| preface i basic probability 1 trials and events 1.1 trials, outcomes, and events 1.2 combinations of events and special events 1.3 indicator functions and combinations of events 1.4 classes, partitions, and boolean combinations 2 probability systems 2.1 probability measures 2.2 some elementary properties 2.3 interpretation and determination of probabilities 2.4 minterm maps and boolean combinations 2a the sigma algebra of events 3 conditional probability 3.1 conditioning and the reassignment of likelihoods 3.2 properties of conditional probability 3.3 repeated conditioning 4 independence of events 4.1 independence as a lack of conditioning 4.2 independent classes .5 conditional independence of events 5.1 operational independence and a common condition 5.2 equivalent conditions and definition 5.3 some problems in probable inference 5.4 classification problems 6 composite trials 6.1 events and component events 6.2 multiple success-failure trials 6.3 bernoulli trials ii random variables and distributions random variables and probabilities 7.1 random variables as functions--mapping concepts 7.2 mass transfer and probability distributions 7.3 simple random variables 7a borel sets, random variables, and borel functions 8 distribution and density functions 8.1 the distribution function 8.2 some discrete distributions 8.3 absolutely continuous random variables and density func- tions 8.4 some absolutely continuous distributions 8.5 the normal distribution 8.6 life distributions in reliability theory 9 random vectors and joint distributions 9.1 the joint distribution determined by a random vector 9.2 the joint distribution function and marginal distributions 9.3 joint density functions 10 independence of random vectors 10.1 independence of random vectors 10.2 simple random variables 10.3 joint density functions and independence 11 functions of random variables 11.1 a fundamental approach and some examples 11.2 functions of more than one random variable 11.3 functions of independent random variables 11.4 the quantile function 11.5 coordinate transformations 11a some properties of the quantile function iii mathematical expectation 12 mathematical expectation 12.1 the concept 12.2 the mean value of a random variable 13 expectation and integrals 13.1 a sketch of the development 13.2 integrals of simple functions 13.3 integrals of nonnegative functions 13.4 integrable functions 13.5 mathematical expectation and the lebesgue integral 13.6 the lebesgue-stieltjes integral and transformation of in- tegrals 13.7 some purther properties of integrals 13.8 the radon-nikodym theorem and fubini's theorem 13.9 integrals of complex random variables and the vector spacel2 13a supplementary theoretical details 13a.1 integrals of simple functions 13a.2 integrals of nonnegative functions 13a.3 integrable functions 14 properties of expectation 14.1 some basic forms of mathematical expectation 14.2 a table of properties 14.3 independence and expectation 14.4 some alternate forms of expectation 14.5 a special case of the radon-nikodym theorem 15 variance and standard deviation 15.1 variance as a measure of spread 15.2 some properties 15.3 variances for some common distributions 15.4 standardized variables and the chebyshev inequality 16 covariance, correlation, and linear regression 16.1 covariance and correlation 16.2 some examples 16.3 linear regression 17 convergence in probability theory 17.1 sequences of events 17.2 almost sure convergence 17.3 convergence in probability 17.4 convergence in the mean 17.5 convergence in distribution 18 transform methods 18.1 expectations and integral transforms 18.2 transforms for some common distributions 18.3 generating functions for nonnegative, integer-valued ran- dom variables 18.4 moment generating function and the laplace transform 18.5 characteristic functions 18.6 the central limit theorem 18.7 random samples and statistics iv conditional expectation 19 conditional expectation, given a random vector 19.1 conditioning by an event 19.2 conditioning by a random vector--special cases 19.3 conditioning by a random vector--general case 19.4 properties of conditional expectation 19.5 regression and mean-square estimation 19.6 interpretation in terms of hilbert space l2 19.7 sums of random variables and convolution 19a some theoretical details 20 random selection and counting processes 20.1 introductory examples and a formal representation 20.2 independent selection from an iid sequence 20.3 a poisson decomposition result--multinomial trials 20.4 extreme values 20.5 bernoulli trials with random execution times 20.6 arrival times and counting processes 20.7 arrivals and demand in an independent random time pe- riod 20.8 decision schemes and markov times 21 poisson processes 21.1 the homogeneous poisson process 21.2 arrivals of m kinds and compound poisson processes 21.3 superposition of poisson processes 21.4 conditional order statistics 21.5 nonhomogeneous poisson processes 21.6 bibliographic note 21a 21a.1 independent increments 21a.2 axiom systems for the homogeneous poisson process 22 conditional independence, given a random vector 22.1 the concept and some basic properties 22.2 the bayesian approach to statistics 22.3 elementary decision models 22a proofs of properties 23 markov sequences 23.1 the markov property for sequences 23.2 some further patterns and examples 23.3 the chapman-kolmogorov equation 23.4 the transition diagram and the transition matrix 23.5 visits to a given state in a homogeneous chain 23.6 classification of states in homogeneous chains 23.7 recurrent states and limit probabilities 23.8 partitioning finite homogeneous chains 23.9 evolution of finite, ergodic chains 23.10 the strong markov property for sequences 23a some theoretical details a some mathematical aids b some basic counting problems index |
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