amit7ul and Traden4Alpha, both of you changed the problem to suit how you "think" that buses should arrive somewhere, by saying that if a bus has just arrived, then it will take the next bus uniform(0,10) time to arrive. To say that the buses arrive according to a Poisson process, as omk originally did, is a well-defined problem (whether or not it approximates reality), and you should do that problem on its merits, rather than say that omk's example is "wrong."(To formulate the problem equivalently, without getting caught up in one's own preconceptions about buses, say that you're away from home, and phone calls are made to your home as some Poisson process with a rate lambda, and you're trying to figure out how long it will take on average, for the phone to ring after you return home.)Without getting into all the gory details (which one can find in, e.g., the first chapter of the second volume of Feller's probability book), the answer to omk's problem is 10. Intuitively, if you think of all the intervals between bus arrivals, you're more likely to come to the bus stop at a long interval rather than a short interval, and thus the average length of the interval in which you come is not 10, but rather 20, which is the number you have to halve to get the correct answer of 10.On a separate note, computer simulations make me believe that the solution to the problem posed by amit7ul and Traden4Alpha (buses come at uniform(0,10) times; how long do you have to wait after you come?) is not 5. Because of length-biasing of intervals as with the Poisson process above, I'd have expected it to be more than 5, but surprisingly, I keep getting answers around 10/3. I don't know why this is, though.EDIT: the mean of a uniform(0,10) is 5, not 10. So the naive answer would be to halve 5, and get 5/2 as the mean waiting time. But again, because of length-biasing of intervals as in the Poisson case, the expected length of the interval should be longer than 5/2, and indeed 10/3 makes sense given this.
Last edited by gentinex
on September 18th, 2006, 10:00 pm, edited 1 time in total.