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MC260 Mathematical Statistics


MC260 Mathematical Statistics

Credits: 10 Convenor: Mr. B. English Semester: 1 (weeks 7 to 12)


Prerequisites: essential: MC160 desirable: MC265
Assessment: Coursework: 20% One and a half hour examination: 80%

Lectures: 18 Classes: 5
Tutorials: none Private Study: 47
Labs: 5 Seminars: none
Project: none Other: none
Total: 75


Explanation of Pre-requisites

The module MC160 provides the basic introduction to probability and univariate random variables upon which this module builds. A number of distributional results, derived formally in this module, were stated and exploited in module MC265; this therefore, provides a practical motivation for the theoretical development presented in this module.

Course Description

In this module we introduce the basic concepts and techniques required for study of the joint behaviour of random variables, and functions of random variables. These techniques are exploited to derive some the classical distributions of mathematical statistics, including the $\chi^2$,t and F distributions. We also introduce the multivariate normal distribution (an extension of the normal distribution to several random variables), and derive some of its basic properties. We also provide a proof of the Central Limit Theorem. The course also provides an introduction to the Minitab package.

Aims

To introduce some of the basic concepts, and provide a practical grounding in techniques required for study of the joint behaviour of random variables, and functions of random variables. To reinforce the student's knowledge of the relationships between the basic distributions of mathematical statistics.

Objectives

On completion of this course, students should:

Transferable Skills

Syllabus

Review of univariate distributions. Moment generating functions; theorem on uniqueness of the corresponding distribution; the moment generating function and convergence in distribution. Application: a proof of the DeMoivre-Laplace Limit Theorem. Probability generating functions. The gamma and beta functions.

Multivariate random variables (mainly restricted to bivariate random variables). Joint distributions; marginal and conditional distributions; independence. Expectations, conditional expectations and regression; covariance and the coefficient of correlation. Theorems: Unconditional means and variance from conditional means and variances; linear regression and the coefficient of correlation. Introduction to bivariate integrals; bivariate continuous random variables, joint probability density functions; marginal and conditional distributions. Joint moment and probability generating functions. The bivariate normal distribution and its basic properties.

Functions of random variables. Functions of a single random variable. Use of the moment and probability generating functions to find the distribution of functions of random variables; examples including the derivation of the $\chi^2$ distribution, the sum of independent Poisson variables, and the Central Limit Theorem. Functions of continuous random variables via the distribution function; the distribution of sums, ratios and products of random variables; examples include the derivation of the F distribution. The distribution of functions of random variables via the Integral Transformation Theorem; examples to include the distribution of linear combinations of independent normal random variables.

Reading list

Recommended:

M. H. DeGroot, Probability and Statistics, 2nd edition, Addison-Wesley, 1986.

J. E. Freund and R. E. Walpole, Mathematical Statistics, 3rd edition, Prentice-Hall.

W. Mendenhall, R. L. Scheaffer and D. D. Wackerly, Mathematical Statistics with Applications, 4th edition, Duxbury Press, 1990.


Details of Assessment

The final assessment of this module will consist of 20% coursework and 80% from a one and a half hour examination during the January exam period. The 20% coursework contribution will be determined by students' solutions to coursework problems. The examination paper will contain 4 questions with full marks on the paper obtainable from 3 complete answers.


next up previous
Next: MC262 Linear Regression Models Up: Year 2 Previous: MC249 Rings and modules
S. J. Ambler
11/20/1999