Refer to the Canadian Journal of Civil Engineering (Jan. 2013) investigation of the use of variable speed limits to control freeway traffic congestion, Exercise 4.9. Recall that the study site was an urban freeway divided into three sections with variable speed limits posted in each section. The probability distribution of the optimal speed limit for each of the three sections was determined. One possible set of distributions is as follows (probabilities in parentheses). Section 1: 30 mph (.06), 40 mph (.24), 50 mph (.24), 60 mph (.46); Section 2: 30 mph (.10), 40 mph (.24), 50 mph (.36), 60 mph (.30); Section 3: 30 mph (.15), 40 mph (.18), 50 mph (.30), 60 mph (.37). Consider a randomly selected vehicle traveling through the study site at a randomly selected time. For this vehicle, let X represent the section and Y represent the speed limit at the time of selection.
a. Which of the following probability distributions is represented by the given probabilities, p(x, y), p₁(x), p₂ (x), p₁(x Ι y), or p₂(y Ι x)? Explain.

b. Assume that the sections are of equal length. Find p₁ (x). Justify your answer.
c. Find the bivariate distribution, p(x, y).

Data from Exercise 4.9

A common transportation problem in large cities is congestion on the freeways. In the Canadian Journal of Civil Engineering (Jan., 2013), civil engineers investigated the use of variable speed limits (VSL) to control the congestion problem. The study site was an urban freeway in Edmonton, Canada. A portion of the freeway was equally divided into three sections, and variable speed limits posted (independently) in each section. Simulation was used to find the optimal speed limits based on various traffic patterns and weather conditions. Probability distributions of the 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).