Expectation identity for the binomial distribution and its application in the calculations of high-order binomial moments

ABSTRACT

For the exponential families normal, gamma, beta, Poisson, and negative binomial, there exists an expectation identity for each of the family. For the binomial family, we discover an expectation identity, which is useful in analytical calculations of its high-order moments.

KEYWORDS:

• Binomial distribution
• Closed-form formula
• Expectation identity
• High-order moments.

MATHEMATICS SUBJECT CLASSIFICATION:

• 62Exx
• 62Qxx
• 05A10
• 05A19

Acknowledgments

The authors gratefully acknowledge the constructive comments offered by the referees. Their comments improve the quality of the paper significantly.

Supporting Information: Additional information for this article is available.

Supplement: Some proofs of the article.

Mathematica folder: Mathematica codes used in the article.

R folder: R codes used in the article.

Additional information

Funding

This work was supported by the Fundamental Research Funds for the Central Universities (CQDXWL-2012-004; 106112016CDJXY100002), China Scholarship Council (201606055028), National Natural Science Foundation of China (11671060), and MOE project of Humanities and Social Sciences on the west and the border area (14XJC910001).