# Module Information

Module Identifier
MA26010
Module Title
Distributions and Estimation
2024/2025
Co-ordinator
Semester
Semester 1
Pre-Requisite
Other Staff

#### Assessment

Assessment Type Assessment length / details Proportion
Semester Exam 2 Hours   (Written Examination)  100%
Supplementary Exam 2 Hours   (Written Examination)  100%

### Learning Outcomes

On completion of this module, a student should be able to:
1. describe the relationships between the joint, marginal, conditional probability (density) functions, cumulative distribution function of bivariate random variables and move between these;
2. test for independence of two random variables;
3. calculate distributions of functions of random variables in univariate and simple bibivariate cases;
4. use moment generating functions to determine moments and distributions of sums of random variables;
5. describe how the chi-squared, t and F distributions arise from a Normal random sample, and use the associated tables;
6. calculate maximum likelihood estimators;
7. apply the criterion of unbiasedness.

### Brief 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.

### Content

1. GAMMA DISTRIBUTION: Gamma function. Gamma distribution.
2. DISCRETE AND CONTINUOUS BIVARIATE DISTRIBUTIONS: Marginal and conditional distributions. Cumulative distribution functions. Independence.
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.

### Module Skills

Skills Type Skills details
Adaptability and resilience Students are expected to develop their own approach to time-management and to use the feedback from marked work to support their learning.
Co-ordinating with others Students will be encouraged to work in groups to solve problems.
Creative Problem Solving The assignments will give the students opportunities to show creativity in finding solutions and develop their problem solving skills.
Digital capability Use of the internet, Blackboard, and mathematical packages will be encouraged to enhance their understanding of the module content and examples of application
Professional communication Students will be expected to submit clearly written solutions to set exercises.
Subject Specific Skills Broadens exposure of students to topics in mathematics, and an area of application that they have not previously encountered.

### Notes

This module is at CQFW Level 5