The skewness and kurtosis of the product of two normally distributed random variables

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.

Keywords:

• Moment generating function
• Pearson’s inequality
• moments

Acknowledgements

We thank the Editor and the reviewer for their careful reading and many insightful comments and suggestions.

Additional information

Funding

This work is partially financed by national funds through FCT - Fundação para a Ciência e a Tecnologia under the project - UIBD/00006/2020.