General description
This module provides a grounding in probability and is a necessary precursor for any subsequent study of mathematical statistics and operational research. The emphasis is on modelling real situations, including probability calculations motivated by statistical problems. The mathematical techniques required will be introduced or revised as an integral part of the course.

Aims
To introduce students to techniques for modelling and understanding randomness and to develop a facility at calculating probabilities and moments of random variables.

Learning outcomes
On completion of this module, a student should be able to:

• use set notation and operations to describe events and to calculate their probabilities;
• calculate probabilities involving equally likely outcomes in both finite and continuous sample spaces;
• use the cumulative distribution function (cdf) of a random variable to calculate probabilities and percentiles;
• find the cdf of simple transformations from first principles;
• describe the relationship between the cdf and (i) the probability mass function (pmf) and (ii) the probability density function (pdf)
• sketch and describe the pmf and pdf;
• calculate the mean and variance of simple distributions and of linear functions of a random variable;
• explain the notion of conditional probability and use it to model more complex situations;
• apply the concept of independence in simple cases including infinite sequences of trials.

Syllabus
1. EVENTS AND PROBABILITY: Elementary set operations; rules for describing events with emphasis on experiments and associated sample spaces; Venn Diagrams; partitions, De Morgan's Laws [MWS 2.3]. The additive rule of probability; probability of the complement. Defining probablties on sample spaces with equally likely outcomes: discrete and continuous. Permutations and combinations. Functions of random variables (monotone only). Conditional probability: definition; simple applications [MWS 2.4, 2.5, 2.8, 2.9]. Tree diagrams; informal applications of the Law of Total Probability and Bayes' Theorem; uses in combinatorial problems; sampling with/without replacement. Independence: Bernoulli trials, infinite games [MWS 2.7-2.10].
2. PROBABILITY DISTRIBUTIONS: Cumulative distribution functions: use in calculating probabilities; medians, percentiles; simple (monotone) transformations. Discrete distributions: probability mass functions; sketching; examples including Geometric [MWS 3.1, 3.2]. Continuous distributions: probability density function; sketching; examples including Pareto, Exponential [MWS 4.2]. Expected values of X and of functions of X; calculation for simple distributions; mean and variance of aX + b [MWS 3.3, 4.3].