Mathematical Statistics
Preface . 7
1. Introduction to probability 9
1.1. General remarks . 9
1.2. Axioms of probability . 10
1.3. Definition and properties of probability 13
2. Random variables 23
2.1. Definitions . 23
2.2. Probability distribution of a discrete random variable 25
2.3. Probability distribution of a continuous random variable . 37
2.4. More about random variables. 46
3. Selected probability distributions. Discrete random variables 48
3.1. The binomial distribution . 48
3.2. The Poisson distribution . 57
4. Selected probability distributions. Continuous random variables .
64
4.1. The uniform distribution 64
4.2. The normal distribution 68
4.3. The chi-squared distribution 77
4.4. Student’s t-distribution . 84
5. Limit Theorems . 92
5.1. De Moivre-Laplace Theorem . 92
5.2. Some operations on random variables . 98
5.3. Central Limit Theorem . 103
6. Sampling. Sampling distributions 113
6.1. General remarks . 113
6.2. Sampling distributions . 115
6.3. Exact sampling distributions . 118
6.4. Approximated sampling distributions 124
7. Estimation. Point estimators . 129
7.1. General remarks . 129
7.2. Point estimators 130
7.3. Properties of point estimators 131
7.4. Examples of point estimators 133
8. Interval estimation . 139
8.1. General remarks . 139
8.2. Confidence interval for the mean value 140
9. Testing parametric hypotheses . 154
9.1. Introduction 154
9.2. Type I and type II error 155
9.3. Significance level, power of the test and -value 158
9.4. Stages of testing hypothesis . 162
9.5. Testing hypotheses about the mean . 167
9.6. Testing hypotheses about two mean values . 172
9.7. Testing hypotheses about proportion and about two proportions. 177
9.8. Testing hypotheses about variance 180
10. Testing nonparametric hypotheses . 185
10.1. Pearson’s chi-squared test for the fit of the distribution 185
10.2. The chi-squared test for independence 189
11. Exercises . 194
Afterword . 209
References . 211
List of figures 212
List of tables 215
215 pages, Paperback
