MAS205 Statistics Core

Semester 1, 2013/14 10 Credits
Lecturer: Dr Jonathan Jordan Home page 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:

MAS273Statistical Modelling
MAS274Statistical Reasoning
MAS275Probability Modelling
MAS361Medical Statistics
MAS362Financial Mathematics
MAS363Linear Models
MAS370Sampling Theory and Design of Experiments
MAS461Medical Statistics
MAS462Financial Mathematics
MAS463Linear Models
MAS465Multivariate 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.

Reading list

Type Author(s) Title Library Blackwells Amazon
B A.M. Mood, F.A. Graybill, D.C. Boes Introduction to the Theory of Statistics 519.5 (M) Blackwells Amazon
B J. E. Freund, I. Miller, M. Miller John E. Freund's Mathematical Statistics with Applications 519.5 (F) Blackwells Amazon

(A = essential, B = recommended, C = background.)

Most books on reading lists should also be available from the Blackwells shop on Mappin Street.