Probability and Statistics - 2019 entry

Uncertainty is a central tenet of almost every complex system, and probability is the branch of mathematics that seeks to study uncertainty in a systematic way. Probabilistic models are used in fields as diverse as quantum physics, statistics, medicine, finance, forensics, machine learning and artificial intelligence and mathematical biology, and are central to the statistical methods that underpin almost every data driven scientific discovery ever made.

This module introduces fundamental concepts in probability, with a focus on their application to statistical theory. The module starts with a review of set theory and the axioms of probability, before exploring the notion of probabilistic outcomes defined in terms of combinatorial experiments. From this basis we introduce discrete random variables and the concept of probability distributions, including their extension into continuous ranges and multiple dimensions. This module provides the essential tools for understanding and conceptualising uncertainty, and is a prerequisite for various subsequent modules.

The aim of the module is to provide a grounding in probability theory and the intrinsic part this plays in the concept of random variables and probability distributions. This provides not only a strong theoretical basis for the understanding of uncertainty, but also introduces the key concepts at the core of many mathematical subjects. This foundation will be key to many of the modules offered later in the course, particularly in statistics and data analytics.

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:

1 understand fundamental concepts in probability theory;

2 demonstrate knowledge about how these abstract ideas can be used to make sense of stochastic systems.

Discipline Specific Skills and Knowledge:

3 demonstrate knowledge of the fundamental concepts and manipulations in probability which are necessary to be able to progress to, and succeed in, further studies in mathematical sciences;

4 understand how mathematical theory can be employed as a means to model and understand uncertainty in real-world processes.

Personal and Key Transferable / Employment Skills and Knowledge:

5 formulate a notion of uncertainty in terms of probabilistic ideas;

6 understand how these ideas can be applied to understand the inherent uncertainty in physical systems;

7 communicate reasoning and solutions effectively verbally and in writing;

8 demonstrate appropriate use of learning resources;

9 demonstrate self management and time management skills.

- Simple set theory and notation;

- Sample spaces and events;

- Permutations and combinations;

- The axioms of probability;

- Probability statements, including the concepts of independence, conditional probabilities and Bayes' Theorem;

- Discrete random variables;

- Continuous random variables;

- Expectations and moments (including probability and moment generating functions);

- Linear combinations of random variables;

- Central limit theorem.

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment. If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination.

For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

Basic reading:

http://vle.exeter.ac.uk/

Other Resources:

Reading list for this module: