CODE  SOR2211  
TITLE  Families of Random Variables and Random Vectors  
LEVEL  02  Years 2, 3 in Modular Undergraduate Course  
ECTS CREDITS  6  
DEPARTMENT  Statistics and Operations Research  
DESCRIPTION  This studyunit 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 x2Distribution  The tdistribution  The Fdistribution 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  Pvariate 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 Studyunit Aims: The main aim of this studyunit 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 studyunit 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 variancecovariance 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 studyunit 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  PreRequisite StudyUnits: SOR1110, SOR1220.  
STUDYUNIT TYPE  Lecture and Tutorial  
METHOD OF ASSESSMENT 


LECTURER/S  Monique Borg Inguanez Fiona Sammut 

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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 studyunits available during the academic year 2020/1. It may be subject to change in subsequent years. 