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# Probability & Markov Chains - MAT00045I

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• Department: Mathematics
• Module co-ordinator: Dr. Yue Zhao
• Credit value: 20 credits
• Credit level: I
• Academic year of delivery: 2023-24

## Module summary

This module will equip students with the theoretical foundations of data science.

## Professional requirements

Used for IFoA exemption purposes.

## Related modules

• None

### Prohibited combinations

• None

• Foundations & Calculus
• Multivariable Calculus & Matrices
• Introduction to Probability and Statistics

## Module will run

Occurrence Teaching period
A Semester 1 2023-24

## Module aims

This module will give students a theoretical and mathematically formal framework for understanding the foundations of data science. Students will learn how to work with multiple random variables in a variety of settings: joint and conditional distributions will be developed, along with estimators and convergence theorems, and Markov chains will be introduced to deal with random variables indexed by discrete time. Further familiarity with the statistical software R will be developed throughout.

## Module learning outcomes

By the end of the module, students will be able to:

1. Perform computations involving the joint and conditional distributions and the related expectations.

2. Compute generating functions of standard distributions, apply them to obtain expectation and variance, and identify the distribution such as that of a sum of independent random variables with the said generating functions.

3. Apply limit theorems such as the Weak Law of Large Numbers and the Central Limit Theorem to deduce the asymptotic properties of a random variable sequence such as unbiasedness, consistency and asymptotic normality.

4. Estimate parameters of standard distributions following the maximum likelihood and the method of moments approach, and judge the quality of the resulting estimators.

5. Calculate absorption probabilities for discrete Markov chains.

6. Calculate, and interpret, stationary distributions for discrete Markov chains

## Module content

• Joint and conditional distributions (covering discrete and continuous distributions, in particular the Multivariate Normal)

• Generating functions (moment and probability generating functions)

• Modes of convergence and limit theorems (including WLLN and CLT)

• Maximum likelihood and method of moments estimation

• Further properties of estimators (for instance, precision measure (e.g., MSE), Cramer-Rao)

• Markov chains, up to convergence to equilibrium and ergodic theorem

• Brief introduction to MCMC (time permitting

## Assessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Probability & Markov Chains
3 hours 100

### Special assessment rules

None

There will be five formative assignments with marked work returned in the seminars. At least one of them will contain a longer written part, done in LaTeX.

### Reassessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Probability & Markov Chains
3 hours 100

## Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.