16.5 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 = α. In a problem like this, it may be reasonable to use the failure rate Z(t) = ⎧ ⎪⎨ ⎪⎩ β  1 − t α for 0 < t < α 0 elsewhere

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

(b) Show that the probability of an initial failure is given by 1 − e−αβ/2