Grimmett G.R., Stirzaker D.R. One Thousand Exercises in Probability

Oxford University Press, 2001. — 448 p. — ISBN: 0198572212.This book contains more than 1000 exercises in probability and random processes, together with their solutions. Apart from being a volume of worked exercises in its own right, it is also a solutions manual for exercises and problems appearing in textbook Grimmett G.R., Stirzaker D.R. Probability and Random Processes, 3rd Edition, 2001, henceforth referred to as PRP. These exercises are not merely for drill, but complement and illustrate the text of PRP, or are entertaining, or both. Despite being intended in part as a companion to PRP, the present volume is as self contained as reasonably possible.Events and their probabilities.
Events as sets, Probability, Conditional probability, Independence,
Completeness and product spaces, Worked examples.
Random variables and their distributions.
Random variables, The law of averages, Discrete and continuous variables, Worked examples,
Random vectors, Monte Carlo simulation.
Discrete random variables.
Probability mass functions, Independence, Expectation, Indicators and matching,
Examples of discrete variables, Dependence, Conditional distributions and conditional expectation,
Sums of random variables, Simple random walk, Random walk: counting sample paths.
Continuous random variables.
Probability density unctions, Independence, Expectation, Examples of continuous variables,
Dependence, Conditional distributions and conditional expectation, Functions of random variables,
Sums of random variables, Multivariate normal distribution,
Distributions arising from the normal distribution, Sampling from a distribution,
Coupling and Poisson approximation, Geometrical probability.
Generating functions and their applications.
Generating functions, Some applications, Random walk, Branching processes, Age-dependent branching processes,
Expectation revisited, Characteristic functions, Examples of characteristic functions, Inversion and continuity theorems,
Two limit theorems, Large deviations.
Markov chains.
Markov processes, Classification of states, Classification of chains, Stationary distributions and the limit theorem,
Reversibility, Chains with finitely many states, Branching processes revisited, Birth processes and the Poisson process,
Continuous-time Markov chains, Uniform semigroups, Birth-death processes and imbedding, Special processes,
Spatial Poisson processes, Markov chain Monte Carlo.
Convergence of random variables.
Modes of convergence, Some ancillary results, Laws of large numbers, The strong law,
The law of the iterated logarithm, Martingales, Martingale convergence theorem, Prediction and conditional expectation, Uniform integrability.
Random processes.
Stationary processes, Renewal processes, Queues, The Wiener process, Existence of processes.
Stationary processes.
Linear prediction, Autocovariances and spectra, Stochastic integration and the spectral representation, The ergodic theorem, Gaussian processes.
Renewals.
The renewal equation, Limit theorems, Excess life, Applications, Renewal-reward processes.
Queues.
Single-server queues, M/M/1, M/G/1, G/M/1, G/G/1, Heavy traffic, Networks of queues.
Martingales.
Martingale differences and Hoeffding's inequality, Crossings and convergence, Stopping times, Optional stopping,
The maximal inequality, Backward martingales and continuous-time martingales, Some examples.
Diffusion processes.
Brownian motion, Diffusion processes, First passage times, Barriers, Excursions and the Brownian bridge,
Stochastic calculus, The Ito integral, Ito's formula, Option pricing, Passage probabilities and potentials.