Notes on Probability

Notes on Probability

Peter J. Cameron

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Preface

Here are the course lecture notes for the course MAS108, Probability I, at Queen

Mary, University of London, taken by most Mathematics students and some others

in the first semester.

The description of the course is as follows:

This course introduces the basic notions of probability theory and develops them to the stage where one can begin to use probabilistic

ideas in statistical inference and modelling, and the study of stochastic

processes. Probability axioms. Conditional probability and independence. Discrete random variables and their distributions. Continuous

distributions. Joint distributions. Independence. Expectations. Mean,

variance, covariance, correlation. Limiting distributions.

The syllabus is as follows:

1. Basic notions of probability. Sample spaces, events, relative frequency,

probability axioms.

2. Finite sample spaces. Methods of enumeration. Combinatorial probability.

3. Conditional probability. Theorem of total probability. Bayes theorem.

4. Independence of two events. Mutual independence of n events. Sampling

with and without replacement.

5. Random variables. Univariate distributions - discrete, continuous, mixed.

Standard distributions - hypergeometric, binomial, geometric, Poisson, uniform, normal, exponential. Probability mass function, density function, distribution function. Probabilities of events in terms of random variables.

6. Transformations of a single random variable. Mean, variance, median,

quantiles.

7. Joint distribution of two random variables. Marginal and conditional distributions. Independence.

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8. Covariance, correlation. Means and variances of linear functions of random

variables.

9. Limiting distributions in the Binomial case.

These course notes explain the naterial in the syllabus. They have been ��fieldtested�� on the class of 2000. Many of the examples are taken from the course

homework sheets or past exam papers.

Set books The notes cover only material in the Probability I course. The textbooks listed below will be useful for other courses on probability and statistics.

You need at most one of the three textbooks listed below, but you will need the

statistical tables.

? Probability and Statistics for Engineering and the Sciences by Jay L. Devore (fifth edition), published by Wadsworth.

Chapters 2�C5 of this book are very close to the material in the notes, both in

order and notation. However, the lectures go into more detail at several points,

especially proofs. If you find the course difficult then you are advised to buy

this book, read the corresponding sections straight after the lectures, and do extra

exercises from it.

Other books which you can use instead are:

? Probability and Statistics in Engineering and Management Science by W. W.

Hines and D. C. Montgomery, published by Wiley, Chapters 2�C8.

? Mathematical Statistics and Data Analysis by John A. Rice, published by

Wadsworth, Chapters 1�C4.

You should also buy a copy of

? New Cambridge Statistical Tables by D. V. Lindley and W. F. Scott, published by Cambridge University Press.

You need to become familiar with the tables in this book, which will be provided

for you in examinations. All of these books will also be useful to you in the

courses Statistics I and Statistical Inference.

The next book is not compulsory but introduces the ideas in a friendly way:

? Taking Chances: Winning with Probability, by John Haigh, published by

Oxford University Press.

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Web resources

the address

Course material for the MAS108 course is kept on the Web at



This includes a preliminary version of these notes, together with coursework

sheets, test and past exam papers, and some solutions.

Other web pages of interest include

aids/

books articles/probability book/pdf.html

A textbook Introduction to Probability, by Charles M. Grinstead and J. Laurie

Snell, available free, with many exercises.



The Virtual Laboratories in Probability and Statistics, a set of web-based resources

for students and teachers of probability and statistics, where you can run simulations etc.



The Birthday Paradox (poster in the London Underground, July 2000).



An article on Venn diagrams by Frank Ruskey, with history and many nice pictures.

Web pages for other Queen Mary maths courses can be found from the on-line

version of the Maths Undergraduate Handbook.

Peter J. Cameron

December 2000

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