# Unit information: Probability 1 in 2021/22

Unit name Probability 1 MATH11300 10 C/4 Teaching Block 1 (weeks 1 - 12) Professor. Johnson Not open An A in A-level Mathematics or equivalent. Analysis 1 (MATH11006) and Calculus 1 (MATH11007), or equivalent. School of Mathematics Faculty of Science

## Description including Unit Aims

Unit aims

To introduce the basic ideas and methods of Probability, developing the concepts of random variables, expectations and variances. To look at some simple applications of these ideas and methods.

General Description of the Unit

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). Finally techniques are developed for evaluating these quantities, including generating functions and conditional expectations.

Relation to Other Units

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

## Intended Learning Outcomes

When you have successfully completed this module you will be able to:

• Define events and sample spaces, describe them in simple examples, and use counting arguments to calculate probabilities when there are equally likely outcomes.
• List the axioms of probability, and use them to prove simple results, including the partition theorem and Bayes’ theorem.
• Define a random variable. Define the probability mass function for discrete random variables, and the probability density function (pdf) and cumulative distribution function (cdf) for continuous random variables. Illustrate links between the pdf and cdf. Calculate the pdf of a function of a random variable.
• Define the following random variables: Bernoulli, Binomial, Geometric, Poisson, *Uniform, Exponential, Gamma, Normal/Gaussian. Recall and illustrate features of these distributions.
• Define and calculate the expectation, variance and covariance of simple random variables, including all of the standard types in the previous objective.
• Define jointly distributed random variables, joint probability mass functions.
• Define the moment generating function of a random variable. Use moment generating functions to analyse sums of random variables.
• Define and explain conditional expectation. Prove the double expectation formula. Use conditional expectation and moment generating functions to analyse random sums.
• Formulate formal probability models from informal descriptions.

Transferable Skills

Model building. Especially the formal mathematical modelling of informal descriptions of events and processes.

## Teaching Information

Lectures supplemented (for first year students) by small group tutorials. Weekly problem sheets, with outline solutions handed out the following week.

## Assessment Information

90% 1.5 hour examination

10% coursework

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

## Resources

If this unit has a Resource List, you will normally find a link to it in the Blackboard area for the unit. Sometimes there will be a separate link for each weekly topic.

If you are unable to access a list through Blackboard, you can also find it via the Resource Lists homepage. Search for the list by the unit name or code (e.g. MATH11300).

How much time the unit requires
Each credit equates to 10 hours of total student input. For example a 20 credit unit will take you 200 hours of study to complete. Your total learning time is made up of contact time, directed learning tasks, independent learning and assessment activity.

Assessment
The Board of Examiners will consider all cases where students have failed or not completed the assessments required for credit. The Board considers each student's outcomes across all the units which contribute to each year's programme of study. If you have self-certificated your absence from an assessment, you will normally be required to complete it the next time it runs (this is usually in the next assessment period).
The Board of Examiners will take into account any extenuating circumstances and operates within the Regulations and Code of Practice for Taught Programmes.