wu :: forums - Print Page


    
      
        wu :: forums
        (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi)
      

        riddles >> medium >> From Littlewood's Book
        
(Message started by: Barukh on Sep 5^th, 2007, 6:33am)

Title: From Littlewood's Book
Post by Barukh on Sep 5^th, 2007, 6:33am

The following passage is taken from Littlewood’s book “Mathematician’s Miscellany”:

“Suppose busses on a given route average 10-minute intervals. If they run at exactly 10 minutes intervals, the average time a passenger arriving randomly at a stop will have to wait is 5 minutes. If the busses are irregular, the average time is greater; for one kind of random distribution it is 10 minutes; and what is more, the average time since the previous bus is also 10 minutes. For a certain other random distribution, both times become infinity.”

This may be easy, may be hard, so I put it in Medium. I don’t know the answer.

Title: Re: From Littlewood's Book
Post by Grimbal on Sep 5^th, 2007, 6:37am

And what is the question? To prove the assertion? To find a distribution where both times go to infinity?

Title: Re: From Littlewood's Book
Post by Barukh on Sep 5^th, 2007, 6:44am

Let's start by finding a distribution where both times equal 10 minutes.

Title: Re: From Littlewood's Book
Post by Grimbal on Sep 5^th, 2007, 6:47am

That would be when 2 buses arrive simultaneously every 20 minutes.

Title: Re: From Littlewood's Book
Post by towr on Sep 5^th, 2007, 7:08am

Isn't the average time to the next bus, and the time to the previous one always the same; it seems symmetric, since you can just reverse time.

Title: Re: From Littlewood's Book
Post by Barukh on Sep 5^th, 2007, 8:15am

on 09/05/07 at 06:47:02, Grimbal wrote:

That would be when 2 buses arrive simultaneously every 20 minutes.

Yes, that does the trick (for both cases). Afterall, it was easy...

on 09/05/07 at 07:08:50, towr wrote:

Isn't the average time to the next bus, and the time to the previous one always the same; it seems symmetric, since you can just reverse time.

:-/

Title: Re: From Littlewood's Book
Post by Grimbal on Sep 5^th, 2007, 9:01am

on 09/05/07 at 07:08:50, towr wrote:

Isn't the average time to the next bus, and the time to the previous one always the same; it seems symmetric, since you can just reverse time.

I thought there might be non-symetric cases, especially if the waiting time raises steadily as the buses get more and more clustered, but I cannot think of a case where the two times converge to a different value when averaged over a larger and larger period.

Title: Re: From Littlewood's Book
Post by Eigenray on Sep 6^th, 2007, 3:47pm

What if we have [hide]k buses arrive at time T(k)=k(k-1)/2, for each k[/hide]? Then (# buses before time n)/n http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/to.gif 1, while E(waiting time | arrive before time n) ~ n/3 http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/to.gif http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/infty.gif.

Or are we supposed to consider a periodic solution, where the bus arrival times have a certain probability distribution? I don't think this is possible, because as long as there is a positive (constant) probability of a bus coming in each period, the expected waiting time is finite.

Title: Re: From Littlewood's Book
Post by Barukh on Sep 6^th, 2007, 10:48pm

on 09/06/07 at 15:47:24, Eigenray wrote:

Or are we supposed to consider a periodic solution, where the bus arrival times have a certain probability distribution?

Yes, that's what I thought at the beginning, feeling it may be hard to find such a distribution.

Quote:

I don't think this is possible, because as long as there is a positive (constant) probability of a bus coming in each period, the expected waiting time is finite.

But is it at least possible for 10-minutes waiting time case?