# Module Information

Module Identifier
MA37410
Module Title
Probability and Stochastic Processes
Academic Year
2023/2024
Co-ordinator
Semester
Semester 2
Pre-Requisite
Reading List
Other Staff

#### Assessment

Assessment Type Assessment length / details Proportion
Semester Exam 2 Hours   (Written Examination)  100%
Supplementary Exam 2 Hours   (Written Examination)  100%

### Learning Outcomes

On completion of this module, a student should be able to:
1. use conditional expectations and variances in a wide range of problems involving stochastic modelling;
2. calculate generating functions of standard univariate distributions and use them to calculate moments and distributions of sums and mixtures;
3. use generating functions in limiting arguments, including the proofs of the Central Limit Theorem and the Weak Law of Large Numbers;
4. explain the need for careful mathematical argument in probability theory;
5. describe and use the recurrence relation for generation sizes in a Branching Process and determine the probability of ultimate extinction;
6. determine the structure of the state space of a Markov chain from its transition matrix;
7. classify the states of Markov chain by period and limiting behaviour, including the calculation of appropriate limiting probabilities;
8. explain the Ergodic Theorem for Markov chains and use it to calculate limiting time averages from limiting probabilities.

### Brief description

Probability theory is one of the great achievements of 20th Century mathematics and a thorough grounding in it is necessary for further study of stochastic modelling and mathematical statistics. This module provides that grounding, and proves the limit theorems which provide foundations for so many large-sample statistical methods. Stochastic Processes are processes that develop in time in a way that is affected by chance, and are used as models for many situations ranging from physics to manpower planning. This module will look at two different sorts of Stochastic Processes, namely Markov chains (whose future development depends only on their present state not their past history) and Branching Processes.

### Aims

To introduce important tools of probability, including conditional expectations and generating functions; to provide students with experience of their use in a variety of problems including proofs of the Central Limit Theorem and the Weak Law of Large Numbers. To introduce students to Branching Processes, to Markov Chains (discrete time, discrete state), and through them to Stochastic Processes in general.

### Content

1. CONDITIONAL EXPECTATIONS: Revision of joint and conditional distributions; existence of expectations; E(X|Y); E[E(X|Y)] = EX; conditional variance formula; random sums and applications.
2. GENERATING FUNCTIONS: Moment generating function (mgf): basic properties, evaluation of moments, distribution of independent sums. Weak Law of Large Numbers; Central Limit Theorem, applications. Probability generating function: basic properties, relationship to mgf, evaluation of probabilities and moments, random sums.
3. BRANCHING PROCESSES: Definition and introduction. Generating functions for the generation sizes. Extinction probabilities.
4. MARKOV CHAINS: Introduction - the transition matrix. Irreducible classes. Periodicity. Classification of states by their limiting behaviour. Stationary distributions. Hitting probabilities and expected hitting times. An ergodic theorem.

### Module Skills

Skills Type Skills details
Adaptability and resilience Students are expected to develop their own approach to time-management and to use the feedback from marked work to support their learning.
Co-ordinating with others Students will be encouraged to work in groups to solve problems.
Creative Problem Solving .The assignments will give the students opportunities to show creativity in finding solutions and develop their problem solving skills.
Digital capability Use of the internet, Blackboard, and mathematical packages will be encouraged to enhance their understanding of the module content and examples of application
Professional communication Students will be expected to submit clearly written solutions to set exercises.
Subject Specific Skills Broadens exposure of students to topics in mathematics, and an area of application that they have not previously encountered.

### Notes

This module is at CQFW Level 6