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## MA20224: Probability 2A

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 Borel-Cantelli 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, inter-arrival times, gamma distributions. Poisson point processes (PPPs). Compound PPPs.
Programme availability:

#### MA20224 is Compulsory on the following programmes:

Department of Mathematical Sciences
• USMA-AFB01 : BSc (hons) Mathematics and Statistics (Full-time) - Year 2
• USMA-AKB02 : BSc (hons) Mathematics and Statistics (Full-time with Thick Sandwich Placement) - Year 2
• USMA-AAB02 : BSc (hons) Mathematics and Statistics with Study Year Abroad (Full-time with Study Year Abroad) - Year 2
• USMA-AFB05 : BSc (hons) Statistics (Full-time) - Year 2
• USMA-AKB06 : BSc (hons) Statistics (Full-time with Thick Sandwich Placement) - Year 2
• USMA-AAB06 : BSc (hons) Statistics with Study Year Abroad (Full-time with Study Year Abroad) - Year 2

#### MA20224 is Optional on the following programmes:

Department of Mathematical Sciences
• USMA-AFB15 : BSc (hons) Mathematical Sciences (Full-time) - Year 2
• USMA-AFB15 : BSc (hons) Mathematical Sciences (Full-time) - Year 3
• USMA-AKB16 : BSc (hons) Mathematical Sciences (Full-time with Thick Sandwich Placement) - Year 2
• USMA-AKB16 : BSc (hons) Mathematical Sciences (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AAB16 : BSc (hons) Mathematical Sciences with Study Year Abroad (Full-time with Study Year Abroad) - Year 2
• USMA-AAB16 : BSc (hons) Mathematical Sciences with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AFB13 : BSc (hons) Mathematics (Full-time) - Year 2
• USMA-AFB13 : BSc (hons) Mathematics (Full-time) - Year 3
• USMA-AKB14 : BSc (hons) Mathematics (Full-time with Thick Sandwich Placement) - Year 2
• USMA-AKB14 : BSc (hons) Mathematics (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AAB14 : BSc (hons) Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 2
• USMA-AAB14 : BSc (hons) Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AFM14 : MMath Mathematics (Full-time) - Year 2
• USMA-AAM15 : MMath Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 2

NB. Programmes and units are subject to change at any time, in accordance with normal University procedures.