School of Mathematics and Statistics (SoMaS)

MAS5051 Probability and Probability Distributions

The module is a distance-learning instrument aimed at graduates who wish to undertake a postgraduate Masters course in Statistics but lack the appropriate mathematical and statistical background. It introduces the concepts of probability theory and distribution theory needed for such a masters course, and gives training in their use. It covers the laws of probability and of conditional probability, the concepts of random variables and random vectors and their distributions, and the methodology for calculating with them. It also discusses Laws of Large Numbers and Central Limit phenomena.

There are no prerequisites for this module.
No other modules have this module as a prerequisite.

Teaching methods

Directed reading from the textbook listed below; discussion boards; coursework assignments.

Assessment

Formal Examination, Coursework

Full syllabus

• Sample spaces and events
• Defining probabilities
• Conditional probability
• Independence
• Discrete random variables
• Expectation (mean), variance and standard deviation
• Independence of random variables
• Mean and variance of linear combinations
• Bernoulli trials and the Binomial distribution
• The Geometric distribution
• Continuous random variables
• Mean and variance for continuous random variables
• Moment generating functions
• The normal distribution
• The Poisson distribution
• The Gamma and exponential distributions
• The Beta distribution
• Transformations of random variables
• The chi-squared, t and F distributions
• Distributions in R
• Normal approximations to Binomial and Poisson
• The Weak Law of Large Numbers
• The Central Limit Theorem
• Bayes' Theorem
• Multivariate random variables
• The Multinomial Distribution
• Multivariate continuous random variables and conditional distributions
• Covariance and correlation
• Conditional expectation
• Multivariate normal distribution
• Transformations of multivariate random variables
• Examples and applications of multivariate transformations
• Introduction to Markov Chains
• Basic Bayesian inference

Reading list