
Academic Year:  2015/6 
Owning Department/School:  Department of Mathematical Sciences 
Credits:  6 
Level:  Intermediate (FHEQ level 5) 
Period: 
Semester 1 
Assessment Summary:  EX 100% 
Assessment Detail: 

Supplementary Assessment: 
MA20224 Mandatory extra work (where allowed by programme regulations) 
Requisites:  Before taking this module you must take MA10207 AND take MA10211 AND take MA10212 
Description:  Aims: To introduce some fundamental topics in probability theory, including conditional expectation as a random variable and three classical limit theorems of probability. To present the main properties of some fundamental stochastic processes, including random walks, branching processes and Poisson processes. To demonstrate the use of generating function techniques. Learning Outcomes: After taking this unit, students should be able to: * work effectively with conditional expectation; * apply the classical limit theorems of probability; * determine whether infinitely or finitely many events occur by applying the BorelCantelli Lemmas; * perform computations on random walks, branching processes and Poisson processes; * use generating function techniques for effective calculations. Skills: Numeracy T/F A Problem Solving T/F A Written and Spoken Communication F (in tutorials) Content: Review of probability measures: event spaces, Borel σalgebra, probability spaces, properties, continuity of probability. Fundamental model: uniform probability measure on [0,1] and LebesgueBorel Theorem (statement). Independence. BorelCantelli Lemmas. Random variables. Expectation. Monotone Convergence Theorem (statement). Conditional probability and expectation. Conditional expectation with respect to a random variable. Generating functions: PGFs, MGFs, Characteristic functions and Laplace transforms. Convergence of generating functions. Types of convergence. Weak Law of Large Numbers. Strong Law of Large Numbers (proof of special case). Central Limit Theorem (sketch proof). Random walks. First return times. First passage times. Gambler's ruin. Reflection principle. Ballot Theorem. Recurrence of random walks. Branching processes: discrete time GaltonWatson process, extinction probabilities, population size. Poisson processes: characterisations, interarrival times, gamma distributions, thinning and conditional uniformity. Poisson point processes (PPPs) on R^{n}. Examples of PPPs with nonconstant intensities. 
Programme availability: 
MA20224 is Compulsory on the following programmes:Department of Mathematical Sciences
MA20224 is Optional on the following programmes:Department of Mathematical Sciences
