2. Let X denote the life (in weeks) of a certain part. Suppose that the PDF of X is

f(x) = 3x2=(400)3; 0 x < 400:

(a) Find the expected time of failing of the part.

(b) What is the probability that the part fails in the rst 200 weeks?

(c) What is the probability that the part will survive for another 200 weeks if it survived

the rst 100 weeks?

(d) To decrease the probability of failing, four independent parts are placed in parallel

so that all must fail if the system is to fail. Let Y be the life (in weeks) of this

parallel system.

i. Find the CDF and the PDF of Y .

ii. Find the probability that the system fails in the rst 200 weeks, and compare it

with the answer in part (b).

iii. When would you expect the system to fail?