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A bus arrives at the bus station with probability $ \frac{1}{10} $ independently.

A family arrives at the bus stop at a random time.

The first son says:

  1. The expected value for the time the next bus will arrive is $ 10 $
  2. the expected for the time since the last bus has arrived in the station (before they got there) is also $10$
  3. thus the expected value by time for the time between the two buses is $20$

The second son says: The expected value for the time the next bus will arrive is $ 10 $ so the expected value between the last bus and the next bus is also $10$.

Which is correct?

Now I tried to approach the problem, I see how the expected if we start at a point and wait for a bus is $10$ (because it is distributed geometrically), however I can't seem to be able to read the data correctly.

Does it matter if we:

  • First plan all the buses (and not tell the family) and then place the family (in which case the second son is obviously correct)

  • First choose a number of buses to go before the family, then place the family, and then the other buses (which would mean the first son seems correct, although I'm not sure)

What is the right way to model this problem?

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Your first multi-line paragraph is unclear. –  ashley Dec 15 '12 at 21:38
    
I edited, is this better? –  Benjamin Gruenbaum Dec 15 '12 at 21:41
    
Is your time discrete or continuous? –  Eckhard Dec 15 '12 at 21:48
1  
The problem you describe seems to apply to a Poisson process, but your first line doesn't describe a Poisson process; in fact it's not clear to me how to interpret it. The question would make sense if you replaced the first line by "Buses arrive at the bus station in a Poisson process with an average rate of $1$ per ten minutes." –  joriki Dec 15 '12 at 21:48
    
The buss arrivals seem to be independent from the time the family going to the stop. so, i don't see the point in the first son's argument. –  ashley Dec 15 '12 at 21:48

1 Answer 1

up vote 2 down vote accepted

The daughter listens to both sons and then explains to them that they're calculating two different quantities:

The first son is calculating the mean duration of an inter-arrival interval selected randomly by uniformly choosing among all points in time and examining the interval that contains that point.

The second son is calculating the mean duration of an inter-arrival interval selected randomly by uniformly choosing among all inter-arrival intervals.

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Thank you. This was my intuition. So it is impossible to determine the expected value between two bus arrivals without knowing in which case we are in? –  Benjamin Gruenbaum Dec 15 '12 at 21:56
    
@Benjamin: You could put it that way, I guess. I'd put it this way: The term "expected value" presupposes a distribution, and uniformly selecting a point and uniformly selecting an interval lead to two different distributions for the intervals, so it's not surprising that they lead to two different expected values. So you have to decide which of the two you want to determine before determining it, and yes, in that sense, it's impossible to determine it without knowing which one you want to determine. –  joriki Dec 15 '12 at 22:04
    
I think I understand, thank you. –  Benjamin Gruenbaum Dec 15 '12 at 22:04

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