# Unit information: Probability and Statistics in 2020/21

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Unit name Probability and Statistics MATH10013 20 C/4 Teaching Block 4 (weeks 1-24) Professor. Johnson Not open A in A Level Mathematics or equivalent None School of Mathematics Faculty of Science

## Description including Unit Aims

Lecturers: Oliver Johnson (TB1) and Feng Yu (TB2)

Unit Aims

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.

Unit Description

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.

Relation to Other Units

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

## Intended Learning Outcomes

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.

## Teaching Information

The unit will be taught through a combination of

• synchronous online and, if subsequently possible, face-to-face lectures
• asynchronous online materials, including narrated presentations and worked examples
• guided asynchronous independent activities such as problem sheets, computing exercises and/or other exercises
• synchronous weekly group problem/example classes, workshops and/or tutorials
• synchronous weekly group tutorials
• synchronous weekly office hours

## Assessment Information

Assessment for learning/Formative assessment:

• problem sheets set by the lecturer and marked by the students' tutors.

Assessment of learning/Summative assessment:

• Two timed, open-book examinations (each worth 45%) after each teaching block
• Coursework (10%)