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- Evaluation of kurtosis into the product of two normally distributed variablesPublication . Oliveira, Amilcar; Oliveira, Teresa; Seijas-Macias, J. AntonioKurtosis (k) is any measure of the "peakedness" of a distribution of a real-valued random variable. We study the evolution of the Kurtosis for the product of two normally distributed variables. Product of two normal variables is a very common problem for some areas of study, like, physics, economics, psychology, ... Normal variables have a constant value for kurtosis (k = 3), independently of the value of the two parameters: mean and variance. In fact, the excess kurtosis is defined as k - 3 and the Normal Distribution Kurtosis is zero. The product of two normally distributed variables is a function of the parameters of the two variables and the correlation between then, and the range for kurtosis is in [0;6] for independent variables and in [0;12] when correlation between then is allowed.
- Distribution function for the ratio of two normal random variablesPublication . Oliveira, Amilcar; Oliveira, Teresa; Seijas-Macias, J. AntonioThe distribution of the ratio of two normal random variables X and Y was studied from [1] (the density function) and [2] (the distribution function). The shape of its density function can be unimodal, bimodal, symmetric, asymmetric, following several type of distributions, like Dirac Distribution, Normal Distribution, Cauchy Distribution or Recinormal Distribution. In this paper we study a different approximation for this distribution Z = X /Y , as a function of four parameters: ratio of the means of the two normal variables, ratio of the standard deviations of the two normal variables, the variation coefficient of the normal variable Y , and the correlation between the two variables. A formula for the Distribution function and the density function of Z is given. In addition, using graphical procedures we established singularity points for the parameters where the approximation given for Z has a non normal shape.