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Answer :
Final answer:
The probabilities were found to be 0.0498 and 0.0005 for the two scenarios and the standard deviation 16.67 hours considering exponential distribution.
Explanation:
The subject of this question is probability, specifically concerning exponential distribution, a common model in stochastic processes. If we denote λ as the failure rate, given in the problem as 0.06 failures/hour, the probability density function (pdf) of an exponential distribution is defined as f(t) = λe^(-λt)
(a) We want to find the probability of failure between 12 and 23 hours. Probability P(12 <= T <= 23) is calculated as ∫[12 to 23] λe^(-λt) dt. After calculation, we get 0.0498.
(b) The memoryless property of the exponential distribution states that the remaining waiting time until the next event is statistically independent of the past. Thus, if no failure has been seen in 24 hours, the probability of the next failure occurring after 14 hours is the same as if no time has passed. So, P(T > 38) = e^(-38λ), which is equal to 0.0005.
(c) The standard deviation for exponential distribution is given by the formula SD = 1/λ, which in this case equals about 16.67 hours.
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