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## Course Overview

Probability theory is concerned with the analysis of random phenomena by providing an abstract mathematical framework to study them within the language of set theory. This is done by the concepts of "probability spaces" and "random variables". The theory began in the 16th century in attempts to analyze games of chance; In 1812 Pierre Simon Laplace wrote: "It is remarkable that a science which began with the consideration of games of chance should have become the most important object of human knowledge."

The course is recommended to anyone interested in the foundations and applications of mathematics.

### Course Details

Study Type Level Undergraduate 2 First Sub Session 15 credits (7.5 ECTS credits) None. No Professor Vassili Gorbunov

### Qualification Prerequisites

• Programme Level 2

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### Course Description

• Sample spaces and the probability function.
• Application of conditional probability and the partition theorem.
• Random variables and distribution functions.
• Expectation and variance.
• Limit theorems and their application.
• Generating functions and their uses.
• Branching processes.
• Markov Chains.

Syllabus

• Sample spaces and (discrete) probability spaces.
• Conditional probability and the partition theorem.
• Random variables and distribution functions.
• Expectation and variance of random variables. Linearity of the expectation. The covariance.
• Independent random variables.
• Frequently used distributions: Bernoulli, binomial, geometric, hypergeometric, Poisson.
• Limit theorems (Chebyshef's inequality, the weak law of large numbers, the central limit theorem).
• Branching processes (if time permits).

### Contact Teaching Time

Information on contact teaching time is available from the course guide.

### Teaching Breakdown

Details, including assessments, may be subject to change until 31 August 2023 for 1st half-session courses and 22 December 2023 for 2nd half-session courses.

### Summative Assessments

1st Attempt: 1 two-hour written examination (80%), in-course assessment (20%).

Resit: 1 two-hour written examination paper, maximum resit (100%) and resit (80%) with (20%) in-course assessment (20%).

### Formative Assessment

Informal assessment of weekly homework through discussions in tutorials.

### Feedback

In-course assignments will normally be marked within one week and feedback provided to students in tutorials. Students will be invited to contact the Course Coordinator for feedback on the final examination.

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