| Unit name | Probability and Statistics for Scientists |
|---|---|
| Unit code | MATH10028 |
| Credit points | 20 |
| Level of study | C/4 |
| Teaching block(s) |
Teaching Block 2 (weeks 13 - 24) |
| Unit director | Dr. Bailey |
| 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 |
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, physics and computer science such as, Mathematical Physics, Statistical Science, Machine learning and AI.
Statistical thinking 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 and predictions. 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.
This unit provides the foundation for all probability and statistics units in later years.
An overview of content
The unit is concerned with the mathematics of randomness. Probability theory is concerned with the modelling of random phenomena involving a population of objects and the calculus of probabilities of specific events occurring in relation to this population. Statistics is concerned with inferring properties of the unobserved population from a random sample and quantifying the associated uncertainty.
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.
Emphasis is put on simulated and thought experiments illustrating the various concepts and helping with the development of an understanding of the subtle concepts involved.
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. Uncertainty, inherent to the fact that the estimator is a function of an observed dataset sampled from an unobserved population, will be illustrated through thought experiments and simulations. Mathematically, 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.
Again, all the concepts involved will be illustrated through thought experiments and simulations.
Learning Outcomes
After completing this unit successfully, students should be able to:
The unit will be taught through a combination of:
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 in-person 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. MATH10028).
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 assessment methods listed in this unit specification are designed to enable students to demonstrate the named learning outcomes (LOs). Where a disability prevents a student from undertaking a specific method of assessment, schools will make reasonable adjustments to support a student to demonstrate the LO by an alternative method or with additional resources.
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.
