The relevation transform and a generalization of the gamma distribution function

— Survival models arising in mathematical demography, in renewal and in replacement models lead to a generalization of the Gamma distribution function. In these models the reîevation product corresponds to the addition of random variables, generally dependent. The reîevation product is non-commutative, non-associative, and only leftdistributive. In the case of auto-relevation, a reîevation product of two identically distributed random variables, a measure of renewal gain or life-extension is described by an expression akin to Shannori*s entropy. A common generalization of both the reîevation and convolution opérations is indicated in Section 8. Section l Définition. s(t) is a survivability function if 1 —s(t) is the cumulative distribution function of a non-negative random variable. s(t) is interpreted as the probability that a newborn individual will survive (at least) till age t; or that a new item will give at least t time units of service. The probability density that a newborn individual will die at age t is — s'(t)l cf. Appendix. Fpr simplicity of exposition we will assume, whenever needed, the differentiability of the functions considered. Theorenu Let ^4(0 and B(t) be survivability functions. Then