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 minutes and 3 minutes

Short answer

The probability for waiting time between 2 minutes and 3 minutes is 0.2.

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

There is a one-to-one association between probability and the area of the curve between the range. It holds under the condition when the total area is equal to 1.

Thus,the probability that the waiting time between 2 minutes and 3 minutes would be the same as the area of the shaded region shown below.

Moreover, the area of the shaded region is the difference between the area to the left of 2 minutes and the area to the left of 3 minutes.

 Step 3: Find the probability

The probability that the waiting time is between 2 minutes and 3 minutes is shown below:

$P\left(2<X<3\right)=\text{Area}\text{to}\text{the}\text{left}\text{of}3-\text{Area}\text{to}\text{the}\text{left}\text{of}2...\left(1\right)$

$$\begin{gathered} \text{Area}\text{to}\text{the}\text{left}\text{of}3=\text{Length}\times \text{width} \\ =0.2\times \left(3-0\right) \\ =0.6...\left(2\right) \end{gathered}$$

$$\begin{gathered} \text{Area}\text{to}\text{the}\text{left}\text{of}2=\text{Length}\times \text{width} \\ =0.2\times \left(2-0\right) \\ =0.4...\left(3\right) \end{gathered}$$

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

$$\begin{gathered} P\left(2<X<3\right)=0.6-0.4 \\ =0.2 \end{gathered}$$

Thus, the probability of selecting a passenger randomly when the waiting time is between 2 minutes and 3 minutes is 0.2.