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Re: Question: arrival-time puzzle



At 10:57 7/10/00 -0400, you wrote:
Regarding:
Suppose a lot of busses whiz past your window. The events are IID, that
is, Independent and Identically Distributed over time. The average rate is
6 busses per hour.

Fact: At any random time, if you ask how long it has been since the last
bus, the answer is 10 minutes, on average.

Is this true???

Jim Green

Yes.

David Bowman
David_Bowman@georgetowncollege.edu

Actually, no.

Jim, there you go again, noticing that the Emperor has no clothes.

The misdirection here is supposing that a succession of real world buses
can be independent and identically distributed in time.
They cannot, if one recalls the universal real world bus-scheduling
constraint: out of the depot, the order of the buses is known.

So a real world answer is that the average delay lies between the
constant frequency case, and the IID case.

This 2Xaverage paradoxical result is characteristic of physical systems
like a bunch of balls stuck under a plane surface with grease, from which
they can fall in any order.

To make over the bus puzzle into a more accurate form, we could take a period
long compared with the average interval of 10 minutes - say 10,000 minutes.
Let's charge 10,000 drivers with spinning a coin successively
(but instantaneously) every 10 minutes, and if they turn up a succession of
thirteen heads, then a succession of 11 tails or heads (but all the same)
they set off round the route.

(This procedure is intended to produce a chance near 1 in 10,000 of success.
I did not in fact construct the procedure with care, so it does not in fact
have a 1 in 10,000 chance of success. Only a nit-picker would notice
though...)

My respects to Jim Green!



brian whatcott <inet@intellisys.net> Altus OK
Eureka!