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To solve the problem of finding the probability that a randomly selected passenger has a waiting time greater than 2.25 minutes, we need to understand the properties of a uniform distribution.

Given:
- The waiting times are uniformly distributed between 0 and 6 minutes.

For a uniform distribution \( U(a, b) \), the probability density function (PDF) is given by:

\[ f(x) = \frac{1}{b - a} \quad \text{for} \quad a \leq x \leq b \]

Here, \( a = 0 \) and \( b = 6 \). So the PDF is:

\[ f(x) = \frac{1}{6 - 0} = \frac{1}{6} \quad \text{for} \quad 0 \leq x \leq 6 \]

To find the probability that the waiting time \( X \) is greater than 2.25 minutes, we calculate:

\[ P(X > 2.25) = 1 - P(X \leq 2.25) \]

First, we find \( P(X \leq 2.25) \):

\[ P(X \leq 2.25) = \int_{0}^{2.25} f(x) \, dx = \int_{0}^{2.25} \frac{1}{6} \, dx \]

\[ P(X \leq 2.25) = \frac{1}{6} \times 2.25 = \frac{2.25}{6} = 0.375 \]

Now, we calculate \( P(X > 2.25) \):

\[ P(X > 2.25) = 1 - 0.375 = 0.625 \]

Therefore, the probability that a randomly selected passenger has a waiting time greater than 2.25 minutes is 0.625.

**Simplified Answer:** 0.625 (rounded to three decimal places).

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. How do you derive the probability density function for a uniform distribution?
2. What is the cumulative distribution function (CDF) for a uniform distribution?
3. How would you calculate the expected value (mean) of a uniformly distributed random variable?
4. What is the variance of a uniformly distributed random variable between \(a\) and \(b\)?
5. How would you calculate the probability that the waiting time is less than 4 minutes?

**Tip:** For any uniform distribution \( U(a, b) \), the area under the curve (PDF) always equals 1, representing the total probability.

The waiting times between a subway departure schedule and the arrival of a passenger are uniformly distributed between 0 and 6 minutes. Find the probability that a randomly selected passenger has a waiting time greater than 2.25 minutes.

Find the probability that a randomly selected subway passenger has a waiting time greater than 2.25 minutes. This problem covers the concept of uniform distribution and shows the step-by-step calculation using the probability density function and cumulative probability.

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