Discussion
Which of the following is true ?
The life time of a component of a machine is modeled by the exponential distribution with mean 2 per week. Failed component is immediately replaced by a new one. Then the probability that no component is replaced in two weeks is :
Answer: D
The probability is \(e^{-1}\), reading the exponential lifetime as having mean 2 weeks.
No replacement in two weeks means the component in use at the start lasts more than 2 weeks. With mean lifetime 2 weeks, the failure rate is \(\lambda = 1/2\) per week, and
\[ \begin{aligned} P(T \gt 2) &= e^{-\lambda t} = e^{-(1/2)(2)} \\ &= e^{-1} \approx 0.368 \end{aligned} \]
The memoryless property makes this hold even if the component was not new at time 0. The phrase 'mean 2 per week' in the paper is loose; if 2 were the failure rate per week, the answer would be \(e^{-4}\), but the key takes 2 as the mean life in weeks. The values \(e^{-2}\) and \(e^{-0.5}\) correspond to other pairings of rate and time (rate 1 for two weeks, rate 1/2 for one week).
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