# Statistics BSc/MSci

## Stochastic Processes STATS4024

**Academic Session:**2020-21**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

To provide a good understanding of the key concepts of stochastic processes in various settings: discrete time and finite state space; discrete time and countable state space; continuous time and countable state space.

### Timetable

Two lectures per week for 10 weeks and fortnightly tutorials.

### Requirements of Entry

STATS2002 Statistics 2R, STATS2005 Statistics 2X

The normal requirement is that students should have been admitted to an Honours- or Master's-level programme in Statistics, and have performed satisfactorily in Statistics 3H or Statistics 3M.

### Excluded Courses

STATS5026 - Stochastic Processes (M)

### Co-requisites

Courses prescribed in the Honours or Master's programme to which the student has been admitted.

### Assessment

90-minute, end-of-course examination (100%)

**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 aims of this course are to provide a good understanding of the key concepts of stochastic processes in various settings: discrete time and finite state space; discrete time and countable state space; continuous time and countable state space.

### Intended Learning Outcomes of Course

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

■ Explain the concept of a homogeneous Markov chain;

■ Define what it means for a matrix to be stochastic and discuss the consequences this has for the eigenvalues;

■ Explain what it means for a state to be absorbing, periodic, persistent, transient, or ergodic;

■ Explain the difference between a stationary and a limiting distribution;

■ Decide whether a Markov chain has a unique limiting distribution;

■ Describe the gambler's ruin problem in terms of a discrete-time Markov chain;

■ Calculate the probability of ruin and the expected duration of a game in the gambler's ruin problem;

■ Explain the difference between a homogeneous and an inhomogeneous linear difference equation and the standard procedure to solve them;

■ Derive the probability distribution of a random walk;

■ Calculate the first return of a symmetric random walk;

■ Define the concept of a homogeneous Poisson process, and derive the form of the distribution of the inter-arrival times;

■ Calculate the expected length and waiting time for a queue in which arrivals form a Poisson process;

■ Define the concepts of a reliability function and a hazard function;

■ Calculate the expected number of renewals in a renewal process;

■ Explain the difference between a discrete-time and a continuous-time Markov chain and explain the concept of a rate matrix;

■ Decide whether a birth-death process has a stationary distribution.

### Minimum Requirement for Award of Credits

N/A