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MA20224: Probability 2A

[Page last updated: 15 October 2020]

Follow this link for further information on academic years Academic Year: 2020/1
Further information on owning departmentsOwning Department/School: Department of Mathematical Sciences
Further information on credits Credits: 6      [equivalent to 12 CATS credits]
Further information on notional study hours Notional Study Hours: 120
Further information on unit levels Level: Intermediate (FHEQ level 5)
Further information on teaching periods Period:
Semester 1
Further information on unit assessment Assessment Summary: EX 100%
Further information on unit assessment Assessment Detail:
  • Examination (EX 100%)
Further information on supplementary assessment Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
Further information on requisites 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 Borel-Cantelli 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 σ-algebra, probability spaces, properties, continuity of probability. Fundamental model: uniform probability measure on [0,1] and Lebesgue-Borel Theorem (statement). Independence. Borel-Cantelli 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 Galton-Watson process, extinction probabilities, population size. Poisson processes: characterisations, inter-arrival times, gamma distributions, thinning and conditional uniformity. Poisson point processes (PPPs) on Rn. Examples of PPPs with non-constant intensities.
Further information on programme availabilityProgramme availability:

MA20224 is Compulsory on the following programmes:

Department of Mathematical Sciences
  • USMA-AFB01 : BSc(Hons) Mathematics and Statistics (Year 2)
  • USMA-AAB02 : BSc(Hons) Mathematics and Statistics with Study year abroad (Year 2)
  • USMA-AKB02 : BSc(Hons) Mathematics and Statistics with Year long work placement (Year 2)
  • USMA-AFB05 : BSc(Hons) Statistics (Year 2)
  • USMA-AAB06 : BSc(Hons) Statistics with Study year abroad (Year 2)
  • USMA-AKB06 : BSc(Hons) Statistics with Year long work placement (Year 2)

MA20224 is Optional on the following programmes:

Department of Economics
  • UHES-AFB04 : BSc(Hons) Economics and Mathematics (Year 2)
  • UHES-AFB04 : BSc(Hons) Economics and Mathematics (Year 3)
  • UHES-AAB04 : BSc(Hons) Economics and Mathematics with Study year abroad (Year 2)
  • UHES-AAB04 : BSc(Hons) Economics and Mathematics with Study year abroad (Year 4)
  • UHES-AKB04 : BSc(Hons) Economics and Mathematics with Year long work placement (Year 2)
  • UHES-AKB04 : BSc(Hons) Economics and Mathematics with Year long work placement (Year 4)
  • UHES-ACB04 : BSc(Hons) Economics and Mathematics with Combined Placement and Study Abroad (Year 2)
  • UHES-ACB04 : BSc(Hons) Economics and Mathematics with Combined Placement and Study Abroad (Year 4)
Department of Mathematical Sciences
  • USMA-AFB15 : BSc(Hons) Mathematical Sciences (Year 2)
  • USMA-AFB15 : BSc(Hons) Mathematical Sciences (Year 3)
  • USMA-AAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 2)
  • USMA-AAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 4)
  • USMA-AKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 2)
  • USMA-AKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 4)
  • USMA-AFB13 : BSc(Hons) Mathematics (Year 2)
  • USMA-AFB13 : BSc(Hons) Mathematics (Year 3)
  • USMA-AAB14 : BSc(Hons) Mathematics with Study year abroad (Year 2)
  • USMA-AAB14 : BSc(Hons) Mathematics with Study year abroad (Year 4)
  • USMA-AKB14 : BSc(Hons) Mathematics with Year long work placement (Year 2)
  • USMA-AKB14 : BSc(Hons) Mathematics with Year long work placement (Year 4)
  • USMA-AFM14 : MMath(Hons) Mathematics (Year 2)
  • USMA-AFM14 : MMath(Hons) Mathematics (Year 3)
  • USMA-AAM15 : MMath(Hons) Mathematics with Study year abroad (Year 2)
  • USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 2)
  • USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 4)

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