The University of Southampton
Courses

# MATH1024 Introduction to Probability and Statistics

## Module Overview

The theory and methods of Statistics play an important role in society, medicine and industry. They enable understanding to be gained and informed decisions to be made, about a population by examining only a sample of the members of that population. For example, to decide whether a new drug improves the symptoms of attention deficit disorder in all those diagnosed as having the condition (the population), a clinical trial might be undertaken in which a sample of people who receive the new drug is compared with a sample receiving no active treatment. Such statistical inferences about a population are subject to uncertainty -what we observe in our particular sample (or samples) may not hold for the whole population. Probability theory and statistical distributions are needed to quantify this uncertainty, and assess the accuracy of our information about the population. This module aims to lay foundations in probability and distribution theory, data analysis and the use of statistical software, which will be built upon in later modules. It begins by defining probability via axioms and develops some of its useful properties. Random variables are introduced, and the properties of probability used to develop distributions of practical importance. Statistical analysis is introduced with simple ideas of summarising data (implemented in R). Basic ideas of statistical inference (including techniques of estimation, confidence intervals and hypothesis testing) are also covered and applied to data sets.

### Aims and Objectives

#### Module Aims

To improve students' IT skills through the use of the comprehensive statistical software system, R, which will be used to illustrate the ideas of summarising and interpreting data. Transferable skills, e.g. report writing, presentation of numerical and statistical information, will be enhanced through the preparation of small reports.

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

• Analyse simple data sets using R and interpret the output
• Understand and recall the basic definitions of probability and statistical inference;
• Manipulate probabilities in practical situations
• Understand the concept of a statistical distribution
• Write a short report on the statistical analysis of some data.
• Derive the mean and variance of a variety of random variables
• Carry out a significance test and construct a confidence interval;
• Understand the Central Limit Theorem and apply it to problems;

### Syllabus

• Probability: Sample space, outcome, events, axioms of probability. Addition and multiplication rules. The law of total probability, conditional probability, independence, Bayes Theorem. Applications including reliability and randomised response in surveys. • Random variables: Discrete and continuous random variables. Probability mass functions, density functions and cumulative distribution functions. Expected value, variance and moments. • Discrete Probability distributions: Bernoulli trials, binomial, geometric, hypergeometric, Poisson. Covariance, correlation, independence. • Continuous probability distributions: The exponential, normal, lognormal and uniform distributions. • Data analysis: measures of location and spread; symmetry and skewness. Basic graphical methods, normal probability plots, factorial effect plots. • Sample and population. • Design of experiments: factorial designs and graphical analysis. • Inference: Sampling distributions. The Central Limit Theorem. Point estimation, confidence intervals. Significance tests and p-values. Practical applications. • Instruction in the use of R for simple data analysis

### Learning and Teaching

#### Teaching and learning methods

Lectures, small group tutorials, computer laboratories, report writing.

TypeHours
Independent Study87
Teaching63
Total study time150

HOGG, R.V. and TANIS, E.. Probability and Statistical Inference.

ROSS, S.A.. First Course in Probability.

DALGAARD, P.. Introductory Statistics with R.

MOORE, D.S. and MCCABE, G.P.. Introduction to the Practice of Statistics.

DEGROOT, M.H. and SCHERVISH, M.J.. Probability and Statistics.

RAWLEY, M.J.. Statistics: An Introduction using R.

MAINDONALD, J. and BRAUN, J.  Data analysis and graphics using R : an example-based approach.

### Assessment

#### Summative

MethodPercentage contribution
Class Test 10%
Coursework 10%
Exam  (120 minutes) 70%
Report 10%

#### Referral

MethodPercentage contribution
Exam %

#### Pre-requisites

To study this module, you will need to have studied the following module(s):

CodeModule
MATH1052Differential Equations
MATH1056Calculus

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Books and Stationery equipment

Course texts are provided by the library and there are no additional compulsory costs associated with the module.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.