Probability 2

 Unit code: MATH20701 Credit Rating: 10 Unit level: Level 2 Teaching period(s): Semester 1 Offered by School of Mathematics Available as a free choice unit?: N

Prerequisite

Aims

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.

Overview

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.

Learning outcomes

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;

Assessment methods

• Other - 20%
• Written exam - 80%

Assessment Further Information

• Coursework weighting within unit 20%
• 2 hour end of semester examination: weighting within unit 80%

Syllabus

Chapter 1: Univariate random variables.
Review of random variables; cumulative distribution functions of discrete, continuous and mixed random variables; probability mass functions and probability density functions; functions of univariate random variables: the equivalent events technique, the distribution function technique. [5]

Chapter 2: Multivariate random variables.
Multivariate random variables; bivariate distributions; marginal distributions; the bivariate Normal distribution; functions of multivariate random variables; the distribution of sums of independent random variables; conditional distributions; independence; the bivariate transformation technique .[5]

Chapter 3: Expectation.
Expectation and variance of univariate random variables; expectation of multivarate random variables; conditional expectation; covariance and correlation; probability generating functions; moments and moment generating functions. [6]

Chapter 4: Sums of random variables.
Moment generating function techniques; the gamma distribution; sums with a random number of terms (compound distributions); expectation and variance of sums of random variables; the Central Limit Theorem. [6]

• 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.

Feedback methods

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.

Study hours

• Lectures - 22 hours
• Tutorials - 11 hours
• Independent study hours - 67 hours

Teaching staff

Ian Hall - Unit coordinator