Exponential distributions on semigroups

Constant Rate Distributions on Partially Ordered Sets

“…Equivalently, the conditional distribution of x −1 X given X x is the same as the distribution of X. Exponential distributions on positive semigroups are studied in [10,11,12,13,14]. In particular, it is shown in [11] that a distribution is exponential if and only if it has constant rate and F (xy) = F (x)F (y), x, y ∈ S However there are constant rate distributions that are not exponential. Moreover, of course, the exponential property makes no sense for a general poset.…”

“…Proof. Suppose first that F is the UPF of a random variable X taking values in S. Then trivially F (e) = 1, and by(11),F (x) − y∈A(x) F (y) = P(X = x) > 0Next, d(X) ≥ n if and only if X x for some x ∈ A n . Moreover the events {X x} are disjoint over x ∈ A n .…”

Relative Aging of Distributions

“…With this definition, ([a,b),*) is a semigroup isomorphic to ([0,oo), +) and R is an isomorphism. It is a special case of a temporal semigroup, as studied in [7] and [5], and amounts to a transformation of time according to R. Of course, all of the standard algebraic operations on [0,oo) have their isomorphic counterparts on [a,b). For ex-.…”

“…Moreover, two functions R and 5 generate the same semigroup operation if and only if S = cR for some constant c > 0. The order on [a, b) induced by the semigroup operation * (in the sense of [7]) is the ordinary order :£. By definition, this means that…”

“…By analogy with the standard setting, we say that X is exponential relative to Y if the distribution of JC" 1 "" < > X given X ^ x is the same as the distribution of X (see [7]).…”