Abstract

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.

摘要

对两个正态分布变量的乘积的分析似乎不遵循任何已知分布。幸运的是，矩生成函数可用，我们可以计算乘积分布的统计量：均值、方差、偏度和峰度（峰态过剩）。在这项工作中，我们考虑了两个正态分布因子（均值和方差）的参数对产品的偏度和峰度值所起的作用。变化范围是为峰度和偏度定义的。确定产品的偏度和峰度值的演变可用于确定产品的正态性以及如何对其分布进行建模。最后，证明了两个正态随机变量乘积的偏度和峰度的皮尔逊不等式。