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Suppose that a certain system contains three components that function independently of each other and are...

Suppose that a certain system contains three components that function independently of each other and are connected in series, so that the system fails as soon as one of the components fails. Suppose that the length of life of the first component, X1, measured in hours, has an exponential distribution with parameter λ = 0.01; the length of life of the second component, X2, has an exponential distribution with parameter λ = 0.03; and the length of life of the third component, X3, has an exponential distribution with parameter λ = 0.06. Denote by Y the length of life of the system and note that Y = min(X1, X2, X3).

(a) Show that Y has an exponential distribution with parameter λ = 0.1. (Hint: Consider the survival function 1 − F(y) = P(Y > y) = P(X1 > y, X2 > y, X3 > y) and show that it coincides with that of an exponential random variable with parameter λ = 0.1.)

(b) Find the probability that the system will fail within 10 hours.

Please state supporting rules and theorems and justify results with them

Math   Statistics-And-Probability   
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