Another Remark on the Alternative Expectation Formula

Abstract

Students in a calculus-based probability course will often see the expectation formula for nonnegative continuous random variables in terms of the survival function. This alternative expectation formula has a wide spectrum of applications. It is natural to ask whether there is a multivariate version of this formula. This note gives an affirmative answer by establishing such a formula using two different approaches. The two approaches employed in this note correspond to the two approaches for the univariate case. Supplementary materials for this article are available online.

• Expectation
• Integration by parts
• Multivariate analysis
• Nonnegative continuous random variable
• Survival function.