Distribution of Functions of Normal Random Variables

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.