This course reviews and expands on the introduction to univariate random variables in Year 1 providing
an understanding of the reasons that certain probability distributions are used in statistical modelling contexts;
the ability to formulate the distributional properties of functions of random variables for use in understanding the characteristics of statistical techniques;
a basis for the subsequent probabilistic study of complex random phenomena.
We review and expand methods for describing the random variations of single discrete or continuous random variable, illustrated by examples from a variety of statistical applications. Then the aim of the course is to extend your knowledge of probability to multiple variables and to transformations of variables, with an introduction to limits of averages of random variables. The computer package R helps with the mathematical calculations and interpretation of mathematical statements. Studying many examples leads to the discovery of how the distributions which are important in statistics are inter-related. Thus, this course provides the mathematical foundations for all the subsequent second and third year statistical courses.
At the end of this course you should be able to
interpret and manipulate the distributions of discrete and continuous univariate and multivariate random variables;
obtain summary measures such as the expectation, variance and covariance, of continuous and discrete random variables;
recognise and relate the distributions of standard random variables;
identify, with justification, which of the standard probability distributions is likely to be most appropriate for any given statistical application;
transform and simulate random variables;
determine the distributional properties of linear combinations of random variables;
understand and use the basic concepts of convergence of random variables.
The course runs for all ten weeks of Michaelmas term, with two lectures every week on Mon 4pm-5pm in Faraday and Tue 4pm-5pm in Faraday together with a workshop on Thursday or Friday; on odd-numbered weeks there is a third lecture on Wed 10-11am in Bowland.
Weekly office hours for this module are: Monday 9-10am and Monday 3pm-4pm.
The deadline for coursework hand-in is: 1pm on Tuesday of Week ().
The deadline for completing Moodle revision quiz is 23:59 on the Sunday at the end of Week .
The deadline for completing the Moodle quiz associated with CW () is 23:59 on the Sunday at the end of Week .
All the core notes for the course are in this printed booklet. The appendix is available in the pdf on Moodle. The working for many of the examples will be given in the lectures. Attendance at lectures is strongly advised.
A 2 hour revision session will run in the summer term.
There is no end-of-module test.
For single majors, combined majors and for minors, the assessment for this course is examination, together with from weekly courseworks/quizzes and the revision quiz.
Math100: calculus, matrices and probability, or equivalent. Math100: statistics would also be helpful.
A good book for this course, with many copies in the library, is
S. Ross, A First Course in Probability (sixth edition - or others), Prentice-Hall, 2002.
Also recommended are
F. Daly et al., Elements of Statistics, Addison-Wesley, 1995.
G. Roussas, An Introduction to Probability and Statistical Inference, 2003.
Hogg and Craig, and Introduction to Mathematical Statistics, 1978 (Ch1-5).
For a more mathematically rigorous version of this material (cf the third year course: Probability and Measure) see:
G. Grimmett and D. Welsh Probability: an Introduction, OUP, 1986.
G. Grimmett and D. R. Stirzaker Probability and Random Processes (2nd edition), OUP, 1992.
Also, many of Wikipedia’s pages on probability are of high quality (e.g. the pages on specific probability distributions) and you may find them useful as a supplementary reference. However, be aware that this material is sometimes subject to incorrect alterations or vandalism, so treat it with some caution.