Module Identifier MA26010 Module Title DISTRIBUTIONS AND ESTIMATION Academic Year 2001/2002 Co-ordinator Mr David Jones Semester Semester 1 Other staff Dr J Basterfield Pre-Requisite MA11310 Mutually Exclusive MX36010 Course delivery Lecture 19 x 1 hour lectures Seminars / Tutorials 3 x 1 hour 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;
• 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;
• calculate maximum likelihood estimators;
• apply the criterion of unbiasedness.

Syllabus

1. DISCRETE AND CONTINUOUS BIVARIATE DISTRIBUTIONS: Marginal and conditional distributions. Cumulative distribution functions. Independence.
2. GAMMA DISTRIBUTION: Gamma function. Gamma distribution.
3. 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.
4. SAMPLING DISTRIBUTIONS FOR NORMAL SAMPLES: The chi-squared, t and F distributions and their relationships to the Normal. The idea of a sample. Sampling distributions for statistics of the Normal sample.
5. POINT ESTIMATORS: The concepts of estimator and estimate. Maximum likelihood. Unbiasedness as a criterion.