Alternative expectation formulas for real-valued random vectors

Abstract

When the elements of a random vector take any real values, formulas of product moments are obtained for continuous and discrete random variables using distribution/survival functions. The random product can be that of strictly increasing functions of random variables. For continuous cases, the derivation based on iterated integrals is employed. It is shown that Hoeffding’s covariance lemma is algebraically equal to a special case of this result. For discrete cases, the elements of a random vector can be non-integers and/or unequally spaced. A discrete version of Hoeffding’s covariance lemma is derived for real-valued random variables.

Keywords:

• Iterated integral
• product moments
• survival function
• Hoeffding’s lemma

Additional information

Funding

This work was partially supported by a Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science and Technology (JSPS KAKENHI, Grant No. 17K00042).