1 introduction to probability 1 1.1 the history of probability 1 1.2 interpretations of probability 2 1.3 experiments and events 5 1.4 set theory 6 1.5 the definition of probability 16 1.6 finite sample spaces 22 1.7 counting methods 25 1.8 combinatorial methods 32 1.9 multinomial coefficients 42 1.10 the probability of a union of events 46 1.11 statistical swindles 51 1.12 supplementary exercises 53
2 conditional probability 55 2.1 the definition of conditional probability 55 2.2 independent events 66 2.3 bayes』 theorem 76 2.4 the gambler』s ruin problem 86 2.5 supplementary exercises 90
3 random variables and distributions 93 3.1 random variables and discrete distributions 93 3.2 continuous distributions 100 3.3 the cumulative distribution function 107 3.4 bivariate distributions 118 3.5 marginal distributions 130 3.6 conditional distributions 141 3.7 multivariate distributions 152 3.8 functions of a random variable 167 3.9 functions of two or more random variables 175 3.10 markov chains 188 3.11 supplementary exercises 202
4 expectation 207 4.1 the expectation of a random variable 207 4.2 properties of expectations 217 4.3 variance 225 4.4 moments 234 4.5 the mean and the median 241 4.6 covariance and correlation 248 4.7 conditional expectation 256 4.8 utility 265 4.9 supplementary exercises 272
5 special distributions 275 5.1 introduction 275 5.2 the bernoulli and binomial distributions 275 5.3 the hypergeometric distributions 281 5.4 the poisson distributions 287
5.5 the negative binomial distributions 297 5.6 the normal distributions 302 5.7 the gamma distributions 316 5.8 the beta distributions 327 5.9 the multinomial distributions 333 5.10 the bivariate normal distributions 337 5.11 supplementary exercises 345
6 large random samples 347 6.1 introduction 347 6.2 the law of large numbers 348 6.3 the central limit theorem 360 6.4 the correction for continuity 371 6.5 supplementary exercises 375
8 sampling distributions of estimators 464 8.1 the sampling distribution of a statistic 464 8.2 the chi-square distributions 469 8.3 joint distribution of the sample mean and sample variance 473 8.4 the t distributions 480 8.5 confidence intervals 485 8.6 bayesian analysis of samples from a normal distribution 495 8.7 unbiased estimators 506 8.8 fisher information 514 8.9 supplementary exercises 528
9 testing hypotheses 530 9.1 problems of testing hypotheses 530 9.2 testing simple hypotheses 550 9.3 uniformly most powerful tests 559 9.4 two-sided alternatives 567 9.5 the t test 576 9.6 comparing the means of two normal distributions 587 9.7 the f distributions 597 9.8 bayes test procedures 605 9.9 foundational issues 617 9.10 supplementary exercises 621
10 categorical data and nonparametric methods 624 10.1 tests of goodness-of-fit 624 10.2 goodness-of-fit for composite hypotheses 633 10.3 contingency tables 641 10.4 tests of homogeneity 647 10.5 simpson』s paradox 653 10.6 kolmogorov-smirnov tests 657 10.7 robust estimation 666 10.8 sign and rank tests 678 10.9 supplementary exercises 686
11 linear statistical models 689 11.1 the method of least squares 689 11.2 regression 698 11.3 statistical inference in simple linear regression 707 11.4 bayesian inference in simple linear regression 729 11.5 the general linear model and multiple regression 736 11.6 analysis of variance 754 11.7 the two-way layout 763 11.8 the two-way layout with replications 772 11.9 supplementary exercises 783
12 simulation 787 12.1 what is simulation? 787 12.2 why is simulation useful? 791 12.3 simulating specific distributions 804 12.4 importance sampling 816 12.5 markov chain monte carlo 823 12.6 the bootstrap 839 12.7 supplementary exercises 850
tables 853 answers to odd-numbered exercises 865 references 879 index 885
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