__Book1: Probability and random variables__

**1. Probability**

1.1 Basic ideas of probability

1.2 Conditional probability

1.3 The Theorem of Total Probability

**2. Discrete random variables**

2.1 Random variables

2.2 Specific probability distributions

2.3 The mean of a discrete random variable

2.4 Variance

**3. Probability generating functions**

3.1 The probability generating function

3.2 The p.g.f.s of standard distributions

3.3 Calculating means and variances

3.4 Sums of independent random variables

3.5 The p.f.g. of a constant multiple

**4. Continuous random variables**

4.1 Cumulative distribution functions and probability density functions

4.2 Expectation

**5. Specific continuous distributions**

5.1 The uniform distribution

5.2 The exponential distribution

5.3 The gamma distribution

5.4 The normal distribution

5.5 The *χ*^{2} distribution

**6. Simulation**

6.1 Simulation for continuous distributions

6.2 Simulation for discrete distributions

**7. Related Variables**

7.1 Discrete bivariate distributions

7.2 Continuous bivariate distributions

7.3 The Theorem of Total Probability

**8. Useful mathematical results and techniques**

8.1 Powers, logarithms and exponential functions

8.2 Geometric series

8.3 Differentiation and integration

8.4 Ordinary differential equations

**9. Further exercises on Book 1**

__Book 2: Modelling events in time and space__

**1. What is a random processes**

1.1 Basic ideas

1.2 The Bernoulli process

**2. Further examples**

**3. The Poisson process**

3.1 Basic ideas and results

3.2 Notation for continuous-time processes

3.3 Simulation

**4. A more formal approach to the Poisson process**

**5. The multivariate Poisson process**

**6. The non-homogeneous Poisson process**

6.1 The model

6.2 Basic results

6.3 Simulation

**7. The compound Poisson process**

**8. Point processes **

8.1 The index of dispersion

8.2 Types of point process

**9. Spatial Patterns**

**10. Random Patterns in space**

10.1 The two-dimensional Bernoulli process

10.2 The two-dimensional Poisson process

10.3 Simulation for Poisson processes

**11. Non-random spatial patterns**

11.1 Patterns with regularity

11.2 Patterns with clustering

**12. Counts and distances**

12.1 Counts of objects and their properties

12.2 Object-to-object and point-to-object distances

**13. Detecting departures from randomness**

13.1 A test based on counts

13.2 A test based on distances

13.3 Significance testing

13.4 Postscript

**14. Exercises on Book 2**

__Book 3: Discrete time random processes__

**1. Introducing the Galton-Watson model**

1.1 The model

1.2 Exploring the model

**2. The size of the n-th generation**

2.1 Probability generating functions

2.2 The sum of a random number of random variables

2.3 The p.g.f. of Zn

2.4 The mean and variance of Z

**3. Extinction**

3.1 Extinction after n generations

3.2 The probability of eventual extinction

**4. Other models for branching processes**

4.1 Extensions to the Galton-Watson model

4.2 Other models

**5. What is a random walk?**

5.1 Simple random walks

5.2 Random walks with barriers

5.3 Random walks and Markov chains

**6. The gambler's ruin**

6.1 The gambler's ruin as a random walk

6.2 The probability of ruin

6.3 The duration of the match

6.4 Gambling against a casino

**7. Unrestricted random walks**

7.1 Return to the origin

7.2 Exploring simple random walks

7.3 The position of the particle

**8. Return probabilities and generating functions**

8.1 Return probabilities

8.2 The recurrence of a random walk

8.3 First return probabilities for simple random walks

**9. Other types of random walks**

9.1 Random walks with reflecting barriers

9.2 The limiting distribution

9.3 More random walks

**10. Basic Ideas**

10.1 What is a Markov Chain?

10.2 Transition Matrices

10.3 Calculating probabilities

**11. Conditional and unconditional probabilities**

11.1 Transition probabilities

11.2 Absolute probabilities

**12. The long-run behaviour of Markov Chains**

12.1 Limiting distributions

12.2 Markov chains with an infinite state space

12.3 Stationary distributions

**13. Properties of Markov Chains**

13.1 Communicating classes

13.2 Recurrence and transience

13.3 Periodicity

13.4 Classifying the states of a finite Markov chain

13.5 Limiting distributions and stationary distributions

**14. Recurrence**

14.1 Return probabilities

14.2 Return times for recurrent states

14.3 Postscript

**15. Exercises on Book 3**

__Book 4: Random processes in continuous time__

**1. Arrivals, births and deaths**

1.1 The Poisson process

1.2 The simple birth process

1.3 The pure birth process

1.4 Markov processes

**2. The distribution of X(t)**

2.1 Differential-difference equations

2.2 Functions of two variables

2.3 Probability generating function approach

2.4 Lagrange's equation

**3. Solving Lagrange's equation**

3.1 The general solution

3.2 Particular solutions

3.3 Lagrange's method

**4. Two growing populations**

4.1 The simple birth process

4.2 The immigration-birth process

**5. Birth and death processes**

5.1 Markov processes

5.2 Birth and death processes

5.3 The immigration-birth-death process

**6. The simple birth-death process**

6.1 The size of the population

6.2 Extinction

6.3 The embedded random walk

**7. The immigration-death process**

7.1 The size of the population

7.2 What happens in the long run?

**8. Deterministic Models**

**9. Models and notation**

**10. The simple queue**

10.1 Queue size

10.2 The equilibrium queue size

10.3 Queuing time

10.4 Idle periods and busy periods

**11. Markov queues**

11.1 The M/M/n queue

11.2 Other Markov queues

**12. General Service times**

12.1 The M/G/1 queue

12.2 More than one server

**13. Modelling the spread of an infectious disease**

**14. The simple epidemic**

14.1 The model

14.2 The deterministic model

14.3 The stochastic model: waiting times

14.4 The stochastic model: distribution of Y(t)

**15. The general epidemic**

15.1 The stochastic model

15.2 The deterministic model

15.3 A variation on the model

**16. The stochastic general epidemic model**

**17. The threshold phenomenon**

17.1 Exploring the stochastic model

17.2 The threshold phenomenon for the stochastic general epidemic model

**18. Other models for epidemics**

**19. Modelling lifetimes**

**20. Life tables**

**21. Stationary populations**

**22. Stable populations**

22.1 The growth of a stable population

22.2 Stable populations and stationary populations

**23. Population pyramids**

**24. Exercises on Book 4**

__Unit 12: Decision Analysis__

**1. What is decision analysis**

1.1 The decision-making process

1.2 Construct a payoff table

1.3 The decision tree

**2. Decision under uncertainty**

2.1 Decision-making criterion

**3. Decision-making using game theory**

3.1 History of game theory

3.2 Game decision models

3.3 Two-person zero-sum game

**4. Statistical decision under risk**

4.1 Introduction to statistical decision theory

4.2 The statistical decision model with distributions

__Book 5: Further applications__

**1. Biological background**

1.1 Chromosomes, genes and alleles

1.2 Cell division

**2. Mendel's laws of genetics**

2.1 Mendel's first law

2.2 Mendel's second law

**3. Population genetics**

3.1 The Hardy-Weinberg law

3.2 Examples of population genetics

3.3 Genes with more than two alleles

**4. Linkage**

4.1 Sex linkage

4.2 Hardy-Weinberg equilibrium for sex-linked genes

4.3 Chromosome linkage

**5. Evolution**

**6. Discrete time renewal processes**

6.1 Some examples

6.2 Outcomes of the event E

6.3 Waiting times

**7. Continuous inter-event times**

7.1 The waiting times between events

7.2 Characterising components

7.3 The hazard function

**8. Renewal processes**

8.1 The ordinary renewal process

8.2 The total lifetime of the component in use on arrival

8.3 The residual lifetime of the component in use on arrival

8.4 The equilibrium renewal process

8.5 Variations

**9. The limit of a random walk**

9.1 Random walks

9.2 A continuous model

9.3 What does Brownian motion look like?

**10. Ordinary Brownian motion**

10.1 The distribution of X(t)

10.2 Using the Markov property

10.3 Conditional distributions

**11. Waiting times**

11.1 Exploring waiting times

11.2 The distribution of Wa

**12. Brownian motion with drift**

**13. Geometric motion with drift**

**14. Simulation**

14.1 Simulation using tables

14.2 Simulation using the software

**15. The definition of a diffusion process**

**16. Exercises on Book 5**