One Thousand Exercises in Probability: Third Edition

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