This module is not currently running in 2024 to 2025.
ONLY AVAILABLE TO INTERNATIONAL MASTERS STUDENTS: This module is a pre-requisite for many of the other statistics modules at Stages 1 and 2, but it can equally well be studied as a module in its own right. It starts by revising the idea of a probability distribution for one or more random variables and looks at different methods to derive the distribution of a function of random variables. These techniques are then used to prove some of the results underpinning hypothesis test and confidence interval calculations, such as for the t-test or the F-test. With these tools to hand, the module moves on to look at how to fit models (probability distributions) to sets of data. A standard technique, known as the method of maximum likelihood, is introduced, which is then used to fit the model to the data to obtain point estimates of the model parameters and to construct hypothesis tests and confidence intervals for these parameters. The module concludes with a directed reading task to explore some of the ideas of Bayesian inference. Outline Syllabus includes: Joint, marginal and conditional distributions of discrete and continuous random variables; Generating functions; Transformations of random variables; Sampling distributions; Point and interval estimation; Properties of estimators; Maximum likelihood; Hypothesis testing; Neyman-Pearson lemma; Maximum likelihood ratio test. Bayesian inference: prior and posterior distributions, conjugate prior, loss function, Bayesian estimators and credible intervals.
42-48 lectures and example classes/workshops
90% Examination, 10% Coursework
MILLER, I. and MILLER, M. (2003) [Recommended]
John E. Freund's Mathematical Statistics with Applications. 7th international edition.
Pearson Education, Prentice Hall, New Jersey.
LINDLEY, D.V. and SCOTT, W.F. (1995) [Recommended]
New Cambridge Statistical Tables. 2nd edition.
HOGG, R., CRAIG, A. and McKEAN, J. (2003) [Background]
Introduction to Mathematical Statistics. 6th international edition.
LARSON, H. J. (1982) [Background]
Introduction to Probability Theory and Statistical Inference. 3rd edition.
SPIEGEL, M. R, SCHILLER, J. and ALU SRINIVASAN, R. (2000) [Background]
Schaum's Outline of Probability and Statistics. 2nd edition.
LEE, P. M. (2012) [Recommended for H-level students]
Bayesian Statistics an Introduction. 4th edition. (ebook)
See the library reading list for this module (Canterbury)
On successful completion of this module, students will be able to demonstrate:
a) a reasonable knowledge of probability theory and of the key ideas of statistical inference, in particular to enable them to study further statistics modules at levels I and H (for which this module is a pre-requisite);
b) a reasonable ability to use mathematical techniques to manipulate joint, marginal and conditional probability distributions, and to derive distributions of transformed random variables;
c) a reasonable ability to use mathematical techniques to calculate point and interval estimates of parameters and to perform tests of hypotheses;
d) some appreciation of the relevance of mathematical statistics to real world problems.
e) a systematic understanding of the areas of probability theory and frequentist statistics covered by this module, in particular to enable them to study further statistics modules at levels H and M (for which this module is a pre-requisite);
f) an ability to explore the statistical literature to extend their experience in frequentist statistics into the area of Bayesian inference.
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