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A Probability and Statistics Companion

ISBN: 978-0-470-47195-1
260 pages
June 2009
A Probability and Statistics Companion (0470471956) cover image
An accessible and engaging introduction to the study of probability and statistics

Utilizing entertaining real-world examples, A Probability and Statistics Companion provides aunique, interesting, and accessible introduction to probability and statistics. This one-of-a-kind book delves into practical topics that are crucial in the analysis of sample surveys and experimentation.

This handy book contains introductory explanations of the major topics in probability and statistics, including hypothesis testing and regression, while also delving into more advanced topics such as the analysis of sample surveys, analysis of experimental data, and statistical process control. The book recognizes that there are many sampling techniques that can actually improve on simple random sampling, and in addition, an introduction to the design of experiments is provided to reflect recent advances in conducting scientific experiments. This blend of coverage results in the development of a deeper understanding and solid foundation for the study of probability and statistics. Additional topical coverage includes:

  • Probability and sample spaces

  • Choosing the best candidate

  • Acceptance sampling

  • Conditional probability

  • Random variables and discrete probability distributions

  • Waiting time problems

  • Continuous probability distributions

  • Statistical inference

  • Nonparametric methods

  • Least squares and medians

  • Recursions and probability

Each chapter contains exercises and explorations for readers who wish to conduct independent projects or investigations. The discussion of most methods is complemented with applications to engaging, real-world scenarios such as winning speeds at the Indianapolis 500 and predicting winners of the World Series. In addition, the book enhances the visual nature of the subject with numerous multidimensional graphical representations of the presented examples.

A Probability and Statistics Companion is an excellent book for introductory probability and statistics courses at the undergraduate level. It is also a valuable reference for professionals who use statistical concepts to make informed decisions in their day-to-day work.

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Preface.

1. Probability and Sample Spaces.

Why Study Probability?

Probability.

Sample Spaces.

Some Properties of Probabilities.

Finding Probabilities of Events.

Conclusions.

Explorations.

2. Permutations and Combinations: Choosing the Best Candidate; Acceptance Sampling.

Permutations.

Counting Principle.

Permutations with Some Objects Alike.

Permuting Only Some of the Objects.

Combinations.

General Addition Theorem and Applications.

Conclusions.

Explorations.

3. Conditional Probability.

Introduction.

Some Notation.

Bayes’ Theorem.

Conclusions.

Explorations.

4. Geometric Probability.

Conclusion.

Explorations.

5. Random Variables and Discrete Probability Distributions—Uniform, Binomial, Hypergeometric, and Geometric Distributions.

Introduction.

Discrete Uniform Distribution.

Mean and Variance of a Discrete Random Variable.

Intervals, σ, and German Tanks.

Sums.

Binomial Probability Distribution.

Mean and Variance of the Binomial Distribution.

Sums.

Hypergeometric Distribution.

Other Properties of the Hypergeometric Distribution.

Geometric Probability Distribution.

Conclusions.

Explorations.

6. Seven-Game Series in Sports.

Introduction.

Seven-Game Series.

Winning the First Game.

How Long Should the Series Last?

Conclusions.

Explorations.

7. Waiting Time Problems.

Waiting for the First Success.

The Mythical Island.

Waiting for the Second Success.

Waiting for the rth Success.

Mean of the Negative Binomial.

Collecting Cereal Box Prizes.

Heads Before Tails.

Waiting for Patterns.

Expected Waiting Time for HH.

Expected Waiting Time for TH.

An Unfair Game with a Fair Coin.

Three Tosses.

Who Pays for Lunch?

Expected Number of Lunches.

Negative Hypergeometric Distribution.

Mean and Variance of the Negative Hypergeometric.

Negative Binomial Approximation.

The Meaning of the Mean.

First Occurrences.

Waiting Time for c Special Items to Occur.

Estimating k.

Conclusions.

Explorations.

8. Continuous Probability Distributions: Sums, the Normal Distribution, and the Central Limit Theorem; Bivariate Random Variables.

Uniform Random Variable.

Sums.

A Fact About Means.

Normal Probability Distribution.

Facts About Normal Curves.

Bivariate Random Variables.

Variance.

Central Limit Theorem: Sums.

Central Limit Theorem: Means.

Central Limit Theorem.

Expected Values and Bivariate Random Variables.

Means and Variances of Means.

A Note on the Uniform Distribution.

Conclusions.

Explorations.

9. Statistical Inference I.

Estimation.

Confidence Intervals.

Hypothesis Testing.

β and the Power of a Test.

p-Value for a Test.

Conclusions.

Explorations.

10. Statistical Inference II: Continuous Probability Distributions II—Comparing Two Samples.

The Chi-Squared Distribution.

Statistical Inference on the Variance.

Student t Distribution.

Testing the Ratio of Variances: The F Distribution.

Tests on Means from Two Samples.

Conclusions.

Explorations.

11. Statistical Process Control.

Control Charts.

Estimating σ Using the Sample Standard Deviations.

Estimating σ Using the Sample Ranges.

Control Charts for Attributes.

np Control Chart.

p Chart.

Some Characteristics of Control Charts.

Some Additional Tests for Control Charts.

Conclusions.

Explorations.

12. Nonparametric Methods.

Introduction.

The Rank Sum Test.

Order Statistics.

Median.

Maximum.

Runs.

Some Theory of Runs.

Conclusions.

Explorations.

13. Least Squares, Medians, and the Indy 500.

Introduction.

Least Squares.

Principle of Least Squares.

Influential Observations.

The Indy 500.

A Test for Linearity: The Analysis of Variance.

A Caution.

Nonlinear Models.

The Median–Median Line.

When Are the Lines Identical?

Determining the Median–Median Line.

Analysis for Years 1911–1969.

Conclusions.

Explorations.

14. Sampling.

Simple Random Sampling.

Stratification.

Proportional Allocation.

Optimal Allocation.

Some Practical Considerations.

Strata.

Conclusions.

Explorations.

15. Design of Experiments.

Yates Algorithm.

Randomization and Some Notation.

Confounding.

Multiple Observations.

Design Models and Multiple Regression Models.

Testing the Effects for Significance.

Conclusions.

Explorations.

16. Recursions and Probability.

Introduction.

Conclusions.

Explorations.

17. Generating Functions and the Central Limit Theorem.

Means and Variances.

A Normal Approximation.

Conclusions.

Explorations.

Bibliography.

Where to Learn More.

Index.

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John J. Kinney, PhD, is Professor Emeritus of Mathematics at Rose-Hulman Institute of Technology. He has extensive academic experience and has conducted research in the areas of probability and statistics. In addition to numerous journal articles, Dr. Kinney is the author of Probability: An Introduction with Statistical Applications, also published by Wiley.
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"Each chapter includes exercises and explorations for interested readers. This book provides teachers and instructors with interesting real-world examples that can be used as supporting material for introductory courses on probability and statistics. Including chapters on more advanced and practical topics, like stratified sampling, analysis of experimental data and statistical process control, it may also be of interest to professionals and engineers who use statistical concepts in their work. Undergraduate students, who want to delve into practical applications, can use this book supplementary to a theoretical introduction." (Zentralblatt MATH, 2010)

"Topics include sampling and sample spaces, basic probability, discrete and continuous data, statistical inferences, regression analysis, and experimental design--thus providing broad-based considerations for introductory course instructors to draw from." (CHOICE, 2009)
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