Introduction to probability with statistical applications

Introduction to Probability with Statistical Applications targets non-mathematics students, undergraduates and graduates, who do not need an exhaustive treatment of the subject. The presentation is rigorous and contains theorems and proofs, and linear algebra is largely avoided so only a minimal amount of multivariable calculus is needed. The book contains clear definitions, simplified notation and techniques of statistical analysis, which combined with well-chosen examples and exercises, motivate the exposition. Theory and applications are carefully balanced. Throughout the book there are references to more advanced concepts if required.

Preface	p. v
Introduction	p. 1
1 The Algebra of Events	p. 3
1.1 Sample Spaces, Statements, Events	p. 3
1.2 Operations with Sets	p. 7
1.3 Relationships between Compound Statements and Events	p. 11
2 Combinatorial Problems	p. 15
2.1 The Addition Principle	p. 15
2.2 Tree Diagrams and the Multiplication Principle	p. 18
2.3 Permutations and Combinations	p. 23
2.4 Some Properties of Binomial Coefficients and the Binomial Theorem	p. 27
2.5 Permutations with Repetitions	p. 32
3 Probabilities	p. 37
3.1 Relative Frequency and the Axioms of Probabilities	p. 37
3.2 Probability Assignments by Combinatorial Methods	p. 42
3.3 Independence	p. 48
3.4 Conditional Probabilities	p. 54
3.5 The Theorem of Total Probability and the Theorem of Bayes	p. 60
4 Random Variables	p. 71
4.1 Probability Functions and Distribution Functions	p. 71
4.2 Continuous Random Variables	p. 80
4.3 Functions of Random Variables	p. 87
4.4 Joint Distributions	p. 96
4.5 Independence of Random Variables	p. 106
4.6 Conditional Distributions	p. 117
5 Expectation, Variance, Moments	p. 127
5.1 Expected Value	p. 127
5.2 Variance and Standard Deviation	p. 140
5.3 Moments and Generating Functions	p. 149
5.4 Covariance and Correlation	p. 156
5.5 Conditional Expectation	p. 163
5.6 Median and Quantiles	p. 169
6 Some Special Distributions	p. 177
6.1 Poisson Random Variables	p. 177
6.2 Normal Random Variables	p. 185
6.3 The Central Limit Theorem	p. 193
6.4 Negative Binomial, Gamma and Beta Random Variables	p. 201
6.5 Multivariate Normal Random Variables	p. 211
7 The Elements of Mathematical Statistics	p. 221
7.1 Estimation	p. 221
7.2 Testing Hypotheses	p. 231
7.3 The Power Function of a Test	p. 239
7.4 Sampling from Normally Distributed Populations	p. 244
7.5 Chi-Square Tests	p. 253
7.6 Two-Sample Tests	p. 263
7.7 Kolmogorov-Smirnov Tests	p. 271
Appendix I Tables	p. 277
Appendix II Answers and Hints for Selected Odd-Numbered Exercises	p. 283
References	p. 307
Index	p. 309

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