Question

Variable speed limit control for freeways. A common transportation problem in large cities is congestion on the freeways. In the Canadian Journal of Civil Engineering (January 2013), civil engineers investigated the use of variable speed limits (VSL) to control the congestion problem. A portion of an urban freeway was divided into three sections of equal length, and variable speed limits were posted (independently) in each section. Probability distributions of the optimal speed limits for the three sections were determined. For example, one possible set of distributions is as follows (probabilities in parentheses). Section 1: 30 mph (.05), 40 mph (.25), 50 mph (.25), 60 mph (.45); Section 2: 30 mph (.10), 40 mph (.25), 50 mph (.35), 60 mph (.30); Section 3: 30 mph (.15), 40 mph (.20), 50 mph (.30), 60 mph (.35).

1. Verify that the properties of a discrete probability distribution are satisfied for Section 1 of the freeway.
2. Repeat part a for Sections 2 and 3 of the freeway.
3. Find the probability that a vehicle traveling at the speed limit in Section 1 will exceed 30 mph.
4. Repeat part c for Sections 2 and 3 of the freeway.

Short answer

1. Yes

2. Section 2: Yes

   Section 3: Yes

3. 0.95

4. Section 2: 0.90

   Section 3: 0.85

Step-by-step solution

(a) Elucidation of properties

A distribution that is said to follow mainly two properties is regarded as a discrete probability distribution. Firstly, the probabilities must be positive, and secondly, they should be 1 or below 1.

Verification of the properties

The probabilities are positive, and the summation of the probabilities of speeds is shown below:

$$\begin{gathered} \mathbf{\text{Summation=0.05+0.25+0.25+0.50}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=1}} \end{gathered}$$

Therefore, as the total probability of the speeds in Section 1 is 1, the properties are satisfied.

(b) Verification of the properties for Section 2

The probabilities for speeds in Section 2 are positive, and the summation of the probabilities of speeds is shown below:

$$\begin{gathered} \mathbf{\text{Summation=0.10+0.25+0.35+0.30}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=1}} \end{gathered}$$

Therefore, as the total probability of the respective speeds in Section 2 is 1, the properties are completely satisfied.

Verification of the properties for Section 3

The probabilities are positive, and the summation of the probabilities of speeds is shown below:

$$\begin{gathered} \mathbf{Summation}\mathbf{=0.15+0.20+0.30+0.35} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=1}} \end{gathered}$$

Therefore, as the total probability of the respective speeds in Section 3 is 1, the properties are completely satisfied.

(c) Definition of a speed limit

Any city or area has certain speed limits for vehicles purring along the roads. If the vehicles at any cost cannot exceed their speed above a certain level, they will be penalised.

Calculation of the probability

The probability for a vehicle to travel above 30 mph is shown below:

$$\begin{gathered} \mathbf{\text{Probability=1}}\mathbf{-}\mathbf{\text{Probability}}\mathbf{(}\mathbf{\text{Speed}}\mathbf{\le }\mathbf{\text{30}}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=1}}\mathbf{-}\mathbf{\text{0.05}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=0.95}} \end{gathered}$$

Therefore, the probability for a vehicle to travel above 30 mph is 0.95.

(d) Calculation of the probability for Section 2

The probability for a vehicle in Section 2 to travel above 30 mph is shown below:

$$\begin{gathered} \mathbf{\text{Probability=1}}\mathbf{-}\mathbf{\text{Probability}}\mathbf{(}\mathbf{\text{Speed}}\mathbf{\le }\mathbf{\text{30}}\mathbf{)} \\ \mathbf{\text{=1}}\mathbf{-}\mathbf{\text{0.10}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=0.90}} \end{gathered}$$

Therefore, the probability for a vehicle in Section 2 to travel above 30 mph is 0.90.

Calculation of the probability for Section 3

The probability for a vehicle in Section 3 to travel above 30 mph is shown below:

$$\begin{gathered} \mathbf{\text{Probability=1}}\mathbf{-}\mathbf{\text{Probability}}\mathbf{(}\mathbf{\text{Speed}}\mathbf{\le }\mathbf{\text{30}}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=1}}\mathbf{-}\mathbf{\text{0.15}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=0.85}} \end{gathered}$$

Therefore, the probability for a vehicle in Section 3 to travel above 30 mph is 0.85.