Unit name | Probability and Statistics |
---|---|

Unit code | MATH10013 |

Credit points | 20 |

Level of study | C/4 |

Teaching block(s) |
Teaching Block 2 (weeks 13 - 24) |

Unit director | Professor. Anthony Lee |

Open unit status | Not open |

Units you must take before you take this one (pre-requisite units) |
None |

Units you must take alongside this one (co-requisite units) |
None |

Units you may not take alongside this one |
None |

School/department | School of Mathematics |

Faculty | Faculty of Science |

**Why is this unit important?**

Probability is an everyday concept that most people have only a vague, intuitive understanding of. It concerns the notion that in a random experiment there are many possible events that could occur and one should quantify how probable these events are. For example, one may wish to compute the probability that if two six-sided dice are thrown the sum of the resulting numbers is 8. Probability has key connections to various other branches of mathematics, such as Analysis, Combinatorics, Functional Analysis, Game Theory, Geometry, Mathematical Physics, and Statistics.

Statistics builds upon probabilistic modelling of phenomena, often using a suitable family of parameterized distributions. Observations are modelled as realizations of random variables, and probability theory can then be used to make suitable inferences about parameters. Statistical ideas are essential for reasoning about uncertainty in a variety of disciplines, e.g. in the Natural Sciences, Social Sciences, Computer Science, Engineering, Machine Learning and AI.

**How does this unit fit into your programme of study?**

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

**An overview of content**

The study of games of chance, such as tossing dice and card games, resulted in early and often fraught attempts to formalize the theory; but a satisfactory rigorous basis for the subject only came with the axiomatic theory of Kolmogorov in 1933.

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 common quantities of interest (probabilities, expected values, variances and covariances) and techniques are developed for evaluating these quantities, including generating functions and conditional expectations.

For Statistics, the unit will introduce the notion of a parametric statistical model: a family of probability models with one probability model for each value of a statistical parameter. Modelling observed data as realizations of random variables from one member of this family then leads to the natural question of estimation of the parameter to infer the distribution associated with the data. Estimators will be introduced according to the method of moments and the maximization of the likelihood function. The sampling distribution of estimators will be covered with a view to assessing estimator quality and constructing confidence intervals and performing hypothesis tests. Finally, a linear regression model, incorporating dependence on explanatory variables, will be developed.

**How will students, personally, be different as a result of the unit**

By the end of the unit, students will be familiar with the basic framework of modern probability theory, including random variables, expectations, probability mass/density functions, conditioning, and independence. They will also understand the principles of parametric statistical modelling and be able to derive and compute parameter estimates for simple models using method-of-moments and maximum likelihood. They will also understand how to reason about the sampling distribution of simple estimators to construct confidence intervals and hypothesis tests. Students will have gained an understanding of how to link explanatory variables to outcomes using a linear model.

**Learning Outcomes**

After completing this unit successfully, students should be able to:

- Calculate probabilities, expectations and variances for random variables with a given probability distribution, including by using techniques such as moment generating functions and conditioning.
- Construct a formal probability model from an informal description.
- Estimate parameters of tractable, parametric statistical models.
- Construct a confidence interval and calibrate a hypothesis test.

The unit will be taught through a combination of:

- Plenary lectures.
- Online materials.
- Guided independent activities such as problem sheets and/or other exercises.
- Weekly problem classes and/or tutorials.
- Weekly office hours.

**Tasks which help you learn and prepare you for summative tasks (formative):**

Guided independent activities such as problem sheets and/or other exercises.

**Tasks which count towards your unit mark (summative):**

90% timed examination, 10% coursework.

**When assessment does not go to plan**

If you fail this unit and are required to resit, then reassessment is by a written examination in the Resit and Supplementary exam period.

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. MATH10013).

**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.

See the University Workload statement relating to this unit for more information.

**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. For appropriate assessments, if you have self-certificated your absence, you will normally be required to complete it the next time it runs (for assessments at the end of TB1 and TB2 this is usually in the next re-assessment period).

The Board of Examiners will take into account any exceptional circumstances and operates
within the Regulations and Code of Practice for Taught Programmes.