# Probability STATS4060

**Academic Session:**2018-19**School:**School of Mathematics and Statistics**Credits:**10**Level:**Level 4 (SCQF level 10)**Typically Offered:**Semester 1**Available to Visiting Students:**Yes**Available to Erasmus Students:**Yes

#### Short Description

This course provides a structured development of probability theory for students in Honours Mathematics. The pace of the course is brisk, as it begins from the assumption that students have little prior exposure to probability yet reaches advanced concepts by the end.

#### Timetable

20 lectures

5 tutorials

#### Requirements of Entry

None.

#### Excluded Courses

STATS5024 Probability (Level M)

STATS2002 Statistics 2R: Probability

STATS2005 Statistics 2X: Probability II

#### Co-requisites

None.

#### Assessment

90-minute, end-of-course examination (85%)

coursework (15%)

**Main Assessment In:** April/May

**Are reassessment opportunities available for all summative assessments?** Not applicable

Reassessments are normally available for all courses, except those which contribute to the Honours classification. For non Honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions for this course are described below.

#### Course Aims

The aim of this course is to provide a structured development of probability theory, with an emphasis on the theory of random variables and random vectors.

#### Intended Learning Outcomes of Course

By the end of this course students will be able to:

■ state the Axioms of Probability and use them to prove basic propositions in probability theory;

■ use probability mass functions and probability density functions in one or two dimensions to compute probabilities and percentiles in particular cases;

■ write down general definitions of moments in one or more dimensions (including the vector of expected values and the variance-covariance matrix), derive general properties from these definitions, and compute moments in particular univariate and bivariate cases;

■ recognise standard discrete and continuous probability distributions in a context, and use them to obtain probabilities, percentiles and moments;

■ use the joint distribution of a random vector to derive the marginal or conditional distribution of one of the component variables;

■ determine whether two or more random variables are independent;

■ write down general definitions of the probability generating function, moment generating function and characteristic function for a random variable, derive general properties from these definitions, and compute the functions in particular cases;

■ find the distribution of functions of random variables in one or two dimensions;

■ use standard methods to derive the exact distribution of the sum of a sequence of random variables;

■ state and use the laws of large numbers and the central limit theorem;

■ state and use properties of the multinomial and Multivariate Normal (MVN) distributions.

#### Minimum Requirement for Award of Credits

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.