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Vital Statistics

Probability and Statistics for Economics and Business

First Edition

William Sandholm and Brett Saraniti

Publication Date - October 2018

ISBN: 9780190668082

840 pages
Paperback
7-1/2 x 9-1/4 inches

Retail Price to Students: $150.95

A probability and statistics text written with the needs of economics students in mind

Description

Unlike most business statistics textbooks, Vital Statistics offers an approachable, calculus-optional introduction to statistics with a careful presentation of basic inference procedures. Sandholm and Saraniti devote substantial effort to helping students develop intuitions about key concepts in probability before providing a deep treatment of core ideas in statistics. They focus on topics and applications that matter to economists but receive little attention in other textbooks, making this the ideal introductory text for economics students.

Features

  • A conversational writing style makes this text accessible to a range of students
  • Chapter 9, "The Psychology of Probability," presents a range of topics from behavioral economics related to probability and inference
  • Chapter 10, "How to Lie with Statistics," provides a useful introduction to common errors in statistics, helping students become critical consumers of statistical arguments
  • Separate chapters on descriptive regression and regression for statistical inference offer students a deeper understanding of each topic
  • Excel workbooks with simulations illustrate fundamental ideas and make probability models easier to understand
  • Templates for Excel calculations perform computations and produce graphics that illustrate the results

About the Author(s)

William Sandholm is the Richard E. Stockwell Professor of Economics at University of Wisconsin - Madison, where he has been teaching since 2002. His research is in the area of game theory, specifically dynamic models of evolution and learning in games. He has published a number of articles and is Associate Editor of Journal of Dynamics and Games, Dynamic Games and Applications, Theoretical Economics, and Journal of Economic Theory.

Brett Saraniti is Clinical Professor of Managerial Economics and Decision Sciences, Kellogg School of Management, Northwestern University, where he has been honored with multiple teaching awards. He also teaches at INSEAD and Hawaii Pacific University. At INSEAD, Professor Saraniti won the Best Teacher Award for the Core Classes in 2008. He has been a frequent visitor at the Sasin Graduate Institute in Bangkok, Thailand; IESE in Barcelona; the Brisbane Graduate School of Business in Queensland, Australia; the Thunderbird School of Global Management; and the Helsinki School of Economics and Business Administration in Finland.

Reviews

"The authors make a strong case for the need for this book as a bridge between two major approaches to teaching statistics; they are not afraid to try and explain complex material to students. The text is a bold attempt to fix what is wrong with Business Statistics textbooks."--Brian Goegan, Arizona State University

"I have no doubt that this is the best probability and statistics book for undergraduate economics, finance, and math majors. This is the book every instructor teaching probability and statistics for economics, finance, and applied math majors have been waiting for."--Jong Kim, Emory University

"This text is a good way to introduce economics majors to fundamental concepts in statistics and probability, which they will use in their future coursework. It actually applies these topics to economics, making them more relevant to students in the major, and it lays better groundwork for more advanced statistics compared to other textbooks I have used."--Lauren Tombari, San Francisco State University

Table of Contents

    *Asterisks indicate sections tangential to the main line of argument

    Preface for Students
    Preface for Instructors
    Acknowledgments


    1. Random Reasoning
    1.1 Introduction
    1.2 Probability
    1.3 Statistics
    1.4 Conclusion

    2. Probability Models
    2.1 Ex Ante vs. Ex Post
    2.2 Sample Spaces
    2.2.1 Sample spaces, outcomes, and events
    2.2.2 New events from old
    2.3 Probability Measures
    2.3.1 The axioms of probability
    2.3.2 Further properties of probability measures
    2.3.3 Interpreting and assigning probabilities
    2.4 Conditional Probability
    2.4.1 What is conditional probability?
    2.4.2 Joint, marginal, and conditional probabilities
    2.4.3 The total probability rule
    2.4.4 Bayes'
    rule
    2.5 Independence
    2.5.1 Independence of pairs of events
    2.5.2 Independence of many events
    2.5.3 Independence of many events: A formal treatment*
    2.6 Constructing Probability Models*
    2.6.1 Two probability problems
    2.6.2 Discussion of the Linda problem
    2.6.3 Discussion of the Monty Hall problem
    2.A Appendix: Finite and Countable Additivity
    2.E Exercises

    3. Random Variables
    3.1 Random Variables
    3.1.1 What exactly is a random variable?
    3.1.2 Ex ante vs. ex post revisited
    3.1.3 The distribution of a random variable
    3.2 Traits of Random Variables
    3.2.1 Expected value
    3.2.2 Variance and standard deviation
    3.2.3 An alternate formula for expected values*
    3.3 Functions of Random Variables
    3.4 Independent Random Variables
    3.4.1 Independence of two random variables
    3.4.2 Independence of many random variables
    3.4.3 Sums of independent random variables
    3.4.4 New independent random variables from old
    3.E Exercises

    4. Multiple Random Variables
    4.1 Multiple Random Variables
    4.1.1 Joint distributions and marginal distributions
    4.1.2 Conditional distributions
    4.1.3 Conditional traits and the law of iterated expectations

    4.2 Traits of Random Variable Pairs
    4.2.1 Covariance
    4.2.2 Correlation
    4.2.3 Some useful facts
    4.2.4 Independence and zero correlation

    4.3 Functions of Multiple Random Variables
    4.4 Portfolio Selection*
    4.4.1 A simple model of a financial market
    4.4.2 Portfolio
    selection and diversification
    4.4.3 Efficient portfolios
    4.4.4 The benefits of diversification
    4.A Appendix
    4.A.1 Definitions, formulas, and facts about random variables
    4.A.2 Derivations of formulas and facts
    4.B The Capital Asset Pricing Model [ONLINE]
    4.E Exercises

    5. Bernoulli Trials Processes and Discrete Distributions
    5.1 Families of Distributions
    5.1.1 Indicator random variables
    5.1.2 Bernoulli distributions
    5.1.3 Traits of Bernoulli random variables
    5.2 Bernoulli Trials Processes
    5.3 How to Count
    5.3.1 Choice sequences
    5.3.2 Orderings
    5.3.3 Permutations
    5.3.4 Combinations
    5.4 Binomial Distributions
    5.4.1 Definition
    5.4.2 Another way to represent binomial distributions

    5.4.3 Traits of binomial random variables
    5.5 Simulation and Mathematical Analysis of Probability Models*
    5.5.1 The birthday problem
    5.5.2 Simulations
    5.5.3 Mathematical analysis
    5.5.4 Simulation versus mathematical analysis
    5.E Exercises

    6. Continuous Random Variables and Distributions
    6.1 Continuous Probability Models
    6.1.1 Why bother with continuous probability models?
    6.1.2 "Probability zero" and "impossible
    6.2 Continuous Random Variables and Distributions
    6.2.1 Cumulative probabilities
    6.2.2 Density functions
    6.2.3 Density functions: Intuition
    6.2.4 Percentiles of continuous distributions
    6.2.5 Traits of continuous random variables
    6.3 Uniform Distributions
    6.3.1 Definitions
    6.3.2
    Traits
    6.3.3 Shifting and scaling
    6.4 Normal Distributions
    6.4.1 Shifting, scaling, and the standard normal distribution
    6.4.2 Standard normal probabilities
    6.4.3 Normal probabilities

    6.5 Calculating Normal Probabilities Using the Table
    6.5.1 The standard normal distribution table
    6.5.2 Calculating standard normal probabilities
    6.5.3 Calculating normal probabilities
    6.6 Sums of Independent Normal Random Variables
    6.6.1 Distributions of sums of independent random variables
    6.6.2 Brownian motion*
    6.A Continuous Distributions (using calculus) [ONLINE]
    6.B Continuous Joint Distributions (using calculus) [ONLINE]
    6.E Exercises

    7. The Central Limit Theorem
    7.1 I.I.D. Random Variables
    7.2 Sums and Sample Means of I.I.D. Random Variables
    7.2.1 Definition
    7.2.2 Traits of sums and sample means of i.i.d. random variables
    7.3 The Law of Large Numbers
    7.3.1 Statement of the law of large numbers
    7.3.2 The law of large numbers and the "law of averages"
    7.3.3 Proving the law of large numbers*
    7.4 The Central Limit Theorem
    7.4.1 Convergence in distribution
    7.4.2 Statement of the central limit theorem
    7.4.3 Simulations with continuous trials
    7.4.4 The continuity correction
    7.4.5 Simulations with discrete trials
    7.5 The Central Limit Theorem: Applications
    7.5.1 Normal approximation of binomial distributions
    7.5.2 Gambling
    7.5.3 Queues
    7.5.4 Statistical inference
    7.A Proof of the Central Limit Theorem [ONLINE]
    7.E Exercises

    8. Poisson and Exponential Distributions
    8.1 Poisson Distributions and the Poisson limit theorem
    8.1.1 e
    8.1.2 Poisson distributions
    8.1.3 The Poisson limit theorem
    8.2 Exponential Distributions
    8.2.1 Definition
    8.2.2 Probabilities and traits
    8.2.3 Peculiar properties
    8.3 The Exponential Interarrival Model and the Poisson Process*
    8.A Appendix
    8.E Exercises

    9. The Psychology of Probability
    9.1 Thought Experiments
    9.2 Framing Effects
    9.3 Overconfidence
    9.4 Misestimating the Impact of Evidence
    9.5 The "Law of Small Numbers"
    9.6 Gambling Systems and Technical Trading Strategies
    9.E Exercises

    10. How to Lie with Statistics
    10.1 Introduction
    10.2 Variation
    10.2.1 Variation within a population
    10.2.2 Variation within subgroups: Simpson's paradox
    10.2.3 Variation in the results of random samples
    10.3 Polls and Sampling
    10.3.1 Sampling from the wrong population
    10.3.2 Designing polls: Wording of questions
    10.3.3 Designing polls: Selection of response alternatives
    10.3.4 Designing polls: Arrangement of questions
    10.3.5 Administering polls: Ensuring honest reporting
    10.3.6 When can I trust a poll?
    10.4 Endogenous Sampling Biases
    10.5 Causal Inference and Extrapolation
    10.5.1 Confounding variables
    10.5.2 Spurious correlation and data mining
    10.5.3 Linear extrapolation of nonlinear data
    10.E Exercises

    11. Data Graphics
    11.1 Data
    11.1.1 Types of variables
    11.1.2 Types of data sets
    11.1.3 Sources of economic and business data
    11.2 Graphics for Univariate Data
    11.2.1 Graphics that display every observation
    11.2.2 Graphics for absolute and relative frequencies
    11.2.3 Graphics for cumulative frequencies
    11.3 Graphics for Multivariate Data
    11.3.1 Graphics for frequencies
    11.3.2 Graphics that display every observation
    11.4 Principles for Data Graphics Design
    11.4.1 First, do no harm
    11.4.2 Infographics
    11.4.3 One step beyond
    11.A Appendix: Creating Data Graphics in Excel [ONLINE]
    11.E Exercises

    12. Descriptive Statistics
    12.1 Descriptive Statistics for Univariate Data
    12.1.1 Measures of relative standing: Percentiles and ranges
    12.1.2 Measures of centrality: Mean and median
    12.1.3 Measures of dispersion: Variance and standard deviation
    12.2 Descriptive Statistics for Bivariate Data
    12.2.1 Measures of linear association: Covariance and correlation
    12.2.2 Visualizing correlations
    12.2.3 Computing correlations: Arithmetic, pictures, or computer
    12.2.4 The road ahead: Regression analysis
    12.E Exercises

    13. Probability Models for Statistical Inference
    13.1 Introduction
    13.2 The I.I.D. Trials Model for Statistical Inference
    13.3 Inference about Inherently Random Processes
    13.3.1 Bernoulli trials
    13.3.2 Trials with an unknown distribution
    13.4 Random Sampling and Inference about Populations
    13.4.1 Random sampling
    13.4.2 The
    trials' traits equal the data set's descriptive statistics
    13.4.3 Bernoulli trials
    13.4.4 Trials with an unknown distribution
    13.5 Random Sampling in Practice
    13.E Exercises

    14. Point Estimation
    14.1 Parameters, Estimators, and Estimates
    14.2 Desirable Properties of Point Estimators
    14.3 The Sample Means
    14.3.1 Unbiasedness and consistency
    14.3.2 Efficiency
    14.3.3 The distribution of the sample mean
    14.4 The Sample Variance
    14.4.1 Defining the sample variance
    14.4.2 Unbiasedness and consistency of the sample variance
    14.5 Classical Statistics and Bayesian Statistics*
    14.A Appendix: A Short Introduction to Bayesian Statistics
    14.B Appendix: Derivations of Properties of the Sample Variance
    14.E Exercises

    15. Interval Estimation and Confidence Intervals
    15.1 What Is Interval Estimation?
    15.2 Constructing Interval Estimators
    15.2.1 The 95% interval estimator for mu when sigma2 is known
    15.2.2 The 95% interval estimator for mu when sigma2 is unknown
    15.2.3 The (1 - alpha) interval estimator for when is unknown
    15.2.4 Looking ahead: Standard errors and t distributions
    15.3 Interval Estimators for Bernoulli Trials
    15.4 Interpreting Confidence
    15.5 Choosing Sample Sizes
    15.5.1 Sample sizes for general i.i.d. trials
    15.5.2 Sample sizes for Bernoulli trials processes

    15.6 A Better Interval Estimator for Bernoulli Trials*
    15.E Exercises

    16. Hypothesis Testing
    16.1 What Is Hypothesis Testing?
    16.2 Hypothesis Testing: Basic Concepts
    16.2.1 The probability model
    16.2.2 Null and alternative hypotheses
    16.2.3 One-tailed and two-tailed tests
    16.2.4 Hypothesis tests and their significance levels
    16.3 Designing Hypothesis Tests
    16.3.1 Hypothesis tests for mu when sigma2 is known
    16.3.2 Hypothesis tests for mu when sigma2 is unknown
    16.3.3 Hypothesis tests for Bernoulli trials
    16.4 Two-Tailed Hypothesis Tests
    16.4.1 Two-tailed tests vs. one-tailed tests
    16.4.2 Comparing two-tailed hypothesis tests and confidence intervals
    16.5 Alternate Ways of Expressing Hypothesis Tests
    16.5.1 z-statistics
    16.5.2 P-values
    16.6 Interpreting Hypothesis Tests
    16.6.1 The meaning of significance
    16.6.2 "Do not reject" vs.
    "accept"
    16.6.3 Statistical significance versus practical significance
    16.6.4 P-value .049 vs. P-value .051
    16.6.5 Hypothesis testing in a
    vacuum
    16.7 Significance and Power
    16.7.1 Type I and Type II errors
    16.7.2 Evaluating error probabilities
    16.7.3 Power and the power curve
    16.7.4 Underpowered studies
    16.8 Choosing Sample Sizes
    16.8.1 Sample sizes for general i.i.d. trials
    16.8.2 Sample sizes for Bernoulli trials processes
    16.9 Summary and Preview
    16.E Exercises

    17. Inference from Small Samples
    17.1 The t-Statistic
    17.2 t Distributions
    17.3 Small-Sample Inference about the Mean of Normal Trials
    17.3.1 The t-statistic and the t distribution
    17.3.2 Interval estimation
    17.3.3 Hypothesis
    testing
    17.4 Sort-of-Normal Trials: The Robustness of the t-Statistic
    17.5 Evaluating Normality of Trials*
    17.A Appendix: Descendants of the Standard Normal Distribution [ONLINE]
    17.E Exercises

    18. Inference about Differences in Means
    18.1 Inference from Two Separate Samples
    18.1.1 The basic two-sample model
    18.1.2 Bernoulli trials
    18.1.3 Small samples, normal trials, equal variances*
    18.2 Inference from Paired Samples
    18.2.1 Constructing paired samples
    18.2.2 The basic paired-sample model
    18.2.3 Small samples, normal trials*
    18.3 Choosing between Separate and Paired Samples
    18.3.1 A general rule
    18.3.2 Paired sampling using two observations per individual
    18.3.3 Pairing samples using observable
    characteristics*
    18.4 Causal Inference: Treatment Effects*
    18.4.1 Randomized controlled experiments and observational studies
    18.4.2 Interventions and causal assumptions
    18.4.3 Potential outcomes and average treatment effects
    18.4.4 A probability model of an observational study
    18.4.5 Selection bias in observational studies
    18.4.6 Random assignment eliminates selection bias
    18.4.7 Controlling for observable confounding variables
    18.A Appendix
    18.B Appendix: The Distribution of the Pooled Sample Variance [ONLINE]
    18.E Exercises

    19. Simple Regression: Descriptive Statistics
    19.1 The Regression Line
    19.1.1 A brief review of descriptive statistics
    19.1.2 The regression line
    19.1.3 Examples, computations, and
    simulations
    19.2 Prediction and Residuals
    19.2.1 Predictors, predictions, and residuals
    19.2.2 Best-in-class predictors
    19.2.3 Further characterizations of the regression line
    19.2.4 Deriving the best constant and best linear predictors*
    19.3 The Conditional Mean Function
    19.3.1 Best unrestricted prediction
    19.3.2 Best linear prediction of conditional means
    19.4 Analysis of Residuals
    19.4.1 Sums of squares and variances of residuals for best-in-class predictors
    19.4.2 Relative quality for best-in-class predictors
    19.4.3 Decomposition of variance for regression
    19.4.4 Sums of squares revisited
    19.5 Pitfalls in Interpreting Regressions
    19.5.1 Nonlinear relationships
    19.5.2 Regression to the mean
    19.5.3
    Correlation and causation
    19.6 Three Lines of Best Fit*
    19.6.1 The reverse regression line
    19.6.2 The neutral line
    19.6.3 The three lines compared

    19.A Appendix
    19.A.1 Equivalence of the characterizations of the regression line
    19.A.2 Best linear prediction of conditional means
    19.A.3 Relative quality for best-in-class predictors: Derivation
    19.A.4 Decomposition of variance for regression: Derivation
    19.B Appendix: Characterization of the Neutral Line [ONLINE]
    19.E Exercises

    20. Simple Regression: Statistical Inference [ONLINE]
    20.1 The Classical and Random Sampling Regression Models
    20.1.1 Fixed x sampling versus random sampling
    20.1.2 Linearity of conditional means
    20.1.3 Constant conditional variances

    20.1.4 How reasonable are the assumptions?

    20.2 The OLS Estimators
    20.2.1 Defining the OLS estimators
    20.2.2 Basic properties of the OLS estimators
    20.2.3 Estimating conditional means
    20.2.4 Approximate normality of the OLS estimators
    20.2.5 Efficiency of the OLS estimators: The Gauss-Markov Theorem
    *
    20.3 The Sample Conditional Variance
    20.4 Interval Estimators and Hypothesis Tests
    20.4.1 Review: Inference about an unknown mean
    20.4.2 Interval estimators and hypothesis tests for beta
    20.4.3 Interval estimators and hypothesis tests for conditional means
    20.4.4 Population regressions versus sample regressions
    20.5 Small Samples and the Classical Normal Regression Model
    20.5.1 The classical normal regression model
    20.5.2 Interval estimators and hypothesis tests for beta
    20.5.3 Interval estimators and hypothesis tests for conditional means
    20.5.4 Prediction intervals*
    20.6 Analysis of Residuals, R2, and F Tests
    20.6.1 Sums of squares and R2
    20.6.2 The F test for beta = 0
    20.6.3 What happens without normality? The robustness of the F-statistic*
    20.7 Regression and Causation
    20.7.1 An alternate description of the classical regression model
    20.7.2 Causal regression models
    20.7.3 Multiple regression
    20.A Appendix
    20.A.1 Analysis of the random sampling regression model
    20.A.2 The unstructured regression model
    20.A.3 Computation of the mean and variance of B
    20.A.4 Proof of the Gauss-Markov Theorem
    20.A.5 Proof that the
    sample conditional variance is unbiased
    20.A.6 Deriving the distribution of the F-statistic
    20.E Exercises

    Index

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