Study-Unit Description

Study-Unit Description


TITLE Families of Random Variables and Random Vectors

LEVEL 02 - Years 2, 3 in Modular Undergraduate Course


DEPARTMENT Statistics and Operations Research

DESCRIPTION This study-unit provides theoretical foundations on random variables and their extension to the multivariate case. The topics covered are listed below.

Functions of Random Variables
- Transformations of Random Variables
- Moment Generating Functions
- Joint Distributions
- Independent Random Variables
- Sequences of Identically Distributed Random Variables
- Sums of Random Variables

Sampling Distributions
- The Sampling Problem
- Sample Mean
- Sample Variance
- Central Limit Theorem
- The x2-Distribution
- The t-distribution
- The F-distribution

Bivariate Random Variables
- Bivariate density functions and distributions
- The Bivariate normal distribution
- Conditional density, expectation and variance with reference to bivariate distributions
- Covariance and Correlation

Random Vectors
- Multivariate Density and Distribution Functions
- Marginal and Conditional Distributions
- Expectation of Random Vector
- Covariance Matrix, Correlation Matrix
- Partial Correlation
- Data Matrices and Transforming them Linearly
- Sample Mean, Sample Covariance, Sample Correlation
- P-variate Normal Distribution and its Properties
- Matrix Transformations of Normal Random Vector
- Distribution of Quadratic Forms

The Wishart Distribution and its Properties
- Distribution of Sample Variance Covariance Matrix

The Hotelling's T2 Distribution and its Properties
- Inference about Mean Vectors and Difference in Mean Vectors

Study-unit Aims:

The main aim of this study-unit is that of providing students with a sound theoretical background on i) random variables, ii) transformations of random variables, iii) sampling distributions, and iv) their extension to the multivariate case.

Learning Outcomes:

1. Knowledge & Understanding
By the end of the study-unit the student will be able to:

- Formulate theoretical models of system of processes or realities, and of ones knowledge about them within the world around us, involving uncertainty in some essential way;
- Comprehend how a finite number of observations coming from the particular system under study can be used to modify, refine and organize our knowledge of the system with reference to the postulated model;
- Recall and apply major theoretical results (such as the Central Limit Theorem), which form the foundations of various topics, which the students cover in their second and third year;
- Explain the important concepts of joint and conditional distributions;
- Discuss how theoretical results on univariate random variables can be extended to multivariate random vectors by applying results from matrix algebra and probability theory;
- Explain how theoretical results involving the multivariate normal distribution, Wishart distribution and Hotelling's T2 distribution can be used to derive the sampling distribution of the mean vector and the variance-covariance matrix and to obtain statistics that can be used in the context of multivariate hypothesis testing of means and differences of means.

2. Skills
By the end of the study-unit the student will be able to:

- Explain how the distribution of a random variable X changes after it is transformed using some function g;
- Derive moments of the distribution of random variables and their transformations;
- Derive the joint and conditional distributions for a sequence of random variables, and the respective expectations and variances;
- Derive the sampling distribution of a number of statistics (e.g. mean, variance, quantiles);
- Use results for the multivariate normal distribution, Wishart distribution and Hotelling's T2 distribution for testing of means and differences of means.

Main Text/s and any supplementary readings:

Main texts:

- Miller, M and Miller, I (2013) John E. Freund's Mathematical Statistics with Applications, Eight Edition, Pearson India
- Knight K., ( 1999 ) Mathematical Statistics, CRC
- Roussas George G., ( 1997 ) A Course in Mathematical Statistics, Academic Press
- Mardia, K.V., Kent, J.T. and Bibby J.M. ( 1995 ) Multivariate Analysis, Academic
- Johnson, R.A. and Wichern, D.W. (1992) Applied Multivariate Statistical Analysis, Prentice Hall Inc.

Supplementary texts:

- Hogg, R.V. and Craig, A.T. ( 1978 ) Introduction to Mathematical Statistics, Macmillan
- Van der Waerden, B.L. ( 1969 ) Mathematical Statistics, Springer Verlag
- Searle,S.R. ( 1971 ) Linear Models, J.Wiley & Sons, New York
- Seber, G.A.F.( 1977 ) Linear Regression Analysis, J.Wiley & Sons, New York
- Flury, B. (1997) A First Course in Multivariate Statistics, Springer
- Hair J., Anderson R., Tatham R. and Black W., (1998) Multivariate Data Analysis, Prentice Hall I.

ADDITIONAL NOTES Pre-Requisite Study-Units: SOR1110, SOR1220.

STUDY-UNIT TYPE Lecture and Tutorial

Assessment Component/s Assessment Due Resit Availability Weighting
Examination (3 Hours) SEM2 Yes 100%

LECTURER/S Monique Borg Inguanez
Fiona Sammut

The University makes every effort to ensure that the published Courses Plans, Programmes of Study and Study-Unit information are complete and up-to-date at the time of publication. The University reserves the right to make changes in case errors are detected after publication.
The availability of optional units may be subject to timetabling constraints.
Units not attracting a sufficient number of registrations may be withdrawn without notice.
It should be noted that all the information in the description above applies to study-units available during the academic year 2020/1. It may be subject to change in subsequent years.