Intervals drinking coffee

Posted on 2012-09-13
Last Modified: 2012-09-17
Now I'm trying to understand the Poisson process.

If over 24 hours I drink 11 cups of coffee, what is the probability that the interval between cups is more than 1.5 hours but less than 2 hours?
Question by:purplesoup
    LVL 27

    Expert Comment

    This can only be answered if it is known that the coffee is consumed at random intervals (unlikely) and that each cup is consumed in a VERY short time (also unlikely, especially if it is hot)

    Author Comment

    ok let's say I'm (a) drinking the coffee over random intervals and (b) the coffee is in a very small cup and is at drinkable temperature, so therefore in effect it is drunk instantly.
    LVL 27

    Expert Comment

    You have stated the requisite assumptions nicely but I still have not formulated the problem correctly in my mine. That is a deficiency on my part. A more knowledgeable person would probably give you a straight answer.
    My difficulty is with the eleven cups.
    Are you looking for the average interval?
    Do you start the 24 hours by drinking a cup?
    You universe which you are sampling goes from 0 to 24 (decimals allowed). Each interval is equally likely. So the average is 2.4 hours. I am having difficulty with the fact that the 10 intervals must add up to 24. I imagine you have some difficulty there too. Hope that a more knowledgeable person comes along.
    LVL 18

    Accepted Solution

    process is poisson with lamda = 11/24 (per hour)

    the poisson process is related to exponential distribution, see

    prob of time between cups given by exponential distribution

    P(x > X) =  e ^ (-lamda x)

    P(x > 1.5)=  e ^ (- (3/2) lamda )

    P(x > 2) =  e ^ (- 2 lamda )

    P (x > 1.5 and x < 2) = P(x > 1.5) - P(x > 2)
      = e ^ (- (3/2) lamda ) - e ^ (- 2 lamda )
    = e ^ (- (3/2) (11/24) ) - e ^ (- 2 (11/24) ) = .103

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