Probability and Statistical Inference

Description

Overview: To present the basic concepts of probability theory and statistical inference; to provide students with the tools to appropriately analyse a given data set and effectively communicate the results of such analysis. Compulsory for: M, MS, MT, MSA, MSE, MSF, MSBA, BA with Mathematics or Statistics Pre-requisites: Overlapping Classes: Syllabus: ANOVA and Non-Parametric Tests. ANOVA: introduction to one-way ANOVA by hand and using R software. Introduction to Two-way and Factorial ANOVA in R, discussion of interaction plots. Non-Parametric Tests: sign test, ranks and rank sums, Mann Whitney test (by hand and using R) and Wilcoxon Signed Rank Test (by hand and using R). Multiple Regression. Multiple Regression: Revision of simple linear regression, overview of multiple regression, worked examples of multiple regression by hand and using R, assessing the assumptions of a multiple regression, relationship between ANOVA and multiple regression, hypothesis testing in a multiple regression setting, prediction intervals and confidence intervals for a multiple regression. Probability models. Random variables and probability distributions: probability mass function, probability density function, cumulative distribution function (by hand and using R). Elementary distributions: Bernoulli, binomial, Poisson, geometric, hypergeometric, negative binomial, Normal, exponential and uniform. Properties of distribution: mean, variance, higher-order moments, linear functions of random variables, central limit theorem, normal approximations (binomial and Poisson). Further probability theory. Bivariate distributions: simple bivariate discrete and continuous distributions, joint, conditional and marginal distributions, probability mass function and probability density function, expectation and variance, covariance, the bivariate Normal distribution. Finding expectations and variances, establishing the distribution of sums of random variables, introduction to moment and probability generating functions. Learning Outcomes: On completion of this Module, the student should be able to • carry out one-way ANOVA by hand and using R; • carry out two-way and factorial ANOVA using R; • undertake sign tests, calculate rank sums, carry out the Mann Whitney test and the Wilcoxon signed-rank test for non-parametric data; • carry out multiple regression using R and check the assumptions of multiple regression; • carry out hypothesis tests of the slope parameter using R; • calculate confidence intervals and prediction intervals of a multiple regression; • understand the concept of a random variable and a probability distribution; • recognise the appropriate probability model to describe particular random variables and derive the mean, variance and higher-order moments for these random variables; • use the central limit theorem in appropriate circumstances and approximate the binomial and Poisson distributions by the normal distribution; • calculate the mean and variance of a linear function of a random variable; • determine the mean, variance and co-variance of simple bivariate distributions; • estimate distribution parameters using least-squares, the method of moments and maximum likelihood estimation; • work collaboratively in group projects. Assessment: Coursework (One group project - 30%), Degree examination in December (70%). July/August summer examination (100%).