### MAS2304 : Foundations of Probability

• Offered for Year: 2015/16
• Module Leader(s): Dr Philip Ansell
• Owning School: Mathematics & Statistics
##### Semesters
 Semester 1 Credit Value: 10 ECTS Credits: 5.0
##### Pre Requisites
Code Title
MAS1341Introduction to Probability
MAS1342Introduction to Statistics

N/A

None

#### Aims

To acquire the mathematical and probabilistic skills necessary for the further study of statistics.

Module Summary

Probability is the branch of mathematics which helps us to describe, analyse and understand chance phenomena. While the development of competence in probability is an essential preparation for the study of modern statistics, probability is also an important object of study for pure mathematicians and plays a key role in many areas of applied mathematics. Perhaps the most remarkable thing we discover is that even random objects demonstrate regular patterns of behaviour which can helpfully be thought of as laws of probability.
Frequently we need to examine two or more variables at a time. Although we could study each random variable of interest separately, it may be more useful to study them jointly in order to discover relationships between them.
In the course we develop properties of probability distributions and present techniques which enable random variables to be transformed or combined. Important applications of probability are discussed and some remarkable general results derived.

#### Outline Of Syllabus

Review of probability; conditional probability and independence; discrete and continuous random variables; simulation and probability integral transform; bivariate distributions; covariance and correlation; expectation; probability and moment generating functions. Illustrations will be carried out using the statistical package R.

#### Learning Outcomes

##### Intended Knowledge Outcomes

Students will gain familiarity with a range of discrete and continuous probability laws, and will learn how to compute moments of random variables and the distributions of transformed random variables. Students will have some knowledge of bivariate distributions, correlation and covariance, expectation, moment and probability generating functions.

##### Intended Skill Outcomes

Students will be able to understand, use and calculate probabilities and expectations for a range of bivariate distributions. They will also be able to work with probability and moment generating functions. They will be able to use bivariate integration for joint distributions. They will be able to use R to analyse joint distributions and transformations of random variables.

• Cognitive/Intellectual Skills
• Numeracy : Assessed
• Self Management
• Personal Enterprise
• Problem Solving : Assessed
• Interaction
• Communication
• Written Other : Assessed

#### Teaching Methods

##### Teaching Activities
Category Activity Number Length Student Hours Comment
Scheduled Learning And Teaching ActivitiesLecture21:002:00Revision lectures
Scheduled Learning And Teaching ActivitiesLecture61:006:00Problem classes
Guided Independent StudyAssessment preparation and completion111:0011:00Revision for unseen Exam
Guided Independent StudyAssessment preparation and completion11:301:30Unseen Exam
Guided Independent StudyAssessment preparation and completion23:006:00Revision for class test
Scheduled Learning And Teaching ActivitiesLecture221:0022:00Formal lectures
Guided Independent StudyAssessment preparation and completion21:002:00Class test
Guided Independent StudyAssessment preparation and completion26:0012:00Written assignments and CBAs
Scheduled Learning And Teaching ActivitiesPractical21:002:00N/A
Scheduled Learning And Teaching ActivitiesDrop-in/surgery61:006:00Drop-ins in the lecture room
Guided Independent StudyIndependent study21:002:00Assignment review
Guided Independent StudyIndependent study127:3027:30Studying, practising and gaining understanding of course material
Total100:00
##### Jointly Taught With
Code Title
MAS3304Foundations of Probability
##### Teaching Rationale And Relationship

Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the studentsâ€™ abilities at applying the theory to solving problems. Drop-ins are used to identify and resolve specific queries raised by students and to allow students to receive individual feedback on marked work. Practicals are used both for solution of problems and work requiring extensive computation and to give insight into the ideas/methods studied. Office hours provide an opportunity for more direct contact between individual students and the lecturer.

#### Assessment Methods

The format of resits will be determined by the Board of Examiners

##### Exams
Description Length Semester When Set Percentage Comment
Written Examination901A80unseen
##### Exam Pairings
Module Code Module Title Semester Comment
MAS3304Foundations of Probability1N/A
##### Other Assessment
Description Semester When Set Percentage Comment
Prob solv exercises1M10Written assignments and computer based assessments
Prob solv exercises1M10Coursework tests
##### Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. Coursework assignments (approximately 2 assignments of approximately equal weight) and two coursework tests (in class) allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; this is thus formative as well as summative assessment. The coursework assignments may be written assignments, computer based assessments or a combination of the two, and in the case of combined assessments the deadlines for the two parts will not necessarily be the same.

N/A