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