6.1 Suppose the probability that an electronic component fails to function in the time interval (t, t + dt) is (t) dt provided the component

6.1 Suppose the probability that an electronic component fails to function in the time interval (t, t + dt) is φ(t) dt provided the component has functioned properly up to time t. Given  ∞ 0 φ(τ ) dτ = ∞, show that p(t) = φ(t)e−  t 0 φ(τ ) dτ (6.45) is the probability density function (for positive t) for the time at which the component first fails to function. [If φ(t) = t/α2, then p(t) = t/α2e−t 2/2α2 is known as the Rayleigh probability density function.]