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- Product of two normal variables: an historical reviewPublication . Seijas-Macias, J. Antonio; Oliveira, Amilcar; Oliveira, TeresaThis paper presents a study of the evolution the study of the product of two normally distributed variables. From first approaches in the middle of the 20th Century until the most recent ones from this decade. Nowadays, existence of an unique formula for the product was not proved. Partial results using di erent approaches: Bessel Functions, Numerical Integration, Pearson Type Function.
- A new R-function to estimate the PDF of the product of two uncorrelated normal variablesPublication . Seijas-Macias, J. Antonio; Oliveira, Amilcar; Oliveira, TeresaThis paper analyses the implementation of a procedure using the software R to calculate the Probability Density Function (PDF) of the product of two uncorrelated Normally Distributed Random Variables. The problem of estimating the distribution of the product of two random variables has been solved for some particular cases, but there is no unique expression for all possible situations. In our study, we chose Rohatgi’s theorem as a basis for approximating the product of two uncorrelated Normally Distributed Random Variables. The numerical approximation of the product PDF was calculated using a function that we implemented in R. Several numerical examples show that the approximations obtained in R fit the theoretical values of the product distributions. The results obtained with our R function are very positive when we compare them with the Monte Carlo Simulation of the product of the two variables.
- The skewness and kurtosis of the product of two normally distributed random variablesPublication . Seijas-Macias, J. Antonio; Oliveira, Amilcar; Oliveira, TeresaThe analysis of the product of two normally distributed variables does not seem to follow any known distribution. Fortunately, the moment-generating function is available and we can calculate the statistics of the product distribution: mean, variance, the skewness and kurtosis (excess of kurtosis). In this work, we have considered the role played by the parameters of the two normal distributions’ factors (mean and variance) on the values of the skewness and kurtosis of the product. Ranges of variation are defined for kurtosis and the skewness. The determination of the evolution of the skewness and kurtosis values of the product can be used to establish the normality of the product and how to modelize its distribution. Finally, the Pearson Inequality is proved for the skewness and kurtosis of the product of two normal random variables.
- Approximating the distribution of the product of two normally distributed random variablesPublication . Seijas-Macias, J. Antonio; Oliveira, Amilcar; Oliveira, Teresa A.; Leiva, VictorThe distribution of the product of two normally distributed random variables has been an open problem from the early years in the XXth century. First approaches tried to determinate the mathematical and statistical properties of the distribution of such a product using different types of functions. Recently, an improvement in computational techniques has performed new approaches for calculating related integrals by using numerical integration. Another approach is to adopt any other distribution to approximate the probability density function of this product. The skew-normal distribution is a generalization of the normal distribution which considers skewness making it flexible. In this work, we approximate the distribution of the product of two normally distributed random variables using a type of skew-normal distribution. The influence of the parameters of the two normal distributions on the approximation is explored. When one of the normally distributed variables has an inverse coefficient of variation greater than one, our approximation performs better than when both normally distributed variables have inverse coefficients of variation less than one. A graphical analysis visually shows the superiority of our approach in relation to other approaches proposed in the literature on the topic.