Expected time until the first of five machines fails

Question

Five identical machines run independently. Each machine's lifetime is exponentially distributed with a mean of 200 hours. What is the expected time until the first machine fails?

Accepted Answer

The first failure happens at the minimum of the five lifetimes. A key fact about exponential lifetimes: the minimum of several independent exponentials is itself exponential, with a rate equal to the sum of the individual rates. Each machine has rate \lambda = \frac{1}{200} per hour, so the first-failure time has rate 5\lambda = \frac{5}{200} = \frac{1}{40} per hour. Its mean is the reciprocal of that rate: E[	ext{first failure}] = \frac{1}{5\lambda} = \frac{200}{5} = 40 	ext{ hours}. Running five machines at once makes breakdowns arrive five times faster than for one machine, so the first …

Expected time until the first of five machines fails

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