2003, Volume 65, Pt. 4 , 799--806
CHARACTERIZATION OF PROBABILITY DISTRIBUTIONS VIA BINARY ASSOCIATIVE OPERATION
PIETRO MULIERE, Bocconi University, Milano, Italy and B.L.S. PRAKASA RAO, Indian Statistical Institute, New Delhi, India
SUMMARY. 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 $\tilde e$ if $x*\tilde e= x.$ 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.
AMS (1991) subject classification. Primary 62E10.
Key words and phrases. Characterization, almost lack of memory property, strong Markov property, periodic failure rate, binary associative operation, exponential distribution.