MA20224: Probability 2A
[Page last updated: 03 August 2022]
Academic Year:  2022/23 
Owning Department/School:  Department of Mathematical Sciences 
Credits:  6 [equivalent to 12 CATS credits] 
Notional Study Hours:  120 
Level:  Intermediate (FHEQ level 5) 
Period: 
 Semester 1

Assessment Summary:  EX 100% 
Assessment Detail:  
Supplementary Assessment: 
 Likeforlike reassessment (where allowed by programme regulations)

Requisites: 
Before taking this module you must take MA10211 AND take MA10212

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.

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.

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
 USMAAFB01 : BSc(Hons) Mathematics and Statistics (Year 2)
 USMAAAB02 : BSc(Hons) Mathematics and Statistics with Study year abroad (Year 2)
 USMAAKB02 : BSc(Hons) Mathematics and Statistics with Year long work placement (Year 2)
MA20224 is Optional on the following programmes:
Department of Economics
 UHESAFB04 : BSc(Hons) Economics and Mathematics (Year 2)
 UHESAFB04 : BSc(Hons) Economics and Mathematics (Year 3)
 UHESAAB04 : BSc(Hons) Economics and Mathematics with Study year abroad (Year 2)
 UHESAAB04 : BSc(Hons) Economics and Mathematics with Study year abroad (Year 4)
 UHESAKB04 : BSc(Hons) Economics and Mathematics with Year long work placement (Year 2)
 UHESAKB04 : BSc(Hons) Economics and Mathematics with Year long work placement (Year 4)
 UHESACB04 : BSc(Hons) Economics and Mathematics with Combined Placement and Study Abroad (Year 2)
 UHESACB04 : BSc(Hons) Economics and Mathematics with Combined Placement and Study Abroad (Year 4)
Department of Mathematical Sciences
 USMAAFB15 : BSc(Hons) Mathematical Sciences (Year 2)
 USMAAFB15 : BSc(Hons) Mathematical Sciences (Year 3)
 USMAAAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 2)
 USMAAAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 4)
 USMAAKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 2)
 USMAAKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 4)
 USMAAFB13 : BSc(Hons) Mathematics (Year 2)
 USMAAFB13 : BSc(Hons) Mathematics (Year 3)
 USMAAAB14 : BSc(Hons) Mathematics with Study year abroad (Year 2)
 USMAAAB14 : BSc(Hons) Mathematics with Study year abroad (Year 4)
 USMAAKB14 : BSc(Hons) Mathematics with Year long work placement (Year 2)
 USMAAKB14 : BSc(Hons) Mathematics with Year long work placement (Year 4)
 USMAAFB20 : BSc(Hons) Mathematics, Statistics, and Data Science (Year 2)
 USMAAAB20 : BSc(Hons) Mathematics, Statistics, and Data Science with Study year abroad (Year 2)
 USMAAKB20 : BSc(Hons) Mathematics, Statistics, and Data Science with Industrial Placement (Year 2)
 USMAAFM14 : MMath(Hons) Mathematics (Year 2)
 USMAAFM14 : MMath(Hons) Mathematics (Year 3)
 USMAAAM15 : MMath(Hons) Mathematics with Study year abroad (Year 2)
 USMAAKM15 : MMath(Hons) Mathematics with Year long work placement (Year 2)
 USMAAKM15 : MMath(Hons) Mathematics with Year long work placement (Year 4)

Notes:  This unit catalogue is applicable for the 2022/23 academic year only. Students continuing their studies into 2023/24 and beyond should not assume that this unit will be available in future years in the format displayed here for 2022/23.
 Programmes and units are subject to change in accordance with normal University procedures.
 Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any prerequisite rules.
 Find out more about these and other important University terms and conditions here.
