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.