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Abstract
We derive a formula for the product moment in terms of the joint survival function when (X
1,…,X
p) is a non-negative random vector. In the course of the derivation, we present an independent approach for deriving the formula for
(which is already known in the literature). These formulas will be useful tools for deriving explicit expressions for the product moments for the various bivariate and multivariate distributions in statistics specified by the joint tails of the cumulative distribution function. We illustrate their use by deriving product moments of general order for the Gumbel's bivariate exponential distribution, the bivariate Weibull distribution, Marshall and Olkin's bivariate exponential distribution, Muliere and Scarsini's bivariate Pareto distribution, and the conditionally specified bivariate Pareto distribution.
Acknowledgments
The second author (Kosto Mitov) is partially supported by NFSI-Bulgaria, Grant No. MM-1101/2001. Both authors wish to thank the Department of Mathematics at the University of South Florida for their hospitality and support. The authors are very grateful to the Editor-in-Chief, an Associate Editor and a referee for their most useful comments and suggestions.