# Bus stop independent events expected value

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
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