Rao B R, Talwalker S
Math Biosci. 1989 Sep;96(1):95-115. doi: 10.1016/0025-5564(89)90085-0.
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.
给出了寿命为随机变量X且X服从瑞利分布或更一般地服从威布尔分布的个体的预期寿命或平均剩余寿命的界限。变量的简单变换给出了米尔斯比率和不完全伽马函数的不等式。还报告了一些数值计算，以比较几个参数值下的上下界与预期寿命函数的精确值。当寿命服从冈珀茨分布时，问题变得复杂，并且无法构建预期寿命函数的界限。证明了冈珀茨分布在正常和肿瘤生长动力学以及鸟类和哺乳动物的胚胎和出生后生长中的重要性，并通过数值方法对多个参数值评估了预期寿命。
