Contents

To the students…………………………………………………….7

Chapter 1. Organization and description of data

1.1. Introduction……………………………………………………9

1.2. The mean………………………………………………………9

1.3. The median ……………………………………………………10

1.4. The mode……………………………………………………...12

Exercises…………………………………………………………...14

1.5. Measures of dispersion for ungrouped data…………………...15

1.5.1. Range………………………………………………………..16

1.5.2. The mean absolute deviation………………………………..17

1.5.3. The variance and the standard deviation……………………18

1.5.4. Interpretation of the population standard deviation…………22

1.5.5. The interquartile range………………………………………24

Exercises…………………………………………………………...26

1.6. Numerical summary of grouped data………………………….29

1.6.1. Mean for data with multiple-observation values…………….29

1.6.2. Median for data with multiple-observation values…………..30

1.6.3. Mode for data with multiple-observation values…………….31

1.6.4. Variance for data with multiple-observation values…………32

Exercises……………………………………………………………35

1.7. Frequency distribution. Grouped data and histograms………...37

1.7.1. Less than method for writing classes………………………...42

Exercises……………………………………………………………45

1.8. Mean for grouped data…………………………………………47

1.9. The Median for grouped data…………………………………..49

1.10. Modal class……………………………………………………51

Exercises…………………………………………………………….52

1.11. Variance and standard deviation for grouped data……………54

1.12. Interquartile range for grouped data…………………………..57

Exercises…………………………………………………………….60

Chapter 2. Probability……………………………………………..65

2.1. Introduction…………………………………………………….65

2.2. Random experiment, outcomes, and sample space…………….65

2.3. Three conceptual approaches to probability……………………69

2.3.1. Classical probability………………………………………….69

2.3.2. Relative frequency concept of probability…………………...69

2.3.3. Subjective probability………………………………………..70

2.4. Probability and its postulates…………………………………..70

2.5. Formula for classical probability………………………………71

2.6. Consequences of the postulates………………………………..72

Exercises……………………………………………………………73

2.7. Counting principle. Permutation and combination…………….76

2.7.1. Permutation…………………………………………………..76

2.7.2. Combination………………………………………………….78

Exercises……………………………………………………………80

2.8. Probability rules………………………………………………..82

Exercises……………………………………………………………83

2.8.1. Conditional probability………………………………………84

2.8.2. The multiplication rule of probability………………………..85

2.8.3. Multiplication rule for independent events…………………..86

Exercises……………………………………………………………88

2.8.4. The law of total probability…………………………………..89

Exercises……………………………………………………………90

2.9. Bayes’ theorem…………………………………………………91

Exercises……………………………………………………………94

2.10. Bivariate probabilities. Joint and marginal probabilities……..96

Exercises……………………………………………………………100

Chapter 3. Discrete random variables and probability distributions………………………………………………………..105

3.1. Random variables………………………………………………105

3.2. Probability distributions for Discrete Random Variables……...106

Exercises……………………………………………………………113

3.3. Expected (mean) value and variance for discrete random variables………………………………………………………….....116

3.3.1. Expected value…………………………………………….....116

3.3.2. Variance and standard deviation of discrete random variable.117

3.3.3. Mean and variance of linear function of a random variable…120

Exercises……………………………………………………………122

3.4. Jointly distributed discrete random variable…………………...124

3.4.1. Conditional probability function…………………………….126

3.4.2. Independence of jointly distributed random variables……….126

3.4.3. Expected value of the function of jointly distributed

random variables…………………………………………...127

3.4.4. Covariance……………………………………………………128

Exercises……………………………………………………………129

3.5. The binomial distribution………………………………………131

3.5.1. Mean and standard deviation of the binomial distribution…...133

Exercises…………………………………………………………….134

3.6. The hypergeometric probability distribution…………………...137

Exercises…………………………………………………………….139

3.7. The Poisson probability distribution…………………………...140

Exercises…………………………………………………………….142

Chapter 4. Continuous random variables and their probability distributions…………………………………….143