# Probability - MAST4009

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## Module delivery information

Location Term Level1 Credits (ECTS)2 Current Convenor3 2022 to 2023
Canterbury
Autumn Term 4 15 (7.5) Owen Lyne

## Overview

Introduction to Probability. Concepts of events and sample space. Set theoretic description of probability, axioms of probability, interpretations of probability (objective and subjective probability).
Theory for unstructured sample spaces. Addition law for mutually exclusive events. Conditional probability. Independence. Law of total probability. Bayes' theorem. Permutations and combinations. Inclusion-Exclusion formula.
Discrete random variables. Concept of random variable (r.v.) and their distribution. Discrete r.v.: Probability function (p.f.). (Cumulative) distribution function (c.d.f.). Mean and variance of a discrete r.v. Examples: Binomial, Poisson, Geometric.
Continuous random variables. Probability density function; mean and variance; exponential, uniform and normal distributions; normal approximations: standardisation of the normal and use of tables. Transformation of a single r.v.
Joint distributions. Discrete r.v.'s; independent random variables; expectation and its application.
Generating functions. Idea of generating functions. Probability generating functions (pgfs) and moment generating functions (mgfs). Finding moments from pgfs and mgfs. Sums of independent random variables.
Laws of Large Numbers. Weak law of large numbers. Central Limit Theorem.

## Details

### Contact hours

Total contact hours: 47
Private study hours: 103
Total study hours: 150

## Method of assessment

80% examination and 20% coursework.

S. Ross, A First Course in Probability (9th ed.), Pearson, 2012.
J.H.McColl, Probability, Butterworth-Heinmann, 1995.

See the library reading list for this module (Canterbury)

## Learning outcomes

The intended subject specific learning outcomes.
On successfully completing the module students will be able to:
1 demonstrate knowledge of the underlying concepts and principles associated with probability
2 demonstrate the capability to make sound judgements in accordance with the basic theories and concepts in the following areas, whilst demonstrating a reasonable level
of skill in calculation and manipulation of the material: set theoretic description of probability, axioms of probability, random variables, examples of discrete and continuous
distributions, generating functions, weak law of large numbers.
3 apply the underlying concepts and principles associated with probability in several well-defined contexts, showing an ability to evaluate the appropriateness of different
approaches to solving problems in this area

The intended generic learning outcomes.
On successfully completing the module students will be able to:
1 manage their own learning and make use of appropriate resources.
2 understand logical arguments, identifying the assumptions made and the conclusions drawn
3 communicate straightforward arguments and conclusions reasonably accurately and clearly
4 manage their time and use their organisational skills to plan and implement efficient and effective modes of working
5 solve problems relating to qualitative and quantitative information
6 make use of information technology skills such as online resources (Moodle), internet communication
7 communicate technical material competently
8 demonstrate an increased level of skill in numeracy and computation.

## Notes

1. Credit level 4. Certificate level module usually taken in the first stage of an undergraduate degree.
2. ECTS credits are recognised throughout the EU and allow you to transfer credit easily from one university to another.
3. The named convenor is the convenor for the current academic session.