Bounds on life expectancy for the Rayleigh and Weibull distributions.

Bounds are presented for the life expectancy or the mean residual life of an individual whose lifetime is a random variable X following a Rayleigh distribution or more generally a Weibull distribution. Simple transformations of the variables give inequalities on the Mills' ratio and the incomplete gamma functions. Some numerical computations are also reported to compare the lower and upper bounds with the exact value of the life expectancy function for several values of the parameter. When the lifetime follows a Gompertz distribution, the problem becomes complicated, and it has not been possible to construct bounds on the life expectancy function. The importance of the Gompertz distribution in the dynamics of normal and tumor growth and in the embryonic and postnatal growth of birds and mammals is demonstrated, and life expectancy is evaluated by numerical methods for a number of parameter values.