Probability and Statistics by Example - Basic Probability and Statistics

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyrighrt......Page 6
Contents......Page 7
Preface......Page 9
Part I Basic probability......Page 15
1.1 A uniform distribution......Page 17
1.2 Conditional Probabilities. The Bayes Theorem. Independent trials......Page 20
1.3 The exclusion–inclusion formula. The ballot problem......Page 41
1.4 Random variables. Expectation and conditional expectation. Joint distributions......Page 47
1.5 The binomial, Poisson and geometric distributions. Probability generating, moment generating and characteristic functions......Page 68
1.6 Chebyshev’s and Markov’s inequalities. Jensen’s inequality. The Law of Large Numbers and the De Moivre–Laplace Theorem......Page 89
1.7 Branching processes......Page 110
2.1 Uniform distribution. Probability density functions. Random variables. Independence......Page 122
2.2 Expectation, conditional expectation, variance, generating function, characteristic function......Page 156
2.3 Normal distributions. Convergence of random variables and distributions. The Central Limit Theorem......Page 182
Part II Basic statistics......Page 205
3.1 Preliminaries. Some important probability distributions......Page 207
3.2 Estimators. Unbiasedness......Page 218
3.3 Sufficient statistics. The factorisation criterion......Page 223
3.4 Maximum likelihood estimators......Page 227
3.5 Normal samples. The Fisher Theorem......Page 229
3.6 Mean square errors. The Rao–Blackwell Theorem. The Cramér–Rao inequality......Page 232
3.7 Exponential families......Page 239
3.8 Confidence intervals......Page 243
3.9 Bayesian estimation......Page 247
4.1 Type I and type II error probabilities. Most powerful tests......Page 256
4.2 Likelihood ratio tests. The Neyman–Pearson Lemma and beyond......Page 257
4.3 Goodness of fit. Testing normal distributions, 1: homogeneous samples......Page 266
4.4 The Pearson x test. The Pearson Theorem......Page 271
4.5 Generalised likelihood ratio tests. The Wilks Theorem......Page 275
4.6 Contingency tables......Page 284
4.7 Testing normal distributions, 2: non-homogeneous samples......Page 290
4.8 Linear regression. The least squares estimators......Page 303
4.9 Linear regression for normal distributions......Page 306
5 Cambridge University Mathematical Tripos examination questions in IB Statistics (1992–1999)......Page 312
Appendix 1. Tables of random variables and probability distributions......Page 360
Appendix 2. Index of Cambridge University Mathematical Tripos examination questions in IA Probability (1992–1999)......Page 363
Bibliography......Page 366
Index......Page 372