Introductory description

The module runs in Term 2 and provides elementary introduction to the theory of probability. The topics include axioms of probability, combinatorics, independent events, conditional probability, random variables, discrete and continuous probability distributions, expectation and variance, joint probability distributions, independence of random variables, sum of independent random variables, covariance and correlation.

This module is core for students with their home department in Statistics and is not available to students from other departments. Students from other departments should consider ST111 Probability A and ST112 Probability B instead.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

1. Experiments with random outcomes: the notions of random experiment, sample space and events. Operations with sets and their interpretation.
2. Axioms of probability. Properties of a probability measure: Boole’s inequality, continuity of a probability measure, inclusion-exclusion formula.
3. Finite sample spaces with equally likely outcomes.
4. Independence of events. Conditional probabilities. Bayes theorem.
5. The notion of a random variable. Examples in both discrete and continuous settings. Indicator random variables.
6. The notion of the distribution of a random variable. Probability mass functions and density functions. Cumulative distribution functions.
7. Expectation of random variables. Properties of expectation.
8. Mean and variance of distributions. Chebyshev's inequality.
9. Independence of random variables. Joint distributions. Covariance and correlation. Cauchy-Schwartz inequality.
10. Addition of independent random variables: convolutions. Moment generating function and use to compute convolutions.
11. Important families of distributions: Binomial, Poisson, negative Binomial, exponential, Gamma and Gaussian. Their properties, genesis and inter-relationships.

Learning outcomes

By the end of the module, students should be able to:

• Understand key notions of mathematical probability including random variables and their distributions
• Appreciate the role of randomness in mathematical modelling of real world situations.
• Use appropriate mathematical techniques to calculate the probabilities of events, and the expectations of random variables

Indicative reading list

Ross, A first course in probability, Prentice Hall, 1994
Pitman, Probability, Springer texts in Statistics
Suhov and Kelbert, Probability and Statistics by Example: Basic Probability and Statistics.

View reading list on Talis Aspire

Subject specific skills

Mathematical, analytical, problem solving

Transferable skills

Analytical, problem solving, investigative skills, communication, good working habits.