Mathematics

Probability Models (L6)

Module code: G1100
Level 6
15 credits in autumn semester
Teaching method: Lecture
Assessment modes: Coursework, Unseen examination

This is the first optional Probability module, where we begin to study stochastic (random) processes. In this module our processes are in discrete time, so a stochastic process is a sequence of random variables where we can view it as a (discrete) time evolution of a random experiment. Stochastic processes are used to model several phenomena with uncertain outcomes, such as stock values, the weather, or the profit evolution of a gambler as they evolve through time.

You will develop basic tools for the study of such processes in discrete time (which makes it less technical). The central objects of study are Markov chains and their various models. These include:

• branching processes
• finite Markov chains
• infinite countable Markov chains
• discrete martingales
• limits of sequences of independent random variables. 

You will also develop your modelling skills. You will pay particular attention to questions such as:

• How can we model a certain problem using a discrete process?
• Can the model be used to estimate probabilities, expected values etc. If so, how?
• How can we understand what happens to the model when we look far into the future?

Module learning outcomes

• Setting up probability spaces, events and random variables to solve real-life probability problems.
• Manipulating distributions, densities, sums of random variables, basic random processes and Markov chains with applications.
• Understanding and using the Laws of Large Numbers and the Central Limit Theorem, with an eye to statistics and probability modelling.
• Acquire and rediscover set-theoretical and calculus skills in the context of probabilistic manipulations.