Probability and Statistics by Example: I Probability and statistics are as much about intuition and problem solving, as they are about theorem proving. Because of this, students can find it very difficult to make a successful transition from lectures to examinations to practice, since the problems involved can vary so much in nature. Since the subject iscritical in many modern applications such as mathematical finance, quantitative management, telecommunications, signal processing, bioinformatics, as well as traditional ones such as insurance, social science and engineering, the authors have rectified deficiencies in traditional lecture-based methods by collecting together a wealth of exercises for which they’ve supplied complete solutions. Thesesolutions are adapted to the needs and skills of students. To make it of broad value, the authors supply basic mathematical facts as and when they are needed, and have sprinkled some historical information throughout the text.
Probability and Statistics by Example
Volume I. Basic Probability and Statistics
University of Cambridge
University of Wales–Swanseacambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge cb2 2ru, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521847667 © Cambridge University Press 2005 This publication is in copyright.Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format isbn-13 isbn-10 isbn-13 isbn-10 isbn-13 isbn-10 978-0-511-13229-2 eBook (EBL) 0-511-13229-8 eBook (EBL) 978-0-521-84766-7 hardback 0-521-84766-4 hardback 978-0-521-61233-3paperback 0-521-61233-0 paperback
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Part I 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 2 2.1 2.2 2.3Basic probability Discrete outcomes A uniform distribution Conditional Probabilities. The Bayes Theorem. Independent trials The exclusion–inclusion formula. The ballot problem Random variables. Expectation and conditional expectation. Joint distributions The binomial, Poisson and geometric distributions. Probability generating, moment generating and characteristic functions Chebyshev’s andMarkov’s inequalities. Jensen’s inequality. The Law of Large Numbers and the De Moivre–Laplace Theorem Branching processes Continuous outcomes Uniform distribution. Probability density functions. Random variables. Independence Expectation, conditional expectation, variance, generating function, characteristic function Normal distributions. Convergence of random variables and distributions. The CentralLimit Theorem
1 3 3 6 27 33 54 75 96 108 108 142 168
Part II Basic statistics 3 3.1 3.2 3.3 3.4 3.5 Parameter estimation Preliminaries. Some important probability distributions Estimators. Unbiasedness Sufficient statistics. The factorisation criterion Maximum likelihood estimators Normal samples. The Fisher Theorem
191 193 193 204 209 213 215 v
vi 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.54.6 4.7 4.8 4.9 5
Contents Mean square errors. The Rao–Blackwell Theorem. The Cramér–Rao inequality Exponential families Confidence intervals Bayesian estimation Hypothesis testing Type I and type II error probabilities. Most powerful tests Likelihood ratio tests. The Neyman–Pearson Lemma and beyond Goodness of fit. Testing normal distributions, 1: homogeneous samples The Pearson 2 test. The...