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14.2 Exercise 30

Avg difficulty 2.5

A system has n components. Time to failure of the ith component is Xi and the class

{Xi:1≤i≤n}\left\{X_{i} : 1 \leq i \leq n\right\} is iid exponential (λ)\left(\lambda\right). The system fails if any one or more of the components fails. Let W be the time to system failure. What is the probability the failure is due to the ith component?

Suggestion. Note that W=XiW = X_{i} iff Xj>XiX_{j} > X_{i} for all j≠ij \neq i. Thus

{ W = X i } = { ( X 1 , X 2 , ⋯ , X n ) ∈ Q } , Q = { ( t 1 , t 2 , ⋯ t n ) : t k > t i , ∀ k ≠ i } \left\{W = X_{i}\right\} = \left\{\left(X_{1} , X_{2} , \hdots , X_{n}\right) \in Q\right\} , Q = \left\{\left(t_{1} , t_{2} , \hdots t_{n}\right) : t_{k} > t_{i} , \forall k \neq i\right\}
P ( W = X i ) = E [ I Q ( X 1 , X 2 , ⋯ , X n ) ] = E { E [ I Q ( X 1 , X 2 , ⋯ , X n ) | X i ] } P \left(W = X_{i}\right) = E \left[I_{Q} \left(X_{1} , X_{2} , \hdots , X_{n}\right)\right] = E \left\{E \left[\right. I_{Q} \left(X_{1} , X_{2} , \hdots , X_{n}\right) \left|\right. X_{i} \left]\right.\right\}

Pfeiffer, Applied Probability, 14.2 Problems on Conditional Expectation, Regression, exercise 30

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Adapted from Applied Probability © 2009 Paul E. Pfeiffer. Licensed under CC BY 3.0. Source: Applied Probability. Imported at revision d7a0420.