This module replaces module MS4213 Probability Theory. It is being created as part of major changes to LM058/LM060, brought about in part by Project Maths. The new first year module MS4222 now contains some probability and this module builds on and extends that knowledge. The intention in this module is to firmly establish the connections between probability theory and its role in statistical applications.

Continuous Random Variables: expectation and variance; uniform, normal, exponential, gamma, beta, Cauchy, Weibull, distribution of a function of a random variable. Jointly Distributed Random Variables: joint distribution functions, sums of independent random variables, conditional densities, functions of jointly distributed random variables, (sum, difference, product, and quotient of two random variables). Properties of Expectation: computing probabilities and expectations by conditioning, conditional variance, conditional expectation and prediction. Sampling Distributions: the central limit theorem, the t-, chi-squared and F distributions and their use as sampling distributions; joint distribution of order statistics, distribution of sample range. Estimation: method-of-moments, fitting standard distributions to discrete and continuous data, pivotal quantities, confidence intervals. Simulation: Monte Carlo methods, variance reduction techniques, applications of simulation.

On successful completion of this module, students should be able to: 1. Calculate expected values and variances and normalizing constants of probability densities. 2. Calculate the distribution of combinations and linear transformations of random variables. 3. Use conditional expectation as a prediction tool. 4. Derive formulae for confidence intervals for parameters from Gaussian data and apply and interpret these interval estimates using examples of real data. 5. Use general approach to pivotal quantities to derive confidence intervals for distributional parameters. 6. Apply computer simulation tools for modelling data.

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Normal lecture/tutorial delivery with final exam and possible midterm exams and homeworks. Recent and relevant research and data to be included where appropriate. Graduates will be expected to demonstrate a high level of mathematical competence deriving, using and applying probability models. This will be developed through weekly homeworks and problem sheets.