# Teaching

## MATH0057 Probability and Statistics - Second term 2021-2022

**Aims of course:**to introduce students to the theory of probability and some of the statistical methods based upon it. Many real-world processes involve random components which can only be modelled using probabilistic methods. Statistical theory is vital for interpreting information when it is necessary to distinguish genuine patterns from random fluctuations. The course begins with the basic ideas of probability theory: events, probabilities, random variables and the notion of independence. It continues with the two crucial principles: the law of large numbers and the central limit theorem. The course then introduces the fundamental concepts of statistical inference (estimation and hypothesis testing), and illustrates these concepts using the most important statistical models.**Course contents:**Revision of basic ideas in probability (axioms, simple laws, discrete and continuous random variables, expectation). Standard univariate distributions. Joint probability distributions: joint and conditional distributions and moments; serial expectation; multinomial and multivariate normal distributions. Moment and probability generating functions; properties; sums of independent random variables. Statement of weak law of large numbers. Statement of the Central Limit Theorem.Introduction to statistics (data, probability models, estimation, hypothesis testing). Normal probability models (1- and 2-sample problems, Chi-squared-distribution, t-distribution, F-distribution). Estimation (point and interval estimation, confidence intervals). Regression and correlation (linear regression, least squares).

**Prerequisites:**Good knowledge of calculus and basic combinatorics, as provided by MATH1401 and MATH1201 (for example) and basic probability (the probability content of MATH1301 or equivalent, as taught in post-exam module).**Texts:**comprehensive lecture notes will be made available for this course via Moodle. For students who wish to consult a text in addition to these notes, the recommendation is

D. Wackerly, W. Mendenhall & R. Scheaffer:*Mathematical Statistics with Applications*(6th edition), Duxbury Press.

This text covers all of the material in the course. It contains a large number of worked examples, as well as supplementary exercises for those students who want extra practice (solutions to the supplementary exercises are available in a companion volume).

The UCL Library stocks a large number of texts on probability and statistics. Two which may be useful for occasional consultation are*A First Course in Probability*by S.M. Ross, and*Introduction to Statistics*by R.E. Walpole.**Assessment:**Assessment is an examination in term 3 (90%) (details of which will be decided later), and in-course assessment (10%). The in-course assessment mark will be taken from FOUR of the weekly exercise sheets set during the course: sheets 2, 4, 5, and 8, one of which will be through STACK.**Other set work:**Weekly problem sheets throughout course.**Timetabled workload:**Recorded lectures: About 8 10-to-25-minute videos per week posted on Moodle during term 2 on Monday and Wednesday mornings. One face-to-face teaching per week (Fridays 9-10am or 10-11am) to be used as necessary for feedback on written work and as a problem class as well as for Q&A.**Moodle:**course material will be made available via the course Moodle page. All students taking the course should be enrolled automatically on this Moodle page; if for some reason the course is not showing in your Moodle home page, you should be able to enrol yourself --- just type MATH0057 in the `search courses' box, and use your initiative. Moodle will be the primary method for making course material available: all students are therefore expected to be enrolled on the course Moodle page and to download course materials as appropriate.