Continuous Uniform Distribution. In Exercises 5–8, refer to the continuous uniform distribution depicted in Figure 6-2 and described in Example 1. Assume that a passenger is randomly selected, and find the probability that the waiting time is within the given range.





Greater than 3.00 minutes

Verified step by step guidance

1. Identify the type of distribution: The problem involves a continuous uniform distribution, which is defined by a constant probability density function (PDF) over a specific interval. From the graph, the interval is [0, 5] and the height of the PDF is 0.2.
2. Recall the formula for the probability in a continuous uniform distribution: The probability of an event occurring within a range [a, b] is given by the formula P(a ≤ X ≤ b) = (b - a) * height of the PDF.
3. Determine the range of interest: The problem asks for the probability that the waiting time is greater than 3.00 minutes. This corresponds to the range [3, 5].
4. Substitute the values into the formula: Use the formula P(a ≤ X ≤ b) = (b - a) * height. Here, a = 3, b = 5, and the height of the PDF is 0.2. Substitute these values into the formula.
5. Simplify the expression: Perform the subtraction (b - a) and multiply the result by the height of the PDF to find the probability. This will give you the final answer.

Key Concepts

Continuous Uniform Distribution

A continuous uniform distribution is a probability distribution where all outcomes are equally likely within a specified range. The probability density function (PDF) is constant across this interval, meaning that the likelihood of any specific value is the same. In the context of waiting times, this distribution can be used to model scenarios where the waiting time is uniformly distributed between a minimum and maximum value.

Probability Density Function (PDF)

The probability density function (PDF) describes the likelihood of a continuous random variable taking on a specific value. For a continuous uniform distribution, the PDF is a horizontal line, indicating that the probability is evenly distributed across the range. The area under the PDF curve represents the total probability, which equals 1, and the height of the line is determined by the range of values.

Calculating Probabilities

To find the probability of a continuous random variable falling within a certain range, one must calculate the area under the PDF over that interval. For the continuous uniform distribution, this is done by multiplying the height of the PDF by the width of the interval. For example, to find the probability that the waiting time is greater than 3 minutes, one would calculate the area from 3 to the maximum value of the distribution.