A further remark on the alternative expectation formula

Abstract

In this note, we give a further remark on the alternative expectation formula. We show that the survival function $\mathbb{P}(X>x)$ involved in the integral tail expectation formula can be replaced by $\mathbb{P}(X\ge x)$ which differs from the formula in the discrete case.

Keywords:

• Expectation
• Fubini’s theorem
• non negative random variable
• survival function

Acknowledgments

We express our gratitude to two referees for their careful readings and useful comments which led to an improved presentation of the article.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Shaochen Wang is supported by the National Natural Science Foundation of China (Grant No. 11801181) and the Guangdong Natural Science Foundation (Grant No. 2018A030310358).