Past exam of the mathematics course of the University of Cambridge / 2005 / ia / Paper 2 / 3F / Solution

Normalize the probability density function using $u=t^c$:The corresponding Weibull distribution has survival functionSince the distribution is continuous, the choices $>$ and $\ge$ have the same probabilities. The conditional probability is consequentlyFor $s,t>0$ and $c>1$, the supplied inequality, applied to $s/t$, gives $(s+t{)}^c>t^c+s^c$. Taking exponentials yieldsThis expresses the increasing hazard function $h_c(t)=c{t}^{c-1}$: surviving to a later age shortens the remaining lifetime in this conditional distribution. At $c=1$ equality holds, as expected from the memoryless property of the exponential distribution.