A stochastic process is one that involves random variables. A large number of practical systems within industry, commerce, finance, biology, nuclear physics and epidemiology can be described as stochastic and analysed using the techniques developed in this module. The systems considered may exist in any one of a finite or possibly countably infinite, number of states. The state of a system may be examined continuously through time or at fixed and regular intervals of time.
You will study processes whose changes of state through time are governed by probabilistic laws, and you will learn how models of such processes can be applied in practice. Module MTH1004 Probability, Statistics and Data is an essential prerequisite, while MTH2006 Statistical Modelling & Inference is desirable.
Prerequisite module: MTH1004 Probability, Statistics and Data or equivalent.
The probability models considered in this module have a common thread running through them: that the behaviour of the system under consideration depends only on the state of the system at a particular point in time and a probablistic description of how the state of the system may change from one point in time to the next. The systems considered may exist in any one of a finite (or possibly countably infinite) number of possible states and the state of the system may be examined continuously through time or at fixed (and regular) intervals of time. A large number of practical systems within industry, commerce, finance, biology, nuclear physics and epidemiology, can be described and analysed using the techniques developed in this module.
INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)
On successful completion of this module you should be able to:
Module Specific Skills and Knowledge:
1. Demonstrate enhanced methodologies for tackling probabilistic problems;
2. Show awareness of a number of processes and systems whose behaviour through time are governed by probabilistic laws;
3. Construct and apply models describing that behaviour.
Discipline Specific Skills and Knowledge
4. Exhibit familiarity with the concept of random behaviour and the facility to analyse queues - skills which will be applied in later modules;
5. Display enhanced facility with the fundamental mathematical techniques of finite and infinite summation, and of differential and integral calculus.
Personal and Key Transferable / Employment Skills and Knowledge:
6. Reveal enhanced analytical skills, numerical skills, reasoning skills, problem-solving skills, time-management skills and facility to understand complex and abstract ideas.
SYLLABUS PLAN - summary of the structure and academic content of the module
- Probability generating functions (PGFs): definition, basic properties and illustrative examples of PGFs;
- moments of random sums of random variables;
- branching processes: definition, PGF and moments of the population in generation n of a branching process;
- probability of ultimate extinction;
- stochastic size of original population;
- Poisson processes: definition;
- memoryless property;
- Erlang distribution of time to the nth event;
- Poisson distribution of number of events in a given period of time;
- binomial distribution of number r of events in t given n in T;
- beta distribution of time t to rth event given n events in T;
- combining and decomposing independent Poisson processes;
- queueing theory: differential equations for the transient behaviour of models with state-dependent Markov arrival and departure processes;
- derivation of the steady state behaviour of this model;
- existence conditions for steady state;
- specific queueing models: fixed arrival rate, finite source population, customer baulking behaviour, one or more servers, finite system capacity, non-queueing systems which can be modelled as queues;
- mean number of customers in the system/queueing;
- mean time spent in the system/queueing;
- statement and proof of Little's formula;
- distribution of time spent in system/queueing given first come first served;
- Markov processes: Markov property;
- time homogeneity;
- stochastic matrices;
- Chapman-Kolmogorov equations;
- classification of states: accessible, communicating, transient, recurrent, periodic, aperiodic;
- Ergodic Markov chains;
- renewal theorem;
- mean recurrence time;
- necessary/sufficient conditions for the system to tend to a steady state;
- random walks: definition of a random walk with absorbing/reflecting/elastic barriers;
- statement of, solution for and mean time to finish for the Gambler's Ruin problem.
