5.98 waiting for a monorail. The problem of passenger conges- tion prompted a la

Question

5.98 waiting for a monorail. The problem of passenger conges- tion prompted a large international airport to insta monorail connecting its main terminal to the three con- courses, A, B, and C. The engineers designed the so that the amount of time a passenger at concourse B must wait for a monorail car has a uniform distribution ranging from 0 to 10 minutes. a. Find the mean and variance of Y, the time a passenger at monor ail concourse B must wait for the monorail. (Assume that the monorail travels sequentially from concourse A, to concourse B, to concourse C, back to concourse B, and then returns to concourse A. The route is then repeated.) b. If it takes the monorail 1 minute to go from concourse to concourse, find the probability that a hurried passen- ger can reach concourse A less than 4 minutes after ar- riving at the monorail station at concourse B

Explanation / Answer

Here, Y denotes the time a passenger at concourse B must wait for the monorail.

Now, Y is uniformly distributed from [0,10] minutes.

(a) Mean of Y = (0 + 10)/2 = 5 minutes

Variance of Y = (1/12)*(10 - 0)^2 = 8.33 minutes

(b) As it takes 1 minute to go from concourse B to course A, the required probability needed to find is the probability to wait less than (4 - 1) = 3 minutes at the monorail station at concourse B.

Now, CDF of Y is (y - 0)/(10 - 0) = y/10 for Y [0.10]

Thus, required probability, P(Y < 3) = F(3) = 3/10 = 0.3

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