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.

Between 2.5 minutes and 4.5 minutes

Short answer

The probability for waiting time between 2.5 minutes and 4.5 minutes is 0.4.

Step-by-step solution

Given information

The graph for a uniform distribution is given with an area enclosed equally to 1.

State the relationship between area and probability

The area under a density curve in a specific interval is equal to the probabilities if the total area under the curve is known to be 1.

Thus, the probability that the waiting time would be between 2.5 minutes and 3.5 minutes is shown as the area of the shaded region below.

The area of the shaded region, in this case, is actually the difference of the areas of two rectangles, that is, to the left of 2.5 minutes from the area to the left of 4.5 minutes.

Find the probability

The probability of waiting time between 2.5 minutes and 4.5 minutes is shown below.

$P\left(2.5<X<4.5\right)=\text{Area}\text{to}\text{the}\text{left}\text{of}4.5-\text{Area}\text{to}\text{the}\text{left}\text{of}2.5...\left(1\right)$

$$\begin{gathered} \text{Area}\text{to}\text{the}\text{left}\text{of}4.5=\text{Length}\times \text{width} \\ =0.2\times \left(4.5-0\right) \\ =0.9...\left(2\right) \end{gathered}$$

$$\begin{gathered} \text{Area}\text{to}\text{the}\text{left}\text{of}2.5=\text{Length}\times \text{width} \\ =0.2\times \left(2.5-0\right) \\ =0.5...\left(3\right) \end{gathered}$$

Substitute (2) and (3) into equation (1).

$$\begin{gathered} P\left(2.5<X<4.5\right)=0.9-0.5 \\ =0.4 \end{gathered}$$

Thus, the probability of selecting a passenger randomly when the waiting time is between 2.5 minutes and 4.5 minutes is 0.4.