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MA20301: Probability 2

[Page last updated: 09 August 2024]

Academic Year: 2024/25
Owning Department/School: Department of Mathematical Sciences
Credits: 12 [equivalent to 24 CATS credits]
Notional Study Hours: 240
Level: Intermediate (FHEQ level 5)
Period:
Academic Year
Assessment Summary: EXCB 100%
Assessment Detail:
  • Examination 1 (EXCB 50%)
  • Examination 2 (EXCB 50%)
Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
Requisites: Before taking this module you must take MA10211 AND take MA10212
In taking this module you cannot take MA20224 OR take MA20225
Learning Outcomes: By the end of the course you will be able to
  • Perform standard estimation procedures and hypothesis tests for parameters in a variety of statistical models
  • Analyse a variety of models for normally distributed data
  • Apply the theory for normal linear models in carrying out data analyses using R



Synopsis: This unit will introduce you to mathematical modelling using stochastic processes. You will explore Markov chains and learn to analyse their long-term behaviour. You will cover applications of these models in areas ranging from card shuffling to biological processes and phenomena in economics. In the second semester, you will learn about the fundamentals of probability theory. These fundamentals allow you to formalize many problems that can be modelled using techniques from probability. Moreover, you will develop various mathematical tools to work with limits in a probabilistic setting. This unit gives you the toolkit necessary for the more advanced probability units.

Aims: On completing the unit, you will be able to:
  • Find transition probabilities, hitting probabilities, expected hitting times, and invariant distributions of discrete time Markov chains
  • Classify the states of a Markov chain
  • Calculate the limiting behaviour of Markov chains
  • Set up simple Markov chain models
  • Formalize problems in a probabilistic framework
  • Understand and prove the convergence of random variables
  • Study standard branching processes
  • Study random walks in dimension one


Skills: Numeracy T/F A Problem Solving T/F A Written and Spoken Communication F (in tutorials)

Content: In Semester 1: Discrete-time Markov property; transition matrix, n-step transition probabilities; basic examples, including random walk; hitting probabilities and expected hitting times; classification of states; convergence to equilibrium and ergodic theorem; symmetrizability. Optional: Numerical exploration of Monte-Carlo method. Examples will be chosen from physical and biological processes, economics, telecommunications, and other application areas. In Semester 2: Theoretical content: Kolmogorov axioms; measure theory essentials: discrete and continuous random variables, expectation of random variables and convergence theorems; modes of convergence of random variables; Borel-Cantelli lemmas; law of large numbers; central limit theorem (without proof); conditional expectation. Main models used as illustration: one-dimensional random walks; branching process;

Course availability:

MA20301 is Optional on the following courses:

Department of Economics
  • UHES-AFB04 : BSc(Hons) Economics and Mathematics (Year 3)
  • 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 4)
  • 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 3)
  • USMA-AAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 4)
  • USMA-AKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 4)
  • USMA-AFB13 : BSc(Hons) Mathematics (Year 3)
  • USMA-AAB14 : BSc(Hons) Mathematics with Study year abroad (Year 4)
  • USMA-AKB14 : BSc(Hons) Mathematics with Year long work placement (Year 4)
  • USMA-AFB20 : BSc(Hons) Mathematics, Statistics, and Data Science (Year 3)
  • USMA-AAB20 : BSc(Hons) Mathematics, Statistics, and Data Science with Study year abroad (Year 4)
  • USMA-AKB20 : BSc(Hons) Mathematics, Statistics, and Data Science with Industrial Placement (Year 4)
  • USMA-AFM14 : MMath(Hons) Mathematics (Year 3)
  • USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 4)

Notes:

  • This unit catalogue is applicable for the 2024/25 academic year only. Students continuing their studies into 2025/26 and beyond should not assume that this unit will be available in future years in the format displayed here for 2024/25.
  • Courses 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 pre-requisite rules.
  • Find out more about these and other important University terms and conditions here.