In some reliability problems we are concerned only with initial failures, treating a component as if (for all practical purposes) it never fails, once it has survived past a certain time \(t=\alpha\). In a problem like this, it may be reasonable to use the failure rate

\[Z(t)= \begin{cases}\beta\left(1-\frac{t}{\alpha}\right) & \text { for } 0

(a) Find expressions for \(f(t)\) and \(F(t)\).

(b) Show that the probability of an initial failure is given by

\[1-e^{-\alpha \beta / 2}\]