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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.

O tempo necessário para a manutenção de uma máquina é uma variável aleatória exponencialmente distribuída com o valor médio de 2 horas. Qual é a probabilidade de que um reparo dure mais de 3 horas?

Learn how to calculate the probability of a repair time exceeding 3 hours given an exponential distribution with a mean of 2 hours. This problem uses key probability concepts and formulas related to the exponential distribution. Suitable for undergraduate statistics or mathematics students.

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