A Probability and Statistics Companion

Preface xv

1. Probability and Sample Spaces 1

Why Study Probability? 1

Probability 2

Sample Spaces 2

Some Properties of Probabilities 8

Finding Probabilities of Events 11

Conclusions 16

Explorations 16

2. Permutations and Combinations: Choosing the Best Candidate; Acceptance Sampling 18

Permutations 19

Counting Principle 19

Permutations with Some Objects Alike 20

Permuting Only Some of the Objects 21

Combinations 22

General Addition Theorem and Applications 25

Conclusions 35

Explorations 35

3. Conditional Probability 37

Introduction 37

Some Notation 40

Bayes’ Theorem 45

Conclusions 46

Explorations 46

4. Geometric Probability 48

Conclusion 56

Explorations 57

5. Random Variables and Discrete Probability Distributions—Uniform, Binomial, Hypergeometric, and Geometric Distributions 58

Introduction 58

Discrete Uniform Distribution 59

Mean and Variance of a Discrete Random Variable 60

Intervals, σ, and German Tanks 61

Sums 62

Binomial Probability Distribution 64

Mean and Variance of the Binomial Distribution 68

Sums 69

Hypergeometric Distribution 70

Other Properties of the Hypergeometric Distribution 72

Geometric Probability Distribution 72

Conclusions 73

Explorations 74

6. Seven-Game Series in Sports 75

Introduction 75

Seven-Game Series 75

Winning the First Game 78

How Long Should the Series Last? 79

Conclusions 81

Explorations 81

7. Waiting Time Problems 83

Waiting for the First Success 83

The Mythical Island 84

Waiting for the Second Success 85

Waiting for the rth Success 87

Mean of the Negative Binomial 87

Collecting Cereal Box Prizes 88

Heads Before Tails 88

Waiting for Patterns 90

Expected Waiting Time for HH 91

Expected Waiting Time for TH 93

An Unfair Game with a Fair Coin 94

Three Tosses 95

Who Pays for Lunch? 96

Expected Number of Lunches 98

Negative Hypergeometric Distribution 99

Mean and Variance of the Negative Hypergeometric 101

Negative Binomial Approximation 103

The Meaning of the Mean 104

First Occurrences 104

Waiting Time for c Special Items to Occur 104

Estimating k 105

Conclusions 106

Explorations 106

8. Continuous Probability Distributions: Sums, the Normal Distribution, and the Central Limit Theorem; Bivariate Random Variables 108

Uniform Random Variable 109

Sums 111

A Fact About Means 111

Normal Probability Distribution 113

Facts About Normal Curves 114

Bivariate Random Variables 115

Variance 119

Central Limit Theorem: Sums 121

Central Limit Theorem: Means 123

Central Limit Theorem 124

Expected Values and Bivariate Random Variables 124

Means and Variances of Means 124

A Note on the Uniform Distribution 126

Conclusions 128

Explorations 129

9. Statistical Inference I 130

Estimation 131

Confidence Intervals 131

Hypothesis Testing 133

β and the Power of a Test 137

p-Value for a Test 139

Conclusions 140

Explorations 140

10. Statistical Inference II: Continuous Probability Distributions II—Comparing Two Samples 141

The Chi-Squared Distribution 141

Statistical Inference on the Variance 144

Student t Distribution 146

Testing the Ratio of Variances: The F Distribution 148

Tests on Means from Two Samples 150

Conclusions 154

Explorations 154

11. Statistical Process Control 155

Control Charts 155

Estimating σ Using the Sample Standard Deviations 157

Estimating σ Using the Sample Ranges 159

Control Charts for Attributes 161

np Control Chart 161

p Chart 163

Some Characteristics of Control Charts 164

Some Additional Tests for Control Charts 165

Conclusions 168

Explorations 168

12. Nonparametric Methods 170

Introduction 170

The Rank Sum Test 170

Order Statistics 173

Median 174

Maximum 176

Runs 180

Some Theory of Runs 182

Conclusions 186

Explorations 187

13. Least Squares, Medians, and the Indy 500 188

Introduction 188

Least Squares 191

Principle of Least Squares 191

Influential Observations 193

The Indy 500 195

A Test for Linearity: The Analysis of Variance 197

A Caution 201

Nonlinear Models 201

The Median–Median Line 202

When Are the Lines Identical? 205

Determining the Median–Median Line 207

Analysis for Years 1911–1969 209

Conclusions 210

Explorations 210

14. Sampling 211

Simple Random Sampling 212

Stratification 214

Proportional Allocation 215

Optimal Allocation 217

Some Practical Considerations 219

Strata 221

Conclusions 221

Explorations 221

15. Design of Experiments 223

Yates Algorithm 230

Randomization and Some Notation 231

Confounding 233

Multiple Observations 234

Design Models and Multiple Regression Models 235

Testing the Effects for Significance 235

Conclusions 238

Explorations 238

16. Recursions and Probability 240

Introduction 240

Conclusions 250

Explorations 250

17. Generating Functions and the Central Limit Theorem 251

Means and Variances 253

A Normal Approximation 254

Conclusions 255

Explorations 255

Bibliography 257

Where to Learn More 257

Index 259