Module Identifier MA26010 Module Title DISTRIBUTIONS AND ESTIMATION Academic Year 2000/2001 Co-ordinator Mr D A Jones Semester Semester 1 Other staff Dr J G Basterfield Pre-Requisite MA11310 Mutually Exclusive MX36010 Course delivery Lecture 19 x 1hour lectures Seminars / Tutorials 3 x 1hour example classes Assessment Exam 2 Hours (written examination) 100% Resit assessment 2 Hours (written examination) 100%

General description
In many situations in statistics and probability it is necessary to handle more than one random variable at the same time. This module covers techniques needed to do this, and also to deal with functions of random variables. Particular attention will be paid to the case of random variables arising from a Normal sample. The module concludes with some material on the theory of estimation.

Aims
This module will provide a thorough grounding in distribution theory for several random variables, and also consolidates the material on estimation introduced in MA11310.

Learning outcomes
On completion of this module, a student should be able to:

• describe the relationships between the joint, marginal, conditional probability (density) functions, cumulative distribution function of bivariate random variables and move between these;
• test for independence of two random variables and describe the generalisation of the test to more than two random variables;
• calculate distributions of functions of random variables in univariate and simple bibivariate cases;
• use moment generating functions to determine moments and distributions of sums of random variables;
• describe how the chi-squared, t and F distributions arise from a Normal random sample, and use the associated tables;
• apply the criteria of unbiasedness, minimum variance, and mean square error to estimators.

Syllabus
1. DISCRETE AND CONTINUOUS BIVARIATE DISTRIBUTIONS: Marginal and conditional distributions. Cumulative distribution functions. Independence. Revision of covariance and correlation.
2. FUNCTIONS OF RANDOM VARIABLES: Calculation of the pdf of a function of one or more random variables by (a) distribution functions, (b) transformation using the Jacobian, (c) moment generating functions.
3. SAMPLING DISTRIBUTIONS AND THE CENTRAL LIMIT THEOREM: The idea of a sample. The chi-squared, t and F distributions and their relationships to the Normal. Sampling distributions for statistics of the Normal sample. The Central Limit Theorem.
4. POINT ESTIMATORS: The concepts of estimator and estimate. Unbiasedness and mean square error as criteria. Maximum likelihood.