Course overview
Probability theory is the branch of mathematics that deals with modelling uncertainty. It is important because of its direct application in areas such as genetics, finance and telecommunications. It also forms the fundamental basis for many other areas in the mathematical sciences including statistics, modern optimisation methods and risk modelling. This course provides an introduction to probability theory, random variables and Markov processes.
Topics covered are: probability axioms, conditional probability; Bayes' theorem; discrete random variables, moments, bounding probabilities, probability generating functions, standard discrete distributions; continuous random variables, uniform, normal, Cauchy, exponential, gamma and chi-square distributions, transformations, the Poisson process; bivariate distributions, marginal and conditional distributions, independence, covariance and correlation, linear combinations of two random variables, bivariate normal distribution; sequences of independent random variables, the weak law of large numbers, the central limit theorem; definition and properties of a Markov chain and probability transition matrices; methods for solving equilibrium equations, absorbing Markov chains.
Course learning outcomes
- Basic probability axioms and rules and the moments of discrete and continous random variables as well as be familiar with common named discrete and continous random variables.
- How to derive the probability density function of transformations of random variables and use these techniques to generate data from various distributions.
- How to calculate probabilities, and derive the marginal and conditional distributions of bivariate random variables.
- discrete time Markov chains and methods of finding the equilibrium probability distributions.
- How to calculate probabilities of absorption and expected hitting times for discrete time Markov chains with absorbing states.
- How to translate real-world problems into probability models.
- How to read and annotate an outline of a proof and be able to write a logical proof of a statement.