One Thousand Exercises in Probability

List of Contents

1 Events and their probabilities

1.1 Introduction
1.2 Events as sets
1.3 Probability
1.4 Conditional probability
1.5 Independence
1.6 Completeness and product spaces
1.7 Worked examples
1.8 Problems

2 Random variables and their distributions

2.1 Random variables
2.2 The law of averages
2.3 Discrete and continuous variables
2.4 Worked examples
2.5 Random vectors
2.6 Monte Carlo simulation
2.7 Problems

3 Discrete random variables

3.1 Probability mass functions
3.2 Independence
3.3 Expectation
3.4 Indicators and matching
3.5 Examples of discrete variables
3.6 Dependence
3.7 Conditional distributions and conditional expectation
3.8 Sums of random variables
3.9 Simple random walk
3.10 Random walk: counting sample paths
3.11 Problems

4 Continuous random variables

4.1 Probability density functions
4.2 Independence
4.3 Expectation
4.4 Examples of continuous variables
4.5 Dependence
4.6 Conditional distributions and conditional expectation
4.7 Functions of random variables
4.8 Sums of random variables
4.9 Multivariate normal distribution
4.10 Distributions arising from the normal distribution
4.11 Sampling from a distribution
4.12 Coupling and Poisson approximation
4.13 Geometrical probability
4.14 Problems

5 Generating functions and their applications

5.1 Generating functions
5.2 Some applications
5.3 Random walk
5.4 Branching processes
5.5 Age-dependent branching processes
5.6 Expectation revisited
5.7 Characteristic functions
5.8 Examples of characteristic functions
5.9 Inversion and continuity theorems
5.10 Two limit theorems
5.11 Large deviations
5.12 Problems

6 Markov chains

6.1 Markov processes
6.2 Classification of states
6.3 Classification of chains
6.4 Stationary distributions and the limit theorem
6.5 Reversibility
6.6 Chains with finitely many states
6.7 Branching processes revisited
6.8 Birth processes and the Poisson process
6.9 Continuous-time Markov chains
6.10 Uniform semigroups
6.11 Birth--death processes and imbedding
6.12 Special processes
6.13 Spatial Poisson processes
6.14 Markov chain Monte Carlo
6.15 Problems

7 Convergence of random variables

7.1 Introduction
7.2 Modes of convergence
7.3 Some ancillary results
7.4 Laws of large numbers
7.5 The strong law
7.6 The law of the iterated logarithm
7.7 Martingales
7.8 Martingale convergence theorem
7.9 Prediction and conditional expectation
7.10 Uniform integrability
7.11 Problems

8 Random processes

8.1 Introduction
8.2 Stationary processes
8.3 Renewal processes
8.4 Queues
8.5 The Wiener process
8.6 Existence of processes
8.7 Problems

9 Stationary processes

9.1 Introduction
9.2 Linear prediction
9.3 Autocovariances and spectra
9.4 Stochastic integration and the spectral representation
9.5 The ergodic theorem
9.6 Gaussian processes
9.7 Problems

10 Renewals

10.1 The renewal equation
10.2 Limit theorems
10.3 Excess life
10.4 Applications
10.5 Renewal--reward processes
10.6 Problems

11 Queues 440

11.1 Single-server queues
11.2 M/M/1
11.3 M/G/1
11.4 G/M/1
11.5 G/G/1
11.6 Heavy traffic
11.7 Networks of queues
11.8 Problems

12 Martingales

12.1 Introduction
12.2 Martingale differences and Hoeffding's inequality
12.3 Crossings and convergence
12.4 Stopping times
12.5 Optional stopping
12.6 The maximal inequality
12.7 Backward martingales and continuous-time martingales
12.8 Some examples
12.9 Problems

13 Diffusion processes

13.1 Introduction
13.2 Brownian motion
13.3 Diffusion processes
13.4 First passage times
13.5 Barriers
13.6 Excursions and the Brownian bridge
13.7 Stochastic calculus
13.8 The It\^o integral
13.9 It\^o's formula
13.10 Option pricing
13.11 Passage probabilities and potentials
13.12 Problems

Bibliography

Index