The module begins with the idea of a probability space, which is how we model the possible outcomes of a random experiment. Concepts such as statistical independence and conditional probability are introduced, and a number of practical problems are studied. We then turn our attention to random variables, and look at some well-known probability distributions. Following this we focus on discrete distributions, and introduce the idea of independence for random variables, and the important concept of mathematical expectation. This leads on to the study of random vectors, where we introduce covariance and correlation, conditional distributions and the law of total expectation. Finally, we show how the ideas developed for discrete distributions can be carried over to continuous distributions, and conclude with some approximation theorems.  

This is a lecture-based module. Students will be required to demonstrate problem-solving skills throughout the module. No previous knowledge of probability theory is assumed.

The module is intended to prepare students for subsequent modules involving probability and statistics within the degree scheme.

Free Standing Module Requirements:  A pass in A-Level Mathematics of at least Grade A

• Appreciate the theoretical foundations of probability theory.
• Apply the concepts of independence and conditional probability to practical problems.
• Understand the concepts of random variables and distribution functions.
• Derive the expected value and variance of several well-known distributions (including the binomial, Poisson, geometric, exponential and normal distributions)
• Apply the concepts of independent random variables and conditional expectation to practical problems.
• Apply various approximation theorems to practical problems.

You will be guided through learning activities appropriate to your module, which may include:

• Weekly face to face classes (e.g. labs, lectures, exercise classes)
• Electronic resources to support the learning (e.g. videos, exercise sheets, lecture notes, quizzes)

Students are also expected to undertake at least 50 hours of self-guided study throughout the duration of the module, including preparation of formative assessments.

Skills:

To choose appropriate ideas and techniques to solve practical problems that involve randomness.

Transferable Skills:

Formulating problems and presenting clear solutions through logical reasoning.

• Set algebra.
• Probability spaces.
• Conditional probability and Bayes’ theorem
• Independent events.
• Random variables and distribution functions.
• Discrete random variables.
• Mathematical expectation.
• Covariance and correlation.
• Conditional distributions and conditional expectation.
• Continuous random variables.
• Medians, quantiles and statistical tables.
• Approximation to the binomial and Poisson distributions.