If the random variable T is the time to failure of a commercial product and the values of its probability density and distribution function at time t are f(t) and F(t), then its failure rate at time t (see also Exercise 5.24 on page 164) is given by f (t) 1– F(t) . Thus, the failure rate at time t is the probability density of failure at time t given that failure does not occur prior to time t.
(a) Show that if T has an exponential distribution, the failure rate is constant.
(b) Show that if T has a Weibull distribution (see Exercise 6.23), the failure rate is given by αβtβ– 1.