Consider a system consisting of three indpendent components

Consider a system consisting of three indpendent components. Supose the lifetimes of these componenets follow an Exponential distribution with mean 200 hours, 250 hours, and 500 hours, respectively.

a) Suppose that the system works only if all three components are functional (e.g.,series connection). Find the probability that the system still works after 300 hours. That is, find the probability that all three components still work after 300 hours. \"Hint\": P(X1>300 AND X2>300 and X3>300).

b)Suppose that the system works only if at least one component is functional (e.g.,parallel connection). Find the probability that the system works after 300 hours. This is, find the probability that at least one component still work after 300 hours. \"Hint\": P(X1>300 OR X2>300 OR X3>300)

Solution

Consider a system consisting of three indpendent components. Supose the lifetimes of these componenets follow an Exponential distribution with mean 200 hours, 250 hours, and 500 hours, respectively.

For an exponential distribution, if the mean is B:

P(X <= x) = 1