# Probability and Statistics

This is a year one module.  It is taught intensively by Birmingham professors over four weeks.  There is continuous assessment and a final examination.

Statistics, often regarded as distinct science rather than a branch of mathematics, is the study of data and uncertainty. Statistical techniques allow us to make conclusions, such as whether or not living near electricity pylons is dangerous, from sets of data. Statistics is also used in the design of effective experiments and in determining what data should be collected. For example, statistical techniques might be used to determine the frequency with which aircraft components should be tested for safety. Underlying these techniques is the assumption that these data are samples of a random variable that follows a probability distribution describing their behaviour.

This module provides an introduction to probability and statistics. Axiomatic probability theory, including Bayes’ Theorem, is discussed briefly. Key discrete and continuous probability modules (such as the binomial, Poisson and normal distributions) are introduced. Properties of expectation and variance are discussed. The Weak Law of Large Numbers and the Central Limit Theorem are covered before basic statistical ideas, such as statistical inference and hypothesis testing are introduced. Real world data are used to illustrate the theory.

## By the end of the module the student should be able to:

• Calculate probabilities and conditional probabilities and apply Bayes’ Theorem in standard situations.
• Know and use the standard discrete and continuous probability models in appropriate situations.
• Know the properties of expectation and variance and apply them to in standard situations.
• Appreciate the significance of the Weak Law of Large Numbers and the Central Limit Theorem.
• Understand and apply basic statistical techniques such as inference, point estimation, confidence intervals, hypothesis testing.