Re: Probability question
- From: Gary Carson <garycarson@xxxxxxxxxxxxxxxxxxxxxxx>
- Date: Tue, 22 Aug 06 1:15:14 GMT
On Aug 21 2006 6:08 PM, Mark-T wrote:
Gary Carson wrote:
A bus arrives at the station at a known average interval.
(random, but with known expected frequency) You
get to the station at some random time. What is your
expected (average) wait time?
That depends on what you mean by random.
The usual interpretation.
And it pretty much has nothing to do with the paradox of
the two envelope problem.
It's another probability paradox, which I
thought the math nerds here might find amusing
The mis-interpretation most people make in the bus stop
question is related to a lack of full understanding of the
Markov property and what independent arrivals means.
Could be...
However, like the envelope paradox, I believe it is ill
posed. Note that the time between bus arrivals is
a random variable, with no upper bound. Then,
what does it mean, to claim you show up at the
station at a random time (uniformly distributed)?
The issue is the same as the envelope problem -
a uniform distribution over an unbounded interval
(or undefined) interval.
Well, no, in the bus stop problem the time intervals aren't uniform, they're
exponential.
That's why it matters what you mean by random. The usual definition in arrival
processes is a randomenss that comes from independence. Each arrival is
independent of each other, and each is independent of time. Uniform arrivals
aren't independent of time because the uniform distribution is only defined over
a finite range.
There's no such thing as a uniform distribution over an unbounded range.
If the number of buses to arrive in a fixed period of time follows a poisson
distribution, then the time between arrivals follows an exponential
distribution, and the expected wait time is just the mean time between buses
independentlhy of what time you get there. In fact if you get there and see a
bus that's just leaving you have the same expected wait time than if you get
there and someone else is waiting and tells you they've been waiting 10 minutes.
That's not really a paradox, the mathematics of it is straight forward.
Counter-intuitive but not paradoxal.
Also, I'm not sure your question means what you think it means
if there's only one bus.
What difference does it make? You could say
all the buses are identical, in which case it may
as well be the same bus every time...
I have no idea what that means.
If you have only one bus going in a circle then you can't have two buses
arriving in a time period to short for the bus to travel the circle.
In the bus problem you have a bus terminal where buses leave every 20 minutes,
at fixed intervals (on a street like Halstead street in Chicago, a really long
street). With stops at every block, after about 30 miles the random delays that
have hit buses will give you close to poisson distributied arrivals, on the
average 20 minutes apart, but frequent occurances of 2-3 buses arriving back to
back and somewhat less frequent occurances of buses an hour apart.
Your expected wait time is 20 minutes no matter what time the last bus came.
The model isn't exact in that delays that hit the buses aren't exactly
independent. If bus 4 is delayed by a wreck at mile 3 then the conditional
probability of bus 5 also being delayed at the point is higher than the
conditional probability of bus 9 being delayed at that point. But if it's a
really long bus route it will be close.
Gary Carson
Mark
http://www.garycarson.com
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