The weakest link in a chain of components

Question

A machine has 4 critical components wired in series, the machine fails as soon as any one component fails. Each component's lifetime is independent and exponentially distributed with mean 2 years. What is the expected time until the machine fails?

Accepted Answer

A series system fails at the moment of the first component failure, so its lifetime is the minimum of the four component lifetimes. Each component has mean 2 years, so its failure rate is \lambda = \frac{1}{2} = 0.5 per year. The minimum of independent exponentials is exponential with rate equal to the sum of the rates: 	ext{rate of first failure} = 4\lambda = 4 	imes 0.5 = 2 	ext{ per year}, so the machine's lifetime is Exponential(2), with E[	ext{machine life}] = \frac{1}{4\lambda} = \frac{1}{2} = 0.5 	ext{ years}. So although each part lasts 2 years on average, the machine as a whole …

The weakest link in a chain of components

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