## MAS205 Statistics Core

Note: This is an old module occurrence.

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 Semester 1, 2013/14 10 Credits Lecturer: Dr Jonathan Jordan Home page Timetable Reading List Aims Outcomes Assessment Full Syllabus

This unit develops tools and ideas underpinning probability and statistics at Level 2 and higher. It introduces some standard distributions beyond those met in MAS113, and uses the package R to study them. It proceeds to a systematic treatment of continuous multivariate distributions, with particular emphasis on the multivariate normal distribution. Transformations of univariate and multivariate continuous distributions are studied, with the derivation of sampling distributions of important summary statistics as applications. The idea of likelihood is developed, including the exploration and visualization of likelihood functions and surfaces using R, and the concept of maximum likelihood estimation.

Prerequisites: MAS113 (Introduction to Probability and Statistics)
Corequisites: MAS201 (Linear Mathematics for Applications); MAS202 (Advanced Calculus)

The following modules have this module as a prerequisite:

 MAS273 Statistical Modelling MAS274 Statistical Reasoning MAS275 Probability Modelling MAS361 Medical Statistics MAS362 Financial Mathematics MAS363 Linear Models MAS370 Sampling Theory and Design of Experiments MAS461 Medical Statistics MAS462 Financial Mathematics MAS463 Linear Models MAS465 Multivariate Data Analysis

## Outline syllabus

• Univariate distribution theory
• Continuous multivariate distributions and the multivariate normal
• Likelihood

## Aims

• Extend students' familiarity with standard probability distributions.
• Give practice in handling discrete and continuous distributions, especially continuous multivariate ones.
• Instil an understanding of the rationale and techniques of likelihood exploration and maximisation.
• Extend students' experience of using R for numerical and graphical tasks.

## Learning outcomes

• handle a wide range of standard distributions, including the multivariate normal.
• calculate joint, marginal and conditional continuous distributions.
• manipulate multivariate means, variances and covariances.
• transform univariate and multivariate continuous random variables.
• derive, manipulate and interpret likelihood functions.

22 lectures, 5 tutorials

## Assessment

One formal 2 hour written examination.

## Full syllabus

• Univariate distribution theory: Revision of sample spaces, events and random variables. Distribution functions. Revision of mean and variance, extension to higher moments, skewness. Revision of standard distributions from Level 1, further standard univariate distributions: hypergeometric, negative binomial, beta. Use of R to visualise univariate probability distributions. Transformations of random variables.
• Continuous multivariate distributions and the multivariate normal: Joint, marginal and conditional distributions for continuous random variables. Covariance, correlation and conditional expectation. Functions of more than one random variable; distributions of summary/test statistics as examples. Covariance matrices, linear transformations of random vectors. The bivariate normal; definition, correlation and covariance, conditional distributions. Extension of definition of bivariate normal to n-dimensional case, basic properties, joint conditional distributions.
• Likelihood: Definition. One-parameter and multi-parameter examples. Plotting and exploring likelihood functions and surfaces. The idea of maximum likelihood estimation; informal ideas of interval estimation based on likelihood. Techniques for manipulating and maximizing likelihood; link with MAS202. Use of Maple/R.