Block C: Random Processes and Simulations

This book is concerned with computer simulations of random processes such as, for example, gambling games and models of the growth of biological populations. Such a subject relies heavily on computing, which is achieved using Maple, although pencil-and-paper analysis also enters into this block. The block starts by introducing some basic notions of randomness and probability, illustrated by examples taken from coin-tossing and die-rolling experiments, followed by some simple models of population dynamics. Then follows a discussion of random walks and the central limit theorem, which relates to properties of random systems with many random variables. This theory is applied to the integration of functions, the so-called Monte Carlo method of integration. The rest of the block confines itself to exploratory simulations of a range of systems starting with the standard map (a type of dynamical system showing chaotic behaviour) and models for infectious disease spreading (modelled either by sets of differential equations or by random processes). Random models of traffic flow are simulated, showing the emergence of traffic jams. Finally the block closes with a detailed investigation of the dynamics of the trebuchet, which is a type of mediaeval siege engine.This block contains exercises and activities which use a commercial software package called Maple , you are advised that you will need access to this software to allow the exercises and activities to be completed.