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 binary operations between random
variables. Further more we characterize distrbutions with the almost lack
of memory property or with strong Markov property or with periodic failure
rate under such a binary operation extending the results on exponential
distributions under addition operation as binary operation.