As indicated in the text, one often distinguishes between initial failures, random failures during the useful life of the product, and wear-out failures. For a given product, suppose the probability of an initial failure (a failure prior to time \(t=\alpha\) ) is \(\theta_{1}\), the probability of a wear-out failure (a failure beyond time \(t=\beta\) ) is \(\theta_{2}\), and that for the interval \(\alpha \leq t \leq \beta\) the failure-time density is given by

\[f(t)=\frac{1-\theta_{1}-\theta_{2}}{\beta-\alpha}\]

(a) Find an expression for \(F(t)\) for the interval \(\alpha \leq t \leq \beta\).

(b) Show that for the interval \(\alpha \leq t \leq \beta\), the failure rate is given by

\[Z(t)=\frac{1-\theta_{1}-\theta_{2}}{(\beta-\alpha)\left(1-\theta_{1}\right)-\left(1-\theta_{1}-\theta_{2}\right)(t-\alpha)}\]

(c) Suppose that the failure of a digital television set is considered to be an initial failure if it occurs during the first 100 hours of usage and a wear-out failure if it occurs after 15,000 hours. Assuming that the model given in this exercise holds and that \(\theta_{1}\) and \(\theta_{2}\) equal 0.05 and 0.75 , respectively, sketch the graph of the failure-rate function from \(t=100\) to \(t=15,000\) hours.