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I dont know how to approach this problem. Suppose that a machine has a probability of failure of 43% at 90 hours of operation. How can you calculate the failure rate and the probability of surviving x hours without having failures?

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$$ \large{R(t)=e^{\lambda t}}\\ \begin{align} R&=\text{Reliability}\\ t&=\text{time}\\ \lambda&=\text{failure rate} \end{align}$$

At $t=0$

$$ R(t=0)=1\\ \implies100\% $$

I think, if the reliability follows exponential distribution, you can use the formula given above. Reliability is the probability that a machine will function normally during a period of time under proper working conditions. So, $$\text{reliability}=1-\text{probability of failure}$$

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    $\begingroup$ Hi, in the future, please use Markup and MathJax to write your equations. $\endgroup$ Commented Oct 8, 2018 at 18:14
  • $\begingroup$ I up voted this answer, because the principal behind it is inspiring and beautiful, but first, we don't know if the distribution follows the exponential function. Second, even if the distribution follows the exponential function, then the time between two failures is not independent of the time between another two failures in future, with other words exponential distribution is memoryless however the nature of failure so depends on the history of previous failures of the machine. $\endgroup$
    – user14407
    Commented Oct 8, 2018 at 18:58
  • $\begingroup$ @Sam Farjamirad I think, the assumption behind this is that the system will be as good as new after maintenance.But as you said, it is not possible in real life. $\endgroup$
    – user17332
    Commented Oct 9, 2018 at 0:45

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