There are two independently operated bus lines, both of which stop in front of y
ID: 3321245 • Letter: T
Question
There are two independently operated bus lines, both of which stop in front of your apartment building. One stops every hour, on the hour. The other also stops once an hour, but not on the hour. That is, the first bus line arrives at ...,6am, 7am, 8am, ...; while the second bus line arrives at ...,6+x am, 7+x am, 8+x am, ... where x is a positive constant. Unfortunately, you don’t know the value of x. Assume x is uniform on (0, 1). What is the average waiting time (if your arrival time to the station is random)?
I would appriciate if you could show all work so I can better understand how to do these types of problems, Thank you.
Explanation / Answer
Here given that, two independent operated bus lines. If first bus line arrives at 6am , while the second bus line arrives at 6+x am . Assume x is uniform on (0,1). Then
The expected waiting time before 1 min, i.e. If first bus line arrives at 6am and the second bus line arrives at 6+x am, then expected time is = (0+1) / 2 = 1/2 = 0.5 . ________(1)
And If first bus line arrives after 06:01am, then expected time is = (0+ 59)/2 = 59/2 = 29.5 _____(2)
i.e 29 min 30 sec.
From eq's (1) and (2) we determine the Whole expected waiting time is (0.5+29.5)/2 = 30/2 = 15 min.