# ON-LINE TEACHING RESOURCES

A short description of the module and a reading list:

## Probability and Distribution Theory 110SOR201

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.

Monday 3-4 pm

Tuesday 3-4 pm

Thursday 1-2 pm

## 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.  