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.

Less than 4.00 minutes

Verified step by step guidance

1. Step 1: Understand the continuous uniform distribution. In this case, the waiting time is uniformly distributed between 0 and 5 minutes, as shown in the graph. The probability density function (PDF) is constant at 0.2, and the total area under the curve equals 1.
2. Step 2: Recall the formula for the probability in a continuous uniform distribution. The probability of a random variable falling within a range [a, b] is given by the area under the curve between those limits. This is calculated as P(a ≤ X ≤ b) = (b - a) × f(x), where f(x) is the constant height of the PDF.
3. Step 3: Identify the range of interest. The problem asks for the probability that the waiting time is less than 4.00 minutes. This corresponds to the range [0, 4].
4. Step 4: Apply the formula. Substitute the values into the formula: P(0 ≤ X ≤ 4) = (4 - 0) × 0.2. This represents the area of the rectangle from x = 0 to x = 4 under the curve.
5. Step 5: Interpret the result. The calculated area represents the probability that the waiting time is less than 4.00 minutes. This probability is proportional to the area under the curve within the specified range.

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 range, resulting in a rectangular shape when graphed. For example, if waiting times are uniformly distributed between 0 and 5 minutes, every time within this interval has the same likelihood of occurring.

Probability Density Function (PDF)

The probability density function (PDF) describes the likelihood of a continuous random variable taking on a specific value. In the case of 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 for a valid distribution.

Calculating Probability

To find the probability of a continuous random variable falling within a certain range, you calculate the area under the PDF over that interval. For example, to find the probability that a passenger's waiting time is less than 4 minutes in a uniform distribution from 0 to 5 minutes, you would determine the area of the rectangle formed by the interval from 0 to 4, which is the height of the PDF multiplied by the width of the interval.