Question

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

Short answer

The probability for waiting time larger than 3.00 minutes is 0.4.

Step-by-step solution

Given information

The graph shows the complete area under the uniform distribution curve, which is equal to 1.

State the relationship between area and probability

As the area under the density curve is 1, it can be inferred that there is a one-to-one correspondence between the area and the probabilities.

Thus, the probability that the waiting time is greater than 3 is equal to the area under the curve in the shaded region, as shown in the graph below.

Find the probability

The probability that the waiting time is greater than 3.00 minutes is the area of the shaded rectangle with length 0.2 and width is $5-3=2$.

Thus,

$$\begin{gathered} P\left(X>3.00\right)=\left(\text{Length}\times \text{width}\right)\text{of}\text{the}\text{shaded}\text{area} \\ =0.2\times 2 \\ =0.4 \end{gathered}$$

So, the probability of selecting a passenger randomly when the waiting time is greater than 3.00 minutes is 0.4.