exponential Distribution

Table 7.18 Properties of Distribution

Table 7.19 Properties of

T

Distribution

Table 7.20 Properties of

F

Distribution

Table 7.21 Properties of Fisher's

Z

Table 7.22 Properties of Weibull Distribution

Table 7.23 Properties of Rayleigh Distribution

Table 7.24 Properties of Maxwell Distribution

Chapter 8: Mathematical Expectation

Table 8.1 Comparison of Discrete and Continuous Random Variables

Table 8.2 Number of Chicken Hatched in 10 Days

Table 8.3 Mean of Noncentral Beta Using Equation (8.22)

Table 8.4 Summary Table of Expressions for Variance

Table 8.5 Summary of Mathematical Expectation

Chapter 9: Generating Functions

Table 9.1 Summary Table of Generating Functions

Table 9.2 Table of Characteristic Functions

Chapter 10: Functions of Random Variables

Table 10.1 Summary Table of Transformation of Variates

Table 10.2 Joint Distribution

Table 10.3 Distribution of

Chapter 11: Joint Distributions

Table 11.1 Common Transformation of Two Variables

Table 11.2 Common Polar Transformation of Three Variables

STATISTICS FOR SCIENTISTS AND ENGINEERS

RAMALINGAM SHANMUGAM

RAJAN CHATTAMVELLI

Copyright © 2015 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

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, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, [phone], fax [phone], or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, [phone], fax [phone], or online at http://www.wiley.com/go/permission.

Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

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Library of Congress Cataloging-in-Publication Data:

Shanmugam, Ramalingam.

Statistics for scientists and engineers / Ramalingam Shanmugam, Rajan Chattamvelli.

pages cm

Includes bibliographical references and index.

ISBN 978-1-118-22896-8 (cloth)

1. Mathematical statistics. I. Chattamvelli, Rajan. II. Title.

QA276.S456 2015

519.5–dc23

2015004898

Preface

This is an intermediate-level textbook on probability and mathematical statistics. This book can be used for one-semester graduate courses in engineering statistics. Prerequisites include a course in college algebra, differential and integral calculus, and some linear algebra concepts. Exposure to computer programming is not a must but will be useful in Chapters 1 and 5. It can also be used as a supplementary reading material for mathematical statistics, probability, and statistical methods. Researchers in various disciplines will also find the book very helpful as a ready reference.

Audience

This book is mainly aimed at three reader groups: (i) Advanced undergraduate and beginning graduate students in engineering and physical sciences and management. (ii) Scientists and engineers who require skill sets to model practical problems in a statistical setting. (iii) Researchers in mathematical statistics, engineering, and related fields who need to make their knowledge up-to-date. Some of the discussions in Chapters 1, 9, and 11 are also of use to computer professionals. It is especially suited to readers who look for nonconventional concepts, problems, and applications. Some of the chapters (especially Chapters 2–8) can also be used by students in other disciplines. With this audience in mind, we have prepared a textbook with the hope that it will serve as a gateway to the wonderful subject of statistics. Readers will appreciate the content, organization, and coverage of the topics.

Purpose

This book discusses the theoretical framework needed to build, analyze, and interpret various models. Our aim is to present the basic concepts in an intuitively appealing and easily understood way. This book will help the readers to choose the correct model or distinguish among various choices that best captures the data or solve the problem at hand. As this book lays a strong foundation of the subject, interested readers will be able to pursue advanced modeling and analysis withease.

Approach

Theoretical concepts are developed or explained in a step-by-step and easy-to-understand manner. This is followed by practical examples. Some of the difficult concepts are exemplified using multiple examples drawn from different fields. Exercises are chosen to test the understanding of concepts. Extensive bibliography appears at the end of the book.

Main Features

Most important feature of the book is the large number of worked out examples from a variety of fields. These are self-explanatory and easily grasped by students. These are drawn from medical sciences and various disciplines of engineering, and so on. In addition, extensive exercises are provided at the end of each chapter. Some novel methods to find the mean deviation (MD) of discrete and continuous distributions are introduced in Chapters 6, 7, and 9.

Mathematics Level

This book is ideal for those who have done at least one course in college algebra and calculus. Sum and product notations given in Chapter 1 are used in Chapters 6 and 8. Set theory concepts are sparingly used in Chapter 5. Basic trigonometric concepts are used in Chapters 7 and 11. Differential calculus concepts are needed to solve some of the problems in Chapters 10 and 11, especially in finding the Jacobian of transformations. Integral calculus is used extensively in Chapters 7, 10, and 11. Some concepts on matrices and linear algebra are needed in Chapters 10 and 11.

Coverage

The book starts with an introductory chapter that fills the gap for those readers who do not have all the prerequisites. Chapter 1 introduces the basic concepts, including several notations used throughout the book. Important among them are the notations for combinations, summation, and products. It briefly discusses the scales of measurement and gives examples of various types of data. The summation notation is extensively discussed, and its variants such as nested sums,fractional steps, symmetric sums, summation over sets, and loop unrolling are thoroughly discussed. The product notation is discussed next and its applications to evaluating powers of the form where x and n are both large are given. Rising and falling factorial notation is briefly discussed. These are extremely helpful for scientists and working engineers in various fields. Data discretization and data transformations are also introduced. The chapter ends with a discussion of testing for normality of data. This chapter can be skipped for senior-level courses. Working engineers and professionals may need to skim through this chapter, as it contains a few useful concepts of immense practical value.

Chapter 2 discusses the measures of location. These are essential tools for anyone working with numeric data. All important measures such as arithmetic mean, geometric and harmonic means, median, and mode are discussed. Some updating formulas for the means are also given. Important among them are the updating formula for weighted mean, geometric mean and harmonic means, and trimmed means, as well as updating formulas for origin and scale changed data and windowed data. The sample median, mode, quartiles, and percentiles are also explained.

Popular measures of spread appear in Chapter 3. A categorization of spread measures helps the reader to distinguish between various measures. These include linear and nonlinear measures, pivotal and pivot-less measures, additive and nonadditive measures, absolute and relative measures, distance-based measures, and so on. The sample range and its advantages and applications are discussed. An illuminating discussion of the “degrees-of-freedom” concept appears in page 3–13. A summary table gives a comparison of various spread measures (pp. 3–8). The average absolute deviation (AAD) (also called sample mean absolute deviation) and its properties are discussed next. Sample variance and standard deviation are the most frequently used measures of spread. These are discussed, and some updating formulae for sample variance are derived. This is followed by the formula for pooling sample variance and covariance, which forms the basis for a divide-and-conquer algorithm. Some bounds on the sample standard deviation in terms of sample range are given. The chapter ends with a discussion of the coefficient of variation and Gini coefficient.

Chapter 4 discusses measures of skewness and kurtosis. Absolute versus relative measures of skewness are discussed, followed by various categories of skewness measures such as location and scale-based measures, quartile-based measures, moment-based measures, measures that utilize inverse of distribution functions, and measures that utilize L-moments. Pearson's and Bowley's measures are given and their ranges are discussed. Coefficient of quartile deviation and its properties are discussed. The range of values of various measures is summarized into a table. This is followed by a discussion of the measures of kurtosis. The kurtosis of other statistical distributions is compared with that of a standard normal with kurtosis coefficient 3, which is derived. A brief discussion of skewness–kurtosis bounds and L-kurtosis appear next. This chapter ends with a discussion of spectral kurtosis and multivariate kurtosis (which may be skipped in undergraduatecourses).

Fundamentals of probability theory are built from the ground up in Chapter 5. As solving some probability problems is a challenge to those without adequate mathematical skills, a majority of this chapter develops the tools and techniques needed to solve a variety of problems. The chapter starts with a discussion of various ways to express probability. Converting repeating and nonrepeating decimal numbers into fractional form p/q is given in algorithmic form. Sample spaces are defined and illustrated using various problems. These are then used to derive the probability of various events. This is followed by building the mathematical background using set theory and Venn diagrams. A discussion of event categories appears next–simple and compound events, mutually exclusive events, dependent and independent events, and so on. Discrete and continuous events are exemplified as well as various laws of events—commutative, associative, distributive laws; the law of total probability; and De'Morgan's laws. Basic counting principle is introduced and illustrated using numerous examples from various fields. This is followed by a lengthy discussion of the tools and techniques such as permutation and combination, cyclic permutation, complete enumeration, trees, principle of inclusion and exclusion, recurrence relations, derangements, urn models, and partitions. Probability measure and space are defined and illustrated. The do-little principle of probability and its applications are discussed. The axiomatic, frequency, and other approaches to probability are given. The chapter ends with a discussion of Bayes theorem for conditional probability and illustrates its use in various problems.

Chapter 6 on discrete distributions builds the concepts by starting with the binomial theorem. As the probabilities of theoretical distributions sum to one, some of them can be easily obtained by putting particular values in the binomial expansion. A novel method to easily find the MD of discrete distributions is introduced in this chapter (Section 6.3, pp. 6–6). Important properties of distributions are succinctly summarized. These include tail probabilities, moments and location measures, dispersion measures, generating functions, and recurrence relations. It is shown that the rate of convergence of binomial distribution to the Poisson law is quadratic in p and linear in n (pp. 6–37). This provides new insight into the classical textbook rule that “binomial tends to the Poisson law when and such that np remains a constant.” Analogous results are obtained for the limiting behavior of negative binomial distributions. Distribution of the difference of successes and failures in Bernoulli trials are obtained in simple form. Other distributions discussed include geometric, Poisson, hypergeometric, negative hypergeometric, logarithmic, beta binomial, and multinomial distributions. Researchers in various fields will find Chapters 6 and 7 to be of immense value.

Chapter 7 introduces important continuous distributions that are often encountered in practical applications. A general method to find the MD of continuous distributions is derived in page 7–4, which is very impressive as it immensely reduces the arithmetic work. This helpful result is extensively used throughout the chapter. A relation between variance of continuous distributions and tail areas is derived.Alternate parametrizations of some distributions are given. List of distributions include uniform (rectangular), exponential, beta-I, beta-II, gamma, arc-sine, cosine, normal, Cauchy, central and chi-, Student's t, Snedecor's F, inverse Gaussian, log-normal, Pareto, Laplace, Weibull, Rayleigh, Maxwell, and Fisher's Z distributions. Important results are summarized and several algorithms for tail areas are discussed. These results are used in subsequent chapters.

Mathematical expectation is discussed in Chapter 8. Expectation and variance using distribution functions are discussed next. Expectation of functions of random variables appears in page 8–20. Properties of expected values, expectation of functions of random variables, variance, covariance, moments, and so on appear next. This is followed by a discussion of conditional expectation, which is used to derive the mean of mixture distributions. Several important results such as expressions for variance (pp. 8–47) and expectation of functions of random variables (pp. 8–50) are summarized. These are needed in subsequent chapters.

Chapter 9 on generating functions gives a brief introduction to various generating functions used in statistics. This includes probability generating function, moment generating function, cumulant generating function, and characteristic functions. These are derived for several distributions and their inter-relationships are illustrated with examples. Two novel generating functions are introduced in this chapter–first one to generate the cumulative distribution function (CDF-GF) in Section 9.3 (pp. 9–10) and second one to generate MD (MD-GF) (Section 9.4, pp. 9–11). Factorial moment generating functions and its relationship to Stirling numbers are briefly mentioned. This chapter is strongly coupled with Chapters 6–8. Readers in prior chapters may want to refer to the results in this chapter as and when needed.

Functions of random variables are discussed in Chapter 10. These are used in deriving distributions of related statistics. This chapter discusses various techniques such as method of distribution functions, Jacobian method, probabilistic methods, and area-based methods, and it also discusses distribution of absolute values of symmetric random variables, distribution of and , and so on. These results are applied to find the MD of continuous distributions using a simple integral of from lower limit to . Other topics discussed include distribution of squares, square roots, reciprocals, sums, products, quotients, integer part, and fractional part of continuous random variables. Distributions of trigonometric and transcendental functions are also discussed. The chapter ends with a discussion of various transformations of normal variables.

Joint distributions are briefly discussed in Chapter 11 and some applications are given. Marginal and conditional distributions are discussed and illustrated with various examples. The concept of the Jacobian is introduced in Section 11.2 in page 11–7. Derivation of joint distributions in a bivariate setup is given in Section 11.2.1, pp. 11–9. An immensely useful summary table of 2D transformations appears in page 11–15 for ready reference. Various polar transformations such as plane polar, spherical polar, and toroidal polar and its inverses arediscussed, and a summary table is given in page 11–28. A good understanding of integration is absolutely essential to grasp some of the examples in this chapter.

Working professionals will find the book to be very handy and immensely useful. Some of the materials in this book were developed during Dr. Ramalingam Shanmugam's teaching of engineering statistics in the University of Colorado and second author's teaching at Frederick University, Cyprus. Any suggestions or comments for improvement are welcome. Please mail them to the first author at [email protected].

The first author would like to thank Wiley editorial staff, especially Ms. Kari Capone, Ms. Amy Henderson, and others in Wiley for tremendous help during the entire production work and their patience. The second author would like to thank his brothers C.V. Santosh Kumar and C.V. Vijayan for all the help and encouragements. The first author dedicates this book to his wife srimathi Malarvizhi Shanmugam, and the second author dedicates it to his late grandfather Mr. Keyath Kunjikannan Nair.

Ram ShanmugamRajan ChattamvelliSan Marcos, TXThanjavur, IndiaNovember, 2014

About the Companion Website

This book is accompanied by a companion website:

www.wiley.com/go/shanmugam/statistics

The website includes:

Solutions Manual available to Instructors.

Chapter 1Descriptive Statistics

After finishing the chapter, students will be able to

Explain the meaning and uses of statistics

Describe the standard scales of measurement

Interpret various summation (

) and product (

) notations

Apply different types of data transformations

Distinguish various data discretization algorithms (DDAs)