Discussiones Mathematicae Probability and Statistics

The 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.

The 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.

The 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 uni-modal, 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.

Kurtosis (κ) 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 (κ = 3), independently of the value of the two parameters: mean and variance. In fact, the excess kurtosis is defined as κ− 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.

Goal: The objective is to conclude, based on a comparative study, if there is a significant difference in sensitivity between the application of MCE and the individual application of the CUSUM or EWMA chart, i.e., greater sensitivity particularly for cases of lesser magnitude of change. Design/Methodology/Approach: These are an applied research and statistical techniques such as statistical control charts are used for monitoring variability. Results: The results show that the MCE chart signals a process out of statistical control, while individual EWMA and CUSUM charts does not detect any situation out of statistical control for the data analyzed. Limitations: This article is dedicated to measurable variables and individual analysis of quality characteristics, without investing in attribute variables. The MCE chart was applied to items that are essential to the productive process development being analysed. Practical Implications: The practical implications of this study can contribute to: the correct choice of more sensitive control charts to detect mainly small changes in the location (mean) of processes; provide clear and accurate information about the fundamental procedures for the implementation of statistical quality control; and encourage the use of this quality improvement tool. Originality/Value: The MCE control chart is a great differential for the improvement of the quality process of the studied company because it goes beyond what CUSUM and EWMA control charts can identify in terms of variability.