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Distribution of Functions of Normal Random Variables Doaa Zuhair Palestine Polytechnic University Deanship of Graduate Studies and Scientific Research Master of Applied Mathematics Distribution of Functions of Normal Random Variables Submitted by: Doaa Zuhair Ibraheem Salim Thesis submitted in partial fulfillment of requirements of the degree Master of Applied Mathematics Feb, 2021 The undersigned hereby certify that they have read, examined and recommended to the Deanship of Graduate Studies and Scientific Research at Palestine Polytechnic University the approval of a thesis entitled: Distribution of Functions of Normal Random Variables, submitted by Doaa Zuhair Ibraheem Salim in partial fulfil- ment of the requirements for the degree of Master in Applied Mathematics. Graduate Advisory Committee: Dr. Monjed H. Samuh (Supervisor), Palestine Polytechnic University. Signature: Date: Dr. Ahmad Khamayseh (Co-supervisor), Palestine Polytechnic University. Signature: Date: Dr. Mohammad Adam (Internal committee member), Palestine Polytechnic University. Signature: Date: Dr. Inad Nawajah (External committee member), Hebron University. Signature: Date: Thesis Approved Dr. Murad Abusubaih Dean of Graduate Studies and Scientific Research Palestine Polytechnic University Signature: Date: i DECLARATION I declare that the Master Thesis entitled "Distribution of Functions of Normal Random Variables" is my original work, and hereby certify that unless stated, all work contained within this thesis is my own independent research and has not been submitted for the award of any other degree at any institution, except where due ac- knowledgement is made in the text. Doaa Zuhair Ibraheem Salim Signature: Date: ii STATEMENT OF PERMISSION TO USE In presenting this thesis in partial fulfilment of the requirements for the master degree in Applied Mathematics at Palestine Polytechnic University, I agree that the library shall make it available to borrowers under rules of the library. Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledgement of the source is made. Permission for extensive quotation from, reproduction, or publication of this thesis may be granted by my main supervisor, or in his absence, by the Dean of Graduate Studies and Scientific Research when, in the opinion of either, the proposed use of the material is for scholarly purposes. Any coping or use of the material in this thesis for financial gain shall not be allowed without my written permission. Doaa Zuhair Ibraheem Salim Signature: Date: iii ACKNOWLEDGEMENT First and foremost, all praise and thanks are due to Allah for helping me to complete this work. I could never have finished this thesis without his help and guidance. I would like to express my sincere gratitude to my supervisor, Dr. Monjed H. Samuh, for the continuous support of my Master study and related research, for his patience, motivation, and immense knowledge. His guidance helped me in all the time of research and writing of this thesis. I also would like to express my gratitude and thanks to Dr. Ahmad Khamayseh for his valuable guidance and all the support he provided me. I am also thankful to Dr. Inad Nawajah and Dr. Mohammad Adam for serving on my thesis committee and for providing constructive comments which have improved my thesis. Last but not the least, I would like to express my gratitude and appreciation to my family, my sisters and brother for supporting me spiritually throughout writing this thesis and my life in general. iv Abstract The need for the distribution of combination of random variables arises in many areas of the sciences and engineering. In this thesis, the distributions of two different combination of Gaussian random variables will be investigated. It is established that such expressions can be represented in their most general form as the sum of chi-square and the product of two normal random variables. The distribution of the product of two normal random variables studied by some authors in the literature in different ways. The first expression assumed two independent and identical random variables with zero mean and the same variance. The second one assumed two dependent random variables with zero mean, same variance, and with a specific correlation coefficient. Closed forms of the probability density function and cumulative distribution function will be derived, as well as some statistical properties such as mean, variance, moments, moment generating function, and order statistics will be studied. All derivation ascertained by using Monte Carlo simulation. Additionally, method of moment estimation and the maximum likelihood estimation will be used to estimate the parameter of the derived distribution. Simulation study carried out using R software. Finally, a practical application well be presented. v Table of Contents 1 Introduction 1 1.1 Motivation and General Purpose ....................... 1 1.2 Statement of the Problem .......................... 2 1.3 Research Questions .............................. 3 1.4 Thesis Objective ............................... 3 1.5 Thesis Organization .............................. 4 2 Background 6 2.1 Preliminaries ................................. 6 2.1.1 Random Variable and Some Properties ............... 6 2.1.2 Methods for Deriving the Probability Distribution of a Function of Random Variables .......................... 12 2.1.3 Method of Estimation ........................ 13 2.1.4 Some Common Discrete and Continuous Random Variables .... 15 2.1.5 Some Special Functions ........................ 16 2.2 Review of the Literature ........................... 20 2.2.1 Distribution of the Product of Two Normal Random Variables .. 20 2.2.2 Distribution of the Product of Independent Rayleigh Random Vari- ables .................................. 27 vi TABLE OF CONTENTS 2.2.3 Exact Distribution of the Product of m Gamma and n Pareto Ran- dom Variables ............................. 28 2.2.4 Slash Maxwell Distribution ...................... 28 2.2.5 Distribution of the Product of Bessel Distribution of Second Kind and Gamma Distribution ...................... 29 2.2.6 Distributions Involving Correlated Generalized Gamma Variables 29 3 Distribution of Function of Independent Normal Random Variables 30 3.1 Introduction .................................. 30 3.2 The Probability Density Function of Z ................... 31 3.3 The Cumulative Distribution Function of Z ................ 39 3.4 Some Statistical Properties of Z ....................... 46 3.4.1 Limit Behaviour ............................ 47 3.4.2 Mean .................................. 48 3.4.3 Variance ................................ 52 3.4.4 Moments and Related Measures ................... 52 3.4.5 Moment Generating Function .................... 57 3.4.6 Order Statistics ............................ 60 3.5 Estimation and Inference ........................... 60 3.5.1 Method of Moments Estimation ................... 61 3.5.2 Maximum Likelihood Estimation .................. 62 3.5.3 Evaluating the Goodness of an Estimator .............. 64 3.5.4 Simulation Study ........................... 64 4 Distribution of Function of Dependent Normal Random Variables 68 4.1 Introduction .................................. 68 4.2 The Probability Density Function of U ................... 69 4.3 Some Statistical Properties of the Distribution .............. 75 vii TABLE OF CONTENTS 4.4 Estimation and Inference ........................... 75 4.4.1 Maximum Likelihood Estimation .................. 76 4.5 Simulation Study ............................... 76 4.6 Application: The Effect of I/Q Imbalance in Advanced Wireless Commu- nication System ................................ 78 4.6.1 Optimal ML Receiver ......................... 81 5 Conclusion and Future Work 84 5.1 Conclusion ................................... 84 5.2 Future Work .................................. 84 A Proof of the Second Moment 86 viii List of Figures 3.1 Plot of Y2 vs Y1 ................................ 33 3.2 Plot of Z vs Y1 ................................ 35 3.3 Plot of fZ (z) when 0 ≤ θ < 2 ........................ 40 3.4 Plot of fZ (z) when θ > 2 ........................... 41 3.5 Plot of FZ (z) when 0 ≤ θ < 2 ........................ 44 3.6 Plot of FZ (z) when θ > 2 ........................... 45 3.7 The coefficient of variation, skewness and kurtosis of Z as a function of θ 56 ^ 3.8 Plot of FZ (z) and FZ (z) ........................... 65 4.1 Plot of Y2 vs Y1 ................................ 71 4.2 Plot of fU (u) for different values of ρ .................... 73 4.3 Plot of FU (u) for different values of ρ .................... 74 ^ 4.4 Plot of FU (u) and FU (u) ........................... 77 4.5 Block diagram of a typical communication system (Alsmadi, 2020) .... 79 4.6 The result of phase imbalance for specific values (Alsmadi, 2020) ..... 80 4.7 The result of transmitter and receiver I/Q imbalances applied to the 4- QAM and 64-QAM signal constellation diagrams for specific values (Als- madi, 2020) .................................. 81 4.8 Wireless communication system under the effect of I/Q imbalance. .... 81 ix List of Tables 2.1 Some discrete distributions (Wackerly et al., 2008) ............. 16 2.2 Some continuous distributions (Wackerly et al., 2008) ........... 16 3.1 Descriptive measures of Z for different values of σ and θ ......... 58 3.2 Mean of MMEs of all parameters, bias, and MSE, for distinct values of n and distinct setting of (σ; θ) ......................... 66 3.3 Mean of MLEs of all parameters, bias, and MSE, for distinct values of n and distinct setting
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