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Characterization of Probability Distributions via Binary Associative Operation

Pietro Muliere and B. L. S. Prakasa Rao
Sankhyā: The Indian Journal of Statistics (2003-2007)
Vol. 65, No. 4 (Nov., 2003), pp. 799-806
Published by: Springer on behalf of the Indian Statistical Institute
Stable URL: http://www.jstor.org/stable/25053314
Page Count: 8
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Characterization of Probability Distributions via Binary Associative Operation
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Abstract

A binary operation * over real numbers is said to be associative if (x * y) * z = x * (y * z) and it is said to be reducible if x * y = x * z or y * w = z * w if and only if z = y. The operation * is said to have an identity element ẽ if x * ẽ = x. We characterize different classes of probability distributions under such binary operations between random variables. Further more we characterize distributions with the almost lack of memory property or with the strong Markov property or with the periodic failure rate under such a binary operation extending the results for exponential distributions under addition operation as binary operation.

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