# SOR Level 1 Modules

**SOR1020 Introduction to Probability & Statistics (Full year)**

*Pre-requisite*: This module is intended for students at stage 1 of either an MSci or a BSc Mathematics pathway, and a mathematical knowledge and ability commensurate with this stage is assumed.

*Lecturers*: Dr G Tribello, Dr R Rollins

**Introduction**

This module introduces students to the fundamentals of probability and statistics. We start at the very beginning with basic probability before introducing both discrete and continuous random variables, and their properties. Typical standard discrete and continuous distributions such as Binomial, Exponential, Poisson and Normal are discussed with particular emphasis on how these distributions relate to systems in real life. This theory is used to provide a foundation for the formulation of statistical models and to introduce methods of parameter estimation. With this knowledge in particular it is possible to carry out hypothesis tests, both parametric and non-parametric, which are frequently used tools to analyse data in real world scenarios, including in important fields such as epidemiology, finance, pharma and in engineering.

**Contents**

Probability: Definitions and laws of probability. Interpretation of probabilities and relationships between probability and statistics. Conditional probability, in particular Bayes Theorem.

Discrete and Continuous Random Variables and Probability Distributions: Key definitions and properties of discrete and continuous random variables and probability distributions. Expected values of discrete and continuous random variables, including properties of expectation and variance operators.

Standard Discrete Distributions: Bernoulli, Geometric, Binomial, Negative Binomial, Hypergeometric and Poisson distributions.

Standard Continuous Random Variables: Uniform, Exponential and Normal distributions to include use of statistical tables, linear combinations of independent normal random variables, central limit theorems and approximations of Binomial and Poisson distributions.

Bivariate Distributions: Key definitions and properties of discrete and continuous bivariate distributions. Properties of independence and expected values; mean, variance, covariance. Correlations coefficients. Means, variances and covariance of linear combinations of random variables.

Statistical Models: Description of "mathematical" modelling. Statistical models. Measurement models, experimental, systematic and random errors; precision and accuracy.

Sampling: Sample surveys. Methods of sampling and probability sampling schemes. Errors in sample surveys. Advantages of sampling. Sampling from infinite populations.

Estimation: Key definitions and properties of estimation. Desirable properties for an estimator and estimation of mean and variance from a single sample and from several samples. Properties of point estimators leading to the method of moments, method of maximum likelihood estimation and method of least squares estimation.

Introduction to Hypothesis Testing: General principles, null and alternative hypotheses, one and two-sided tests, test statistics, critical regions, P-values, significance level, type I and type II errors and power function. Interpretation of results of a significance test, including confidence intervals.

Hypothesis Tests: Parametric tests based upon Normal distribution, t-distribution, F-distribution and chi-squared distribution. Non parametric tests including the Wilcoxon signed-rank test and Mann-Whitney test.

Statistical Quality Control: Process control for systems including Shewhart control charts, upper and lower control limits, upper and lower warning limits. Analysis of patterns and construction of a control chart for attributes and variables.

**Assessment**

3 Hour examination 60%

Class tests (x2 tests) 15%

Coursework (Computer Quiz) 10%

**SOR1021 Introduction to Statistical and Operational Research Methods (2nd semester)**

*Pre-requisite*: This module is intended for students at stage 1 of either an MSci or a BSc Mathematics pathway, and a mathematical knowledge and ability commensurate with this stage is assumed.

*Lecturer*: Dr R Rollins

**Introduction**

Introduction to statistical software for applying the following topics in Operational Research and Statistical Methods:

**Contents**

Linear Programming: Characteristics of linear programming models, general form.

Decision Theory:

Random Sampling and Simulation:

Initial Data Analysis: Scales of measurement.

**Assessment**

Computer Practical Assignments 100%