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).
Author(s): Geoffrey Grimmett, David Stirzaker
Edition: 3
Publisher: Oxford University Press
Year: 2020
Language: English
Commentary: Vector PDF
Pages: 592
City: Oxford, UK
Tags: Probability Theory; Queues; Markov Models; Assignments
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
