# Probability and Statistics

## Unit description

This unit introduces student to basic ideas and methods in the areas of Probability and Statistics. It aims to develop the concepts of random variables, expectations and variances and look at some simple applications of these ideas and methods. It will also introduce the role of statistics in contemporary applications and aims to develop an elementary understanding of, and fluency in, the statistical paradigm of data collection, exploration, modelling and inference.

Probability is an everyday concept of which most people have only a vague intuitive understanding. Study of games of chance, such as tossing dice and card games, resulted in early attempts to formalise the theory; but a satisfactory rigorous basis for the subject only came with the axiomatic theory of Kolmogorov in 1933. Today probability is a well established and actively researched area of mathematics with lively links to Analysis, Combinatorics, Functional Analysis, Game Theory, Geometry, Mathematical Physics, Statistics. It also serves as a very important basis which various disciplines build on (Biology, Computer Science, Economics, Engineering, Linguistics, Physics, Sociology, just to mention a few).

The unit starts with the idea of a probability space, which is how we model the outcome of a random experiment. Probability models are then introduced in terms of random variables (which are functions of the outcomes of a random experiment), and the simpler properties of standard discrete and continuous random variables are discussed. Motivation is given for studying the common quantities of interest (probabilities, expected values, variances and covariances) and techniques are developed for evaluating these quantities, including generating functions and conditional expectations.

Computer technology has revolutionised both the scope and method of statistics, and the second half of this unit aims to give a basic grounding in statistical methodology that reflects this contemporary view. The role of statistics in the modern world is becoming ever-wider and applications can be found in almost all fields of human endeavour - in science, medicine, industry, social science, commerce and government. Taking real-life examples as motivation, this unit aims to develop an understanding of the basic principles of statistics, combining exploratory techniques and the machinery of probability theory to build a toolkit that can be used to uncover and identify relationships in the presence of random variation.

This unit provides the foundation for all probability and statistics units in later years.

## Learning objectives

Students should be able to:

- Understand the basic framework of modern probability theory, including random variables, expectations, probability mass/density functions, conditioning, and independence.
- Define the following random variables: Bernoulli, Binomial, Geometric, Poisson, Uniform, Exponential, Gamma, Normal/Gaussian. Recall and illustrate features of these distributions.
- Define jointly distributed random variables, joint probability mass functions.
- Understand how to analyse sums of independent random variables, including using moment generating functions and conditioning.
- Formulate formal probability models from informal descriptions.Formulate simple statistical models as appropriate to particular applications;
- Use exploratory techniques to identify simple relationships in data;
- Understand the principles of parametric modelling, and be able to derive parameter estimates for simple models using method-of-moments and maximum likelihood;
- Derive the simple linear regression model and implement it in appropriate situations;
- Understand estimators and sample variability, confidence intervals, and hypothesis tests, using both closed-form expressions for simple models, and simulation methods.
- Use the statistical software system R to support each of the above tasks.

## Reading and References

Recommended reading:

- Probability 1 (compiled from the first 8 chapters of A First Course in Probability by S. Ross). Pearson Custom Publishing.
- J. A. Rice, Mathematical statistics and data analysis, Wadsworth and Brooks Cole.
- P. Dalgaard, Introductory Statistics with R, Springer

**Unit code:** MATH10013

**Level of study:** C/4 (Honours)

**Credit points:** 20

**Teaching block (weeks):** 1 and 2 (1-24)

**Lecturers:** Professor Oliver Johnson and Professor Christophe Andrieu

## Pre-requisites

An A in A-level Mathematics or equivalent.

## Co-requisites

None

## Methods of teaching

Lectures, supported by lecture notes with problem sets and model solutions, problems classes, computer labs and small group tutorials.

## Methods of Assessment

The pass mark for this unit is 40.

Formative assessment:

- Problem sheets set by the lecturer and marked by the students’ tutors.

Summative assessment:

- Two 1.5h exams (45% each)
- Coursework (10%)

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.