MAT1017

Module Catalogue

Module Code: MAT1017

Module Title: PROOF, PROBABILITY AND EXPERIMENT

Module Provider: Mathematics	Short Name: MS125	Previous Short Name: MS125
Level: HE1	Module Co-ordinator: GODOLPHIN JD Dr (Maths)
Number of credits: 30	Number of ECTS credits: 15

Module Availability

Autumn / Spring

Assessment Pattern

Unit(s) of Assessment

Weighting Towards Module Mark( %)

Coursework 1 (Proof) 2 tests in Autumn semester

25%

Coursework 2 (Experiment) 2 assignments in Spring semester (two assignments in Matlab, 1st counts 40%, 2^nd 60%)

25%

Coursework 3 (Probability) 2-3 tests in Spring semester

10%

Exam (Probability) End of Spring semester

40%

Qualifying Condition(s)

A weighted aggregate mark of 40% is required to pass the module.

Module Overview

Prerequisites/Co-requisites

None

Module Aims

This module introduces student to the concept and techniques of proof in mathematics, to mathematical probability theory and its applications to statistics, and to the ideas and techniques of mathematical experimentation.

Learning Outcomes

By the end of this module a student should be able to:

·Recognise and construct simple proofs in set theory and number theory by induction, contradiction, contrapositive and construction.

Apply basic results in probability, distribution theory and statistical inference to straightforward problems.
Use the statistical package R to perform statistical calculations and to produce numerical and graphical data summaries.
Use computer algebra packages for symbolic manipulation, numerical calculation and graphical visualisation.
Formulate simple mathematical models and investigate their behaviour using appropriate computational methods.

Explore, through computer-aided experimentation, unfamiliar problems and topics from various areas of mathematics and its applications.

Module Content

Proof

Axiomatic systems and the need for proof;
Implication and negation;
Concept of proof, proofs by induction, contradiction contrapositive and construction;
Definition of a set;
Order;
Subsets; union and intersections;
Injections, surjections and bijections between sets;
Cartesian products;
Revision of integers, rational and real numbers;
Countability of the rationals

Divisibility; LCM and HCF;

Primes; Euclid's proof of infinite number of primes.

Continued Fractions

Probability

Probability theory, including Bayes Theorem.
Some standard discrete distributions: binomial, Poisson and hypergeometric and continuous distributions: normal, exponential.

Expectation and moments.
Moment generating functions.
Sums of random variables.
Statement and applications of the central limit theorem.
Sampling distributions.
Inference for means and proportions.
Estimation: unbiased estimators; maximum likelihood and moment estimators.
Hypothesis testing: Type I and II errors, power.
Chi-squared Goodness of fit.
An Introduction to R.

Experiment

Mathematical modelling: Principles and tools for constructing simple models.Techniques for computing, visualising, and interpreting solutions to problems formulated as mathematical models.

Computational packages: Using packages such as MATLAB and/or MAPLE to perform algebraic and numerical computational tasks needed to develop and understand mathematical models.

Experimental mathematics: A selection of topics from pure and applied mathematics will be explored. Some of these should be familiar from previous modules; others may be preparatory for work to be covered in later modules. The emphasis throughout will be on an investigative approach: exploring various systems by developing mathematical models of them and discovering their properties via computer-aided experimentation.

Methods of Teaching/Learning

Teaching is by lectures, tutorials, computer lab sessions and tests. Learning takes place through lectures, peer instruction, tutorials, computer lab sessions, tests and exercise sheets.

Proof: 20 hours of lectures and tutorials over 10 weeks in the Autumn semester.

Probability: 40 hours of lectures and tutorials over 12 weeks in the Spring semester. A significant proportion of the lecture time will be devoted to learning via the technique of peer instruction.

Experiment: 20 hours of lectures and tutorials over 12 weeks in the Spring semester.

Selected Texts/Journals

Recommended Reading:

Proof

* D.L.Johnson, Elements of Logic via Numbers and Sets, Springer, 1998.

Probability

* G.M. Clarke and D. Cooke, A Basic Course in Statistics, Arnold

Experiment

* Frank Garvan: The MAPLE Book (Chapman & Hall 2001)

* André Heck: Introduction to Maple (Springer, 2003)

* D.J. Higham, N.J. Higham: Matlab Guide (SIAM)

* B.R. Hunt, R.L. Lipsman, J.M. Rosenberg, K.R. Coombes, John E. Osborn, G.J. Stuck: A Guide to