This problem is about the exponential distribution with a given mean, and it asks for the probability that the maintenance time exceeds 3 hours. Let me guide you through the solution.

Key Information:

• The maintenance time follows an exponentially distributed random variable.
• The mean of the exponential distribution is 2 hours.
• You need to find the probability that the repair time lasts more than 3 hours.

Step-by-Step Solution:

1. Exponential Distribution Formula: The probability density function (PDF) for an exponential distribution with parameter $\lambda$ (rate parameter) is: $f(t) = \lambda e^{-\lambda t}$ where $t$ is the time, and $\lambda = \frac{1}{\text{mean}}$. In this case, the mean is 2 hours, so: $\lambda = \frac{1}{2}$

2. Required Probability: The probability that the repair lasts more than 3 hours is given by: $P(T > 3) = 1 - P(T \leq 3)$ For an exponential distribution, this simplifies to: $P(T > 3) = e^{-\lambda \cdot 3}$ Substituting $\lambda = \frac{1}{2}$: $P(T > 3) = e^{-\frac{3}{2}}$

3. Comparing with Options: The probability expression $e^{-\frac{3}{2}}$ matches the option b.

Final Answer:

The correct option is b: $\frac{1}{e^{3/2}}$.

Would you like more details or have any further questions? Here are some related questions to deepen your understanding:

1. How does the exponential distribution differ from other continuous distributions?
2. What does the rate parameter $\lambda$ represent in an exponential distribution?
3. How would the result change if the mean time was 3 hours instead of 2?
4. What is the cumulative distribution function (CDF) for an exponential distribution?
5. How can exponential distribution be applied in real-life scenarios, such as reliability analysis?

Tip: Remember, the exponential distribution is memoryless, which means the probability of something lasting an additional time does not depend on how long it has already lasted.