MMath Mathematics with Financial Mathematics

This course continues the development of probability and statistics from the first year so that all students on the single honours programme have the basic grounding in this area which would be expected of a mathematics graduate. It provides a solid basis for a wide variety of options later in the programme for students who wish to take their studies in probability and/or statistics further.

The course unit unit aims to develop a solid foundation in the calculus of probabilities and indicate the relevance and importance of this to tackling real-life problems.

On completion of this unit successful students will:

• understand the concept of both univariate and multivariate random variables;
• be familiar with a range of parametric families to model their probability distribution;
• be able to calculate expectations and conditional expectations;
• be able to evaluate the distribution of functions of random variables;

Chapter 1: Random Variables (4 lectures) Definition of Events and their probabilities; definition of Random variables and their distributions; features of Discrete and Continuous random variables; Functions of random variables and mixed random variables.

Chapter 2: Multivariate random variables (6 lectures) Bivariate Distributions; Independence; Sums of several variables; Conditional distributions; the bivariate transform.

Chapter 3: Expectation (6 lectures) Expectation of a univariate random variable; Variance and higher Moments; Expectation of a bivariate random variable and conditional expectation; Probability generating functions; Moment generating functions; Sums of random variables using generating functions.

Chapter 4: Sampling and convergence; The sample mean; Central limit theorem; Chebyshev's Inequality; Poisson Limit Theorem and characteristic functions.

• Coursework weighting within unit 20%
• End of semester examination: weighting within unit 80%

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

• Mood, A. M., Graybill, F. A. and Boes, D. C., Introduction to the Theory of Statistics, 3rd edition, McGraw-Hill 1974
• S. Ross, A First Course in Probability, 4th edition, Macmillan.
• D. Stirzaker, Elementary Probability, Cambridge University Press. Available electronically
• Neil A. Weiss, A Course in Probability, Pearson.

The independent study hours will normally comprise the following. During each week of the taught part of the semester:

·         You will normally have approximately 60-75 minutes of video content. Normally you would spend approximately 2-2.5 hrs per week studying this content independently

·         You will normally have exercise or problem sheets, on which you might spend approximately 1.5hrs per week

·         There may be other tasks assigned to you on Blackboard, for example short quizzes or short-answer formative exercises

·         In some weeks you may be preparing coursework or revising for mid-semester tests

Together with the timetabled classes, you should be spending approximately 6 hours per week on this course unit.

The remaining independent study time comprises revision for and taking the end-of-semester assessment.