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Class notes MATH 545 Mathematical Statistics and Probability Theory
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Notes on ProbabilityPeter J. Cameron
,ii
,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 de-
velops 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 indepen-
dence. 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, uni-
form, normal, exponential. Probability mass function, density function, dis-
tribution 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 distri-
butions. Independence.
iii
, iv
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 “field-
tested” 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 text-
books 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. De-
vore (fifth edition), published by Wadsworth.
Chapters 2–5 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 2–8.
• Mathematical Statistics and Data Analysis by John A. Rice, published by
Wadsworth, Chapters 1–4.
You should also buy a copy of
• New Cambridge Statistical Tables by D. V. Lindley and W. F. Scott, pub-
lished 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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