Question: A system consists of three components connected a) in series; c) the system is designed so that at least 2 components must operate for the system to operate properly.

Suppose that i^th component has a lifetime that is exponentially distributed with \({{\lambda }_{1}}=0.001\), \({{\lambda }_{2}}=0.002\) and \({{\lambda }_{3}}=0.003\), and that components fail independently.

Let X = the time at which the system fails, X, lifetime of i^th component, i = 1, 2, 3.

a) Calculate the c.d.f. F(x) = P (X < x), the p.d.f. of X and the expected value of X for all cases.

b) Determine the probability that the system will not fail before 800 hours for all cases.