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 (t) and F(t), then its failure rate at time t is given by  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, the failure rate is given by αβt ^β-1.