The University of Southampton
Courses

# MATH1024 Introduction to Probability and Statistics

## Module Overview

The theory and methods of Statistics play an important role in all walks of life, society, medicine and industry. They enable important understanding to be gained and informed decisions to be made, about a population by examining only a small random sample of the members of that population. For example, to decide whether a new drug improves the symptoms of a disease 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 inference about the population. This module aims to lay foundations in probability and distribution theory, data analysis and the use of a statistical software package, which will be built upon in later modules. The module begins by introducing statistical data analysis by using the freely available R package, https://cran.r-project.org/. Statistical analysis and report writing are discussed along with the use of the R software package for summarising and interpreting data. It then formally defines probability and studies the key properties. The concepts of random variables as outcomes of random experiments are introduced and the key properties of the commonly used standard univariate random variables are studied. Emphasis is placed on learning the theories by proving key properties of each distribution. Basic ideas of statistical inference, including techniques of point and interval estimation and hypothesis testing, are introduced and illustrated with practical examples. One of the pre-requisites for MATH2010, MATH2011, MATH2013, MATH2040

### Aims and Objectives

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

• A good understanding of exploratory data analysis.
• Ability to write a short-report describing a simple statistical data set.
• A good understanding of elementary probability theory and its application
• A good understanding of the laws of probability and the use of Bayes theorem
• A good understanding of the concept of a statistical distribution
• A good understanding of the standard univariate distributions and their properties
• A good understanding of the Central Limit Theorem and its application
• A good understanding of the basic concepts of statistical inference

### Syllabus

Exploratory data analysis: measures of location and spread; symmetry and skewness. Introduction to the R package for exploratory data analysis using graphics Presentation and interpretation of data and report writing. Probability: Sample space, events, outcome, and axioms of probability. Addition and multiplication rules. The law of total probability, conditional probability, independence, Bayes Theorem. Practical applications. Random variables: Discrete and continuous random variables. Probability mass function, probability density function and cumulative distribution function. Expectation, variance and moments. Discrete probability distributions: Bernoulli trials, binomial, geometric, hyper-geometric, Poisson. Continuous probability distributions: uniform, exponential, normal, and log-normal. Joint probability distribution, covariance, correlation, independence. Sample and population : Sampling distributions. The Central Limit Theorem. Statistical modelling and Inference: Point and interval estimation, Hypothesis testing and P-value; various one sample and two sample t-tests. . Collection of data and design of experiments.

### Learning and Teaching

#### Teaching and learning methods

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

TypeHours
Independent Study100
Teaching50
Total study time150

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

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

DALGAARD, P.. Introductory Statistics with R.

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

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

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

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

### Assessment

#### Summative

MethodPercentage contribution
Coursework 30%
Written assessment 70%

#### Referral

MethodPercentage contribution
Written assessment 100%

#### Repeat Information

Repeat type: Internal & External

### 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.