# ON-LINE TEACHING RESOURCES

A short description of the module and a reading list:
2002/2003 is the first year I teach this module. I have taken over from
Dr Donald Davison who gave this module in 1996-2001, and will make wide use
of his excellent set of lecture notes and homework problem sheets.

## Lectures timetable

Monday 3-4 pm
Tuesday 3-4 pm
Thursday 1-2 pm
### All lectures in rm 1022, David Bates Building

## Tutorial (Examples Class): Monday 2-3 pm, rm 1022 DBB

except week 5, when it is in rm 3001 DBB

## Open Class: Friday 3-4 pm, rm 1022 DBB

This is intended to enable you to come and ask questions about
the lectures and/or homework problems. Please use this opportunity
and do not leave anything, which you have not fully understood, till late.

## Lecture notes

Introduction. Problems in combinatorial
analysis.
Chapter 1 (part a). Basic probability concepts.
Sample and event spaces. Probability axioms and properties. Problems.
Chapter 1 (part b). Conditional probability.
Law of total probability and Bayes' theorem. Independence. Examples.
Chapter 2 (part a). Discrete random variables.
Probability and distribution functions, expectation, variance. Important
discrete distributions.
Chapter 2 (part b). Bivariate distributions.
Independence. Conditional expectation. Multivariate distributions. Indicator
random variables.
Chapter 3. Probability generating functions.
Momenta. Sums of independent random variables. Branching processes.
Chapter 4. Markov Chains.
Classification of states. The limiting distribution.
Absorption in a finite Markov chain (Gambler's ruin problem).
Chapter 5 (part a). Continuous random variables.
Distribution function and probability density function. Expectation,
variance, skewness and kurtosis. Transformations. Important continuous
distributions. Reliability.
Chapter 5 (part b). Bivariate distributions.
Joint probability density function. Independence. Expectation. The bivariate
normal distribution. Transformation rule and examples, including Student's
t-distribution. Orthogonal transformations. Applications to sampling theory.
Order statistic random variables.
Chapter 6. Moment generating functions. Definition
and properties. Sums of independent variables. Bivariate MGF. Sequences of
random variables. The central limit theorem.
Chapter 7. Continuous time processes. Counting
process. Poisson process, arrival and interarrival times. Markov processes.
Birth-and-death process, steady-state distribution. Simple queueing systems.

## Homework problems (Examples)

Problem sheet 1 (due 10 October 2002)
Combinations of events. Probability "calculus". Matching, collecting and other
problems.
Problem sheet 2 (due 17 October 2002) Conditional
probability. Multiplication rule. Bayes' rule. Independent events.
Problem sheet 3 (due 28 October 2002) Discrete
random variables. Expectation and variance. Bivariate and multivariate
distributions: covariance, independence, conditional expectation.
Problem sheet 4 (due 4 November 2002) Indicator
random variables. Probability generating function. Discrete branching process.
Problem sheet 5 (due 11 November 2002) Markov
chains.
Problem sheet 6 (due 18 November 2002) Continuous
random variables 1.
Problem sheet 7 (due 28 November 2002) Continuous
random variables 2.
Problem sheet 8 (due 5 December 2002) Order
statistics. Moment generating functions.
Problem sheet 9 (due 12 December 2002) Continuous
time random processes.

## Solutions

(copies are in the David Bates and Science Libraries)
Problem sheet 1
Problem sheet 2
Problem sheet 3
Problem sheet 4
Problem sheet 5
Problem sheet 6
Problem sheet 7
Problem sheet 8
Problem sheet 9

## Past exam papers

(copies and some solutions in the libraries)
Jan 2000,
Aug 2000,
Jan 2001,
Aug 2001,
Jan 2002
( solutions ),
Aug 2002.