Department of Mathematical Sciences, Unit Catalogue 2011/12 

Credits:  6 
Level:  Intermediate (FHEQ level 5) 
Period: 
Semester 1 
Assessment:  EX 100% 
Supplementary Assessment:  MA20224 Mandatory extra work (where allowed by programme regulations) 
Requisites:  Before taking this unit 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. To give a basic introduction to martingales and demonstrate their use. Learning Outcomes: After taking this unit, students should be able to: * Work effectively with conditional expectation. * Apply the classical limit theorems of probability. * 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: Generating functions. Convergence of generating functions. Central Limit Theorem.Weak Law of large numbers. Strong Law of Large Numbers. Modes of convergence. Infinitely many events and BorelCantelli lemmas. Conditional expectation with respect to a random variable. Simple martingales: definition, statement of convergence theorem, discussion of optional stopping theorem. Random walks. First return times. First passage times. Reflection principle. Gambler's ruin. Recurrence of random walks. Branching processes: extinction probabilities, population growth. Poisson processes: characterisations, interarrival times, gamma distributions. Poisson point processes (PPPs). Compound PPPs. 
Programme availability: 
MA20224 is Compulsory on the following programmes:Department of Mathematical Sciences
MA20224 is Optional on the following programmes:Department of Mathematical Sciences
