Statistics - David W Scott

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Résumé :

Preface xiii

1 Data Analysis and Understanding 1

1.1 Exploring the Distribution of Data 1

1.1.1 Pearson's Father-Son Height Data 2

1.1.2 Lord Rayleigh's Data 3

1.1.3 Discussion 4

1.2 Exploring Prediction Using Data 4

1.2.1 Body and Brain Weights of Land Mammals 5

1.2.2 Space Shuttle Flight 25 5

1.2.3 Pearson's Father-Son Height Data Revisited 7

1.2.4 Discussion 8

Problems 8

2 Classical Probability 11

2.1 Experiments with Equally Likely Outcomes 11

2.1.1 Simple Outcomes 12

2.1.2 Compound Events and Set Operations 12

2.2 Probability Laws 13

2.2.1 Union and Intersection of Events A and B 13

2.2.1.1 Case (i) 14

2.2.1.2 Cases (ii) and (iii) 14

2.2.1.3 Case (iv) 14

2.2.2 Conditional Probability 14

2.2.2.1 Definition of Conditional Probability 15

2.2.2.2 Conditional Probability With More Than Two Events 15

2.2.3 Independent Events 16

2.2.4 Bayes Theorem 17

2.2.5 Partitions and Total Probability 18

2.3 Counting Methods 19

2.3.1 With Replacement 20

2.3.2 Without Replacement (Permutations) 20

2.3.3 Without Replacement or Order (Combinations) 21

2.3.4 Examples 21

2.3.5 Extended Combinations (Multinomial) 22

2.4 Countable Sets: Implications as n ? ? 22

2.4.1 Selecting Even or Odd Integers 22

2.4.2 Selecting Rational Versus Irrational Numbers 23

2.5 Kolmogorov's Axioms 23

2.6 Reliability: Series Versus Parallel Networks 24

2.6.1 Series Network 24

2.6.2 Parallel Network 25

Problems 26

3 Random Variables and Models Derived From Classical Probability and Postulates 27

3.1 Random Variables and Probability Distributions: Discrete Uniform Example 27

3.1.1 Toss of a Single Die 28

3.1.2 Toss of a Pair of Dice 28

3.2 The Univariate Probability Density Function: Continuous Uniform Example 30

3.2.1 Using the PDF to Compute Probabilities 32

3.2.2 Using the PDF to Compute Relative Odds 33

3.3 Summary Statistics: Central and Non-Central Moments 33

3.3.1 Expectation, Average, and Mean 34

3.3.2 Expectation as a Linear Operator 35

3.3.3 The Variance of a Random Variable 35

3.3.4 Standardized Random Variables 36

3.3.5 Higher Order Moments 37

3.3.6 Moment Generating Function 37

3.3.7 Measurement Scales and Units of Measurement 38

3.3.7.1 The Four Measurement Scales 38

3.3.7.2 Units of Measurement 39

3.4 Binomial Experiments 39

3.5 Waiting Time for a Success: Geometric PMF 42

3.6 Waiting Time for r Successes: Negative Binomial 43

3.7 Poisson Process and Distribution 43

3.7.1 Moments of the Poisson PMF 44

3.7.2 Examples 45

3.8 Waiting Time for Poisson Events: Negative Exponential PDF 45

3.9 The Normal Distribution (Also Known as the Gaussian Distribution) 46

3.9.1 Standard Normal Distribution 48

3.9.2 Sums of Independent Normal Random Variables 49

3.9.3 Normal Approximation to the Poisson Distribution 49

Problems 50

4 Bivariate Random Variables, Transformations, and Simulations 51

4.1 Bivariate Continuous Random Variables 51

4.1.1 Joint CDF and PDF Functions 51

4.1.2 Marginal PDF 52

4.1.3 Conditional Probability Density Function 52

4.1.4 Independence of Two Random Variables 54

4.1.5 Expectation, Correlation, and Regression 54

4.1.5.1 Covariance and Correlation 55

4.1.5.2 Regression Function 55

4.1.6 Independence of n Random Varia...

Biographie:
2(n ? 1) 103

7.3.3.3 Where We Finally Derive the T PDF 103

7.3.4 The One-Sample T-test 104

7.3.5 Example 105

7.3.6 Other T-tests 106

7.3.6.1 Paired T-test 106

7.3.6.2 Two-Sample T-test 107

7.3.6.3 Example Two-Sample T-test: Lord Rayleigh's Data 107

7.4 Reporting Results: p-values and Power 108

7.4.1 Example When the Null Hypothesis Is Rejected 109

7.4.2 When the Null Hypothesis is Not Rejected 109

7.4.3 The Power Function 110

7.5 Multiple Testing and the Bonferroni Correction 111

Problems 111

8 Confidence Intervals and Other Hypothesis Tests 113

8.1 Confidence Intervals 113

8.1.1 Confidence Interval for ...