The distribution of the sum of two dependent randomly weighted random variables with applications

Abstract

There has been much work on the distribution of independent or dependent random variables. But we are not aware of any work giving exact results for the distribution of the sum of randomly weighted random variables. In this paper, we derive exact results for the randomly weighted sum of two dependent random variables. The derived expressions are for the cumulative distribution function, conditional expectation, moment generating function, value at risk, expected shortfall and the limiting tail behavior of the randomly weighted sum of two dependent random variables. Two numerical illustrations are given.

Keywords:

• Copula
• dependence
• expected shortfall

2020 AMS SUBJECT CLASSIFICATION:

• 62E10
• 60E05
• 62P05

Acknowledgments

The authors would like to thank the two referees and the Editor for careful reading and comments which greatly improved the paper.