FROM THE CLASSICAL BETA DISTRIBUTION TO GENERALIZED BETA DISTRIBUTIONS Title A project submitted to the School of Mathematics, University of Nairobi in partial fulfillment of the requirements for the degree of Master of Science in Statistics. By Otieno Jacob I56/72137/2008 Supervisor: Prof. J.A.M Ottieno School of Mathematics University of Nairobi July, 2013 FROM THE CLASSICAL BETA DISTRIBUTION TO GENERALIZED BETA DISTRIBUTIONS Half Title ii Declaration This project is my original work and has not been presented for a degree in any other University Signature ________________________ Otieno Jacob This project has been submitted for examination with my approval as the University Supervisor Signature ________________________ Prof. J.A.M Ottieno iii Dedication I dedicate this project to my wife Lilian; sons Benjamin, Vincent, John; daughter Joy and friends. A special feeling of gratitude to my loving Parents, Justus and Margaret, my aunt Wilkista, my Grandmother Nereah, and my late Uncle Edwin for their support and encouragement. iv Acknowledgement I would like to thank my family members, colleagues, friends and all those who helped and supported me in writing this project. My sincere appreciation goes to my supervisor, Prof. J.A.M Ottieno. I would not have made it this far without his tremendous guidance and support. I would like to thank him for his valuable advice concerning the presentation of this work before the Master of Science Project Committee of the School of Mathematics at the University of Nairobi and for his encouragement and guidance throughout. I would also like to thank Dr. Kipchirchir for his questions and comments which were beneficial for finalizing the project. I would like to recognize the financial support from the Higher Education Loans Board (HELB) without which this study might not have been accomplished. Most of all, I would like to thank the Almighty God for always being there for me. v Table of Contents Title.......................................................................................................................................................... i Half Title ................................................................................................................................................. ii Declaration ........................................................................................................................................... iii Dedication ............................................................................................................................................. iv Acknowledgement ................................................................................................................................. v Table of Contents .................................................................................................................................. 1 Executive Summary............................................................................................................................. 16 1 Chapter I: General Introduction ................................................................................................. 18 1.1 Introduction ........................................................................................................................... 18 1.2 Problem Statement ................................................................................................................ 19 1.3 Objective................................................................................................................................ 19 1.4 Literature Review ................................................................................................................... 20 2 Chapter II: The Classical Beta Distribution and its Special Cases ........................................... 27 2.1 Beta and Gamma Functions.................................................................................................... 27 2.2 Construction of the Classical Beta Distribution ....................................................................... 31 2.2.1 From the beta function ................................................................................................... 31 2.2.2 From stochastic processes .............................................................................................. 32 2.2.3 From ratio transformation of two independent gamma variables ................................... 32 2.2.4 From the rth order statistic of the Uniform distribution ................................................... 33 2.3 Properties of the Classical Beta Distribution ........................................................................... 34 2.3.1 rth-order moments .......................................................................................................... 34 2.3.2 The first four moments ................................................................................................... 34 2.3.3 Mode ............................................................................................................................. 36 2.3.4 Variance ......................................................................................................................... 37 2.3.5 Skewness ........................................................................................................................ 37 2.3.6 Kurtosis .......................................................................................................................... 39 2.4 Shapes of Classical Beta Distribution ...................................................................................... 41 2.5 Applications of the Classical Beta Distribution ........................................................................ 45 2.6 Special Cases of the Classical Beta Distribution ....................................................................... 45 2.6.1 Power Distribution Function ........................................................................................... 45 2.6.2 Properties of power distribution ..................................................................................... 46 2.6.3 Shape of Power distribution ........................................................................................... 47 2.6.4 Uniform Distribution Function ........................................................................................ 48 2.6.5 Properties of the Uniform distribution ............................................................................ 48 2.6.6 Shape of Uniform distribution ........................................................................................ 50 2.6.7 Arcsine Distribution Function.......................................................................................... 51 2.6.8 Properties of the arcsine distribution .............................................................................. 51 2.6.9 Shape of Arcsine distribution .......................................................................................... 54 2.6.10 Triangular shaped distributions (a=1, b=2) ...................................................................... 54 2.6.11 Properties of the Triangular shaped distribution (a=1, b=2) ............................................ 54 2.6.12 Shape of triangular distribution (a=1, b=2) ...................................................................... 57 2.6.13 Triangular shaped distributions (a=2, b=1) ...................................................................... 57 2.6.14 Properties of the Triangular shaped distribution (a=2, b=1) ............................................ 57 2.6.15 Shape of triangular distribution (a=2, b=1) ...................................................................... 59 2.6.16 Parabolic shaped distribution ......................................................................................... 60 2.6.17 Properties of parabolic shaped distribution .................................................................... 60 2.6.18 Shape of Parabolic distribution ....................................................................................... 62 2.7 Transformation of Special Case of the Classical Beta Distribution ........................................... 62 2.7.1 Wigner Semicircle Distribution Function ......................................................................... 62 2 2.7.2 Properties of Wigner Semicircle ...................................................................................... 63 2.7.3 Shape of Wigner semicircle distribution .......................................................................... 65 3 Chapter III: Type II Two Parameter Inverted Beta Distribution .............................................. 66 3.1 Construction of Beta Distribution of the Second Kind ............................................................. 66 3.2 Properties of Beta Distribution of the Second Kind ................................................................. 67 3.2.1 rth-order moments .......................................................................................................... 67 3.2.2 Mean .............................................................................................................................. 69 3.2.3 Mode ............................................................................................................................. 70 3.2.4 Variance ........................................................................................................................