A system consists of three components connected so that at l

A system consists of three components connected 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 1 = .001, 2=.002, and 3 = .004, and that components fail independently. Let X = the time at which the system fails, Xi = the life time of i-th component, i = 1,2,3.

a) Calculate the cdf F(x)=P(X<=x), the pdf of X and the expected value of X.

b) Determine the probability that the system will not fail before 900 hours.

Hint: Introduce rv Y(x) = # of components working after x and relate F(x) with pmf of Y(x).

Solution