Probability Theory I

Summary module description:
The module rigorously introduces basic concepts of probability from a mathematical perspective. It aims to equip the students with a basic knowledge in probability which will reveal the interplay between probability theory and fundamental areas of mathematics such as analysis and algebra, will allow students to formulate general real or abstract problems in a probabilistic model and will unravel the fundamentals which statistical methods are built on. In more detail the module will be developed around the concepts of probability distributions, random variables, independence, sums of random variables, limit laws and their application (Central Limit Theorem and laws of large numbers), and structures that depend on the present to study the future evolution of stochastic phenomena (Markov chains).

Aims:
This module aims to introduce students to some of the fundamental concepts and results of probability. It covers random variables together with probability distributions as the fundamental objects of probability theory, the concept of dependence/independence, which lead then to fundamental asymptotic results as well as a first introduction of stochastic processes such as Markov chains.

Assessable learning outcomes:
By the end of the module the students are expected to be able to:
- Identify and demonstrate understanding of the main concepts and definitions in probability theory
- Without the help of notes to state all and prove some of the main results
- Identify and formulate problems in terms of probability and solve them to build up a simple stochastic model
- Use the main results to do various approximations.

Additional outcomes:
At the end of the module students will have some insight in the interrelation between analysis, linear algebra and probability and their relevance for applications.

Outline content:
Random variables with uniform distribution, continuous and discrete, distributions with densities and weights, expectation of random variables, the concept of independence, sums of independent random variables, concepts of convergence of random variables, dependent random variables and conditional distributions, random transitions and Markov chains.

Brief description of teaching and learning methods:
Lectures, supported by problem sheets and lecture-based tutorials.

Summative Assessment Methods:
Written exam 75%
Set exercise 25%

Other information on summative assessment:
Two assignments and one examination

Formative assessment methods:
Problem sheets.

Penalties for late submission:
The Module Convener will apply the following penalties for work submitted late, in accordance with the University policy.
where the piece of work is submitted up to one calendar week after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for the piece of work will be deducted from the mark for each working day (or part thereof) following the deadline up to a total of five working days;
where the piece of work is submitted more than five working days after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.

The University policy statement on penalties for late submission can be found at: http://www.reading.ac.uk/web/FILES/qualitysupport/penaltiesforlatesubmission.pdf
You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Length of examination:
2 hours

Requirements for a pass:
A mark of 40% overall