# Probability and Stochastic Modelling

## Module description

In this module we provide you with a solid grounding in the fundamentals of random variables and their distributions, together with an introduction to axiomatic probability theory and the convergence of sequences and sums of random variables. These form the foundations of statistics. We will then discuss the theory underlying modern statistics as well as (mathematical) statistics and the principles of statistical inference. Emphasis is placed on demonstrating the applicability of the theory and techniques in practical applications.

Lectures on time series and forecasting and Markov chains aim to extend the static ideas of the probability lectures to a dynamic framework in which randomness unfolds over time. They will introduce you to the properties of ARMA models and the principles of ARIMA modelling, including model identification, estimation and forecasting. A variety of models are discussed together with examples of their application, and you will use advanced statistical software to analyse time series data.

### Indicative syllabus

• Probability and Distribution Theory
• Statistical inference
• Time series and forecasting
• Discrete time Markov chains

## Learning objectives

By the end of this module, you should be able to:

• understand the theory of random variables and their distributions, and a wide range of standard distributions
• understand the axiomatic approach to probability
• recognise the appropriate distributions to use when modelling data that arise in different contexts and applications
• understand the principles and theory of statistical inference and use the theory and available data to estimate model parameters and formulate and test statistical hypotheses
• understand the theory and properties of ARMA and ARIMA models, and apply the theory to the analysis of times series data, to model fitting, model choice, interpretation and forecasting
• use advanced statistical software for the analysis of time series data
• understand the theory of homogeneous discrete time Markov chains.