Japan and the World / 日本留学試験対策問題集 ハイレベル総合科目 [改訂第二版]

This volume contains more than 1300 exercises in probability and random processes togetherwith their solutions. Apart from being a volume of worked exercises in its own right, it is alsoa solutions manual for exercises and problems appearing in the fourth edition of our textbookProbability and RandomProcesses, published by OxfordUniversity Press in 2020, henceforthreferred to as PRP. These exercises are not merely for drill, but complement and illustrate thetext of PRP, or are entertaining, or both. The current edition extends the previous edition bythe inclusion of numerous new exercises, and several new sections devoted to further topicsin aspects of stochastic processes. Since many exercises have multiple parts, the total numberof interrogatives exceeds 3000.Despite being intended in part as a companion to PRP, the present volume is as self-contained as reasonably possible. Where knowledge of a substantial chunk of bookwork isunavoidable, the reader is providedwith a reference to the relevant passage in PRP. Expressionssuch as ‘clearly’ appear frequently in the solutions. Although we do not use such terms intheir Laplacian sense to mean ‘with difficulty’, to call something ‘clear’ is not to imply thatexplicit verification is necessarily free of tedium.The table of contents reproduces that of PRP. The covered range of topics is broad,beginning with the elementary theory of probability and random variables, and continuing,via chapters on Markov chains and convergence, to extensive sections devoted to stationarityand ergodic theory, renewals, queues, martingales, and diffusions, including an introductionto the pricing of options. Generally speaking, exercises are questionswhich test knowledge ofparticular pieces of theory, while problems are less specific in their requirements. There arequestions of all standards, the great majority being elementary or of intermediate difficulty.We have found some of the later ones to be rather tricky, but have refrained from magnifyingany difficulty by adding asterisks or equivalent devices. To those using this book for self-study,our advice is not to attempt more than a respectable fraction of these at a first read.We offer two caveats to readers. While a great deal of care has been devoted to ensuringthese exercises are as correct as possible, there inevitably remain a few slips which have sofar escaped detection, and in this regard the readers’ patience is invited. Secondly, there willalways be debate on justwhat constitutes a proper or full solution. We have aimed at conveyingthe nubs of arguments, with as many details as needed to satisfy most readers. Cover One Thousand Exercises in Probability Copyright Epigraph Preface to the Third Edition Contents Questions 1 Events and their probabilities 1.2 Exercises. Events as sets 1.3 Exercises. Probability 1.4 Exercises. Conditional probability 1.5 Exercises. Independence 1.7 Exercises. Worked examples 1.8 Problems 2 Random variables and their distributions 2.1 Exercises. Random variables 2.2 Exercises. The law of averages 2.3 Exercises. Discrete and continuous variables 2.4 Exercises. Worked examples 2.5 Exercises. Random vectors 2.7 Problems 3 Discrete random variables 3.1 Exercises. Probability mass functions 3.2 Exercises. Independence 3.3 Exercises. Expectation 3.4 Exercises. Indicators and matching 3.5 Exercises. Examples of discrete variables 3.6 Exercises. Dependence 3.7 Exercises. Conditional distributions and conditional expectation 3.8 Exercises. Sums of random variables 3.9 Exercises. Simple random walk 3.10 Exercises. Random walk: counting sample paths 3.11 Problems 4 Continuous random variables 4.1 Exercises. Probability density functions 4.2 Exercises. Independence 4.3 Exercises. Expectation 4.4 Exercises. Examples of continuous variables 4.5 Exercises. Dependence 4.6 Exercises. Conditional distributions and conditional expectation 4.7 Exercises. Functions of random variables 4.8 Exercises. Sums of random variables 4.9 Exercises. Multivariate normal distribution 4.10 Exercises. Distributions arising from the normal distribution 4.11 Exercises. Sampling from a distribution 4.12 Exercises. Coupling and Poisson approximation 4.13 Exercises. Geometrical probability 4.14 Problems 5 Generating functions and their applications 5.1 Exercises. Generating functions 5.2 Exercises. Some applications 5.3 Exercises. Random walk 5.4 Exercises. Branching processes 5.5 Exercises. Age-dependent branching processes 5.6 Exercises. Expectation revisited 5.7 Exercises. Characteristic functions 5.8 Exercises. Examples of characteristic functions 5.9 Exercises. Inversion and continuity theorems 5.10 Exercises. Two limit theorems 5.11 Exercises. Large deviations 5.12 Problems 6 Markov chains 6.1 Exercises. Markov processes 6.2 Exercises. Classification of states 6.3 Exercises. Classification of chains 6.4 Exercises. Stationary distributions and the limit theorem 6.5 Exercises. Reversibility 6.6 Exercises. Chains with finitely many states 6.7 Exercises. Branching processes revisited 6.8 Exercises. Birth processes and the Poisson process 6.9 Exercises. Continuous-time Markov chains 6.10 Exercises. Kolmogorov equations and the limit theorem 6.11 Exercises. Birth–death processes and imbedding 6.12 Exercises. Special processes 6.13 Exercises. Spatial Poisson processes 6.14 Exercises. Markov chain Monte Carlo 6.15 Problems 7 Convergence of random variables 7.1 Exercises. Introduction 7.2 Exercises. Modes of convergence 7.3 Exercises. Some ancillary results 7.4 Exercise. Laws of large numbers 7.5 Exercises. The strong law 7.6 Exercise. The law of the iterated logarithm 7.7 Exercises. Martingales 7.8 Exercises. Martingale convergence theorem 7.9 Exercises. Prediction and conditional expectation 7.10 Exercises. Uniform integrability 7.11 Problems 8 Random processes 8.2 Exercises. Stationary processes 8.3 Exercises. Renewal processes 8.4 Exercises. Queues 8.5 Exercises. TheWiener process 8.6 Exercises. L ́evy processes and subordinators 8.7 Exercises. Self-similarity and stability 8.8 Exercises. Time changes 8.10 Problems 9 Stationary processes 9.1 Exercises. Introduction 9.2 Exercises. Linear prediction 9.3 Exercises. Autocovariances and spectra 9.4 Exercises. Stochastic integration and the spectral representation 9.5 Exercises. The ergodic theorem 9.6 Exercises. Gaussian processes 9.7 Problems 10 Renewals 10.1 Exercises. The renewal equation 10.2 Exercises. Limit theorems 10.3 Exercises. Excess life 10.4 Exercises. Applications 10.5 Exercises. Renewal–reward processes 10.6 Problems 11 Queues 11.2 Exercises. M/M/1 11.3 Exercises. M/G/1 11.4 Exercises. G/M/1 11.5 Exercises. G/G/1 11.6 Exercise. Heavy traffic 11.7 Exercises. Networks of queues 11.8 Problems 12 Martingales 12.1 Exercises. Introduction 12.2 Exercises. Martingale differences and Hoeffding’s inequality 12.3 Exercises. Crossings and convergence 12.4 Exercises. Stopping times 12.5 Exercises. Optional stopping 12.6 Exercise. The maximal inequality 12.7 Exercises. Backward martingales and continuous-time martingales 12.9 Problems 13 Diffusion processes 13.2 Exercise. Brownian motion 13.3 Exercises. Diffusion processes 13.4 Exercises. First passage times 13.5 Exercises. Barriers 13.6 Exercises. Excursions and the Brownian bridge 13.7 Exercises. Stochastic calculus 13.8 Exercises. The Itˆo integral 13.9 Exercises. Itˆo’s formula 13.10 Exercises. Option pricing 13.11 Exercises. Passage probabilities and potentials 13.12 Problems Solutions 1 Events and their probabilities 1.2 Solutions. Events as sets 1.3 Solutions. Probability 1.4 Solutions. Conditional probability 1.5 Solutions. Independence 1.7 Solutions. Worked examples 1.8 Solutions to problems 2 Random variables and their distributions 2.1 Solutions. Random variables 2.2 Solutions. The law of averages 2.3 Solutions. Discrete and continuous variables 2.4 Solutions. Worked examples 2.5 Solutions. Random vectors 2.7 Solutions to problems 3 Discrete random variables 3.1 Solutions. Probability mass functions 3.2 Solutions. Independence 3.3 Solutions. Expectation 3.4 Solutions. Indicators and matching 3.5 Solutions. Examples of discrete variables 3.6 Solutions. Dependence 3.7 Solutions. Conditional distributions and conditional expectation 3.8 Solutions. Sums of random variables 3.9 Solutions. Simple random walk 3.10 Solutions. Random walk: counting sample paths 3.11 Solutions to problems 4 Continuous random variables 4.1 Solutions. Probability density functions 4.2 Solutions. Independence 4.3 Solutions. Expectation 4.4 Solutions. Examples of continuous variables 4.5 Solutions. Dependence 4.6 Solutions. Conditional distributions and conditional expectation 4.7 Solutions. Functions of random variables 4.8 Solutions. Sums of random variables 4.9 Solutions. Multivariate normal distribution 4.10 Solutions. Distributions arising from the normal distribution 4.11 Solutions. Sampling from a distribution 4.12 Solutions. Coupling and Poisson approximation 4.13 Solutions. Geometrical probability 4.14 Solutions to problems 5 Generating functions and their applications 5.1 Solutions. Generating functions 5.2 Solutions. Some applications 5.3 Solutions. Random walk 5.4 Solutions. Branching processes 5.5 Solutions. Age-dependent branching processes 5.6 Solutions. Expectation revisited 5.7 Solutions. Characteristic functions 5.8 Solutions. Examples of characteristic functions 5.9 Solutions. Inversion and continuity theorems 5.10 Solutions. Two limit theorems 5.11 Solutions. Large deviations 5.12 Solutions to problems 6 Markov chains 6.1 Solutions. Markov processes 6.2 Solutions. Classification of states 6.3 Solutions. Classification of chains 6.4 Solutions. Stationary distributions and the limit theorem 6.5 Solutions. Reversibility 6.6 Solutions. Chains with finitely many states 6.7 Solutions. Branching processes revisited 6.8 Solutions. Birth processes and the Poisson process 6.9 Solutions. Continuous-time Markov chains 6.10 Solutions. Kolmogorov equations and the limit theorem 6.11 Solutions. Birth–death processes and imbedding 6.12 Solutions. Special processes 6.13 Solutions. Spatial Poisson processes 6.14 Solutions. Markov chain Monte Carlo 6.15 Solutions to problems 7 Convergence of random variables 7.1 Solutions. Introduction 7.2 Solutions. Modes of convergence 7.3 Solutions. Some ancillary results 7.4 Solutions. Laws of large numbers 7.5 Solutions. The strong law 7.6 Solution. The law of the iterated logarithm 7.7 Solutions. Martingales 7.8 Solutions. Martingale convergence theorem 7.9 Solutions. Prediction and conditional expectation 7.10 Solutions. Uniform integrability 7.11 Solutions to problems 8 Random processes 8.2 Solutions. Stationary processes 8.3 Solutions. Renewal processes 8.4 Solutions. Queues 8.5 Solutions. TheWiener process 8.6 Solutions. L ́evy processes and subordinators 8.7 Solutions. Self-similarity and stability 8.8 Solutions. Time changes 8.10 Solutions to problems 9 Stationary processes 9.1 Solutions. Introduction 9.2 Solutions. Linear prediction 9.3 Solutions. Autocovariances and spectra 9.4 Solutions. Stochastic integration and the spectral representation 9.5 Solutions. The ergodic theorem 9.6 Solutions. Gaussian processes 9.7 Solutions to problems 10 Renewals 10.1 Solutions. The renewal equation 10.2 Solutions. Limit theorems 10.3 Solutions. Excess life 10.4 Solutions. Applications 10.5 Solutions. Renewal–reward processes 10.6 Solutions to problems 11 Queues 11.2 Solutions. M/M/1 11.3 Solutions. M/G/1 11.4 Solutions. G/M/1 11.5 Solutions. G/G/1 11.6 Solution. Heavy traffic 11.7 Solutions. Networks of queues 11.8 Solutions to problems 12 Martingales 12.1 Solutions. Introduction 12.2 Solutions. Martingale differences and Hoeffding’s inequality 12.3 Solutions. Crossings and convergence 12.4 Solutions. Stopping times 12.5 Solutions. Optional stopping 12.6 Solution. The maximal inequality 12.7 Solutions. Backward martingales and continuous-time martingales 12.9 Solutions to problems 13 Diffusion processes 13.2 Solution. Brownian motion 13.3 Solutions. Diffusion processes 13.4 Solutions. First passage times 13.5 Solutions. Barriers 13.6 Solutions. Excursions and the Brownian bridge 13.7 Solutions. Stochastic calculus 13.8 Solutions. The Itˆo integral 13.9 Solutions. Itˆo’s formula 13.10 Solutions. Option pricing 13.11 Solutions. Passage probabilities and potentials 13.12 Solutions to problems Bibliography Index This third edition is a revised, updated, and greatly expanded version of previous edition of 2001. The 1300+ exercises contained within are not merely drill problems, but have been chosen to illustrate the concepts, illuminate the subject, and both inform and entertain the reader. A broad range of subjects is covered, including elementary aspects of probability and random variables, sampling, generating functions, Markov chains, convergence, stationary processes, renewals, queues, martingales, diffusions, L�vy processes, stability and self-similarity, time changes, and stochastic calculus including option pricing via the Black-Scholes model of mathematical finance. The text is intended to serve students as a companion for elementary, intermediate, and advanced courses in probability, random processes and operations research. It will also be useful for anyone needing a source for large numbers of problems and questions in these fields. In particular, this book acts as a companion to the authors' volume, Probability and Random Processes, fourth edition (OUP 2020). This third edition is a revised, updated, and greatly expanded version of previous edition of 2001. The 1300+ exercises contained within are not merely drill problems, but have been chosen to illustrate the concepts, illuminate the subject, and both inform and entertain the reader. A broad range of subjects is covered, including elementary aspects of probability and random variables, sampling, generating functions, Markov chains, convergence, stationary processes, renewals, queues, martingales, diffusions, Lévy processes, stability and self-similarity, time changes, and stochastic calculus including option pricing via the Black-Scholes model of mathematical finance. The text is intended to serve students as a companion for elementary, intermediate, and advanced courses in probability, random processes and operations research. It will also be useful for anyone needing a source for large numbers of problems and questions in these fields. In particular, this book acts as a companion to the authors'volume, Probability and Random Processes, fourth edition (OUP 2020). This volume of more than 1300 exercises and solutions in probability theory has two roles. It is both a freestanding book of exercises and solutions in probability theory, and a manual for students and teachers covering the exercises and problems in the companion volume Probability and Random Processes (4th edition).