Re: Probability question

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?

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.

The time intervals between bus arrivals, you mean.
But that misses the point...

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.

OK

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.

Yes, that is the point...

I know the bus arrival frequency is exponential distributed,
I wasn't referring to that.

Rather, the passenger arrival time is described as
'random', which implies uniformly random... but over
what interval? Bus A left at noon, and Bus B could
arrive any time between now and doomsday, mathematically
speaking. This interval is unknown, with no upper bound,
so how does one define the random variable representing
the passenger's arrival?

That is what I mean by ill posed... a uniform distribution
over a (potentially) unbounded range. Same as the
envelope paradox...

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
independently 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.

Mark

.

Relevant Pages

  • Re: Probability question
    ... even if the question is about you coming to the bus stop rather than just ... Markov property and what independent arrivals means. ... distribution is only defined over a finite range. ... There's no such thing as a uniform distribution over an unbounded range. ...
    (rec.gambling.poker)
  • Re: Probability question
    ... Markov property and what independent arrivals means. ... Well, no, in the bus stop problem the time intervals aren't uniform, they're ... aren't independent of time because the uniform distribution is only defined over ... If the number of buses to arrive in a fixed period of time follows a poisson ...
    (rec.gambling.poker)
  • Re: Renewal process
    ... be played by a geometric distribution with parameter p? ... studied "compound random variables" yet? ... if N = number of arrivals until ... we put the arriving customer on hold; ...
    (sci.math)
  • Re: Gamma Distribution application for modeling arrivals
    ... trying to use gamma distribution to model the arrival ... namely the shape and the scale parameter. ... same in the create block to which represents arrivals ...
    (sci.stat.math)
  • Re: Poisson and exponential distributions
    ... i'm having some trouble seeing the relationship between the Poisson and the ... Because it is not a probability density, so does not need to integrate ... Thus, the complementary cumulative distribution of T is exp, ... if at least n arrivals occur by t. ...
    (sci.math)