Answer :
Final answer:
The probability that a repair time exceeds 2 hours is approximately 13.5%, while the conditional probability that a repair takes at least 3 hours, given that its duration exceeds 2 hours, is approximately 63.2%.
Explanation:
The random variable in this question is time required to repair a machine, which is exponentially distributed with parameter λ = 1. This implies that the mean and standard deviation of this distribution are both 1 hour.
a) The probability that a repair time exceeds 2 hours is calculated by the exponential distribution's cumulative density function (CDF) which is 1 - exp(-λt). Plugging in λ = 1 and t = 2 gives us 1 - exp(-2) ≈ 0.865. Thus, the probability is 1 - 0.865 = 0.135 or 13.5%.
b) The conditional probability that a repair takes at least 3 hours, given that its duration exceeds 2 hours, is calculated by the exponential distribution's memoryless property. This property states that P(X > s + t | X > s) = P(X > t). Hence the required conditional probability is just the probability that a repair takes more than 1 hour, which would be 1 - exp(-1) ≈ 0.632 or 63.2%.
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