The running time in minutes, X, of a computer program on a PC has a distribution function F(t) = 1 − (3t + 1)e−3t, t ≥ 0.

(i) John has started running this program half a minute ago and it is still running.

What is the probability that it will stop within the next 15 seconds?

(ii) Obtain the expected value of the running time, E(X), in two ways:

(a) by first finding the density function of X;

(b) by using the result of Proposition 6.7.