Cofee shop customers

  1. I've been working on this problem for quite a long time now but I'm not being able to solve it. It goes as follows :

    A coffee shop opens his doors from 9.00 to
    supposedly 8 customers will enter this shop and stay for a period of half an hour.
    Let X be the number of customers present in this shop at a certain time t, what is the probabilty mass function of X ?

    I would be very much grateful if you could please help me in solving this problem.
    Thank you.
  2. jcsd
  3. HallsofIvy

    HallsofIvy 41,259
    Staff Emeritus
    Science Advisor

    ??? All you say is that 8 customers will enter this shop and stay for a period of half an hour. Without more information, we can't say what the probability function is- it might well be that the 8 customers are guarenteed to come at 9 and leave and 9:30!

    I could see two plausible probability functions here: the simpler (in concept, probably not in calculation) is that the time each customer enters the shop is uniform over the 3 1/2 hour period from 9 to 12:30.

    However, it is more common for problems like this to obey a Poisson distribution. I suspect what you intend here is a Poisson distribution for the number of customers entering, with an average of 8 customers, each staying for 1/2 hour.
  4. Thank you for your reply.
    The problem here is that the number of customers that are present at the same time (which is X) could take the following values : 0,2,3,4,5,6,7 and 8.
    Hence they could as you said be all present at the same time (hence, X=8), as well as they could enter each for half an hour and leave , without anyone meeting the other (Hence X=0).
    The problem actually is to find the probability mass fuction of X for n customers over a period t of time, with each choosing a interval of time of length l , where sum of all intervals = t. But I just gave a simple example here.
    Thank you again for your reply and hope I have made the problem clearer now.
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