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    POISSON distribution question




    Hey everybody,

    I'm working (struggling?) through a book about care logistics for my work. Sadly my background in statistics is rather limited. In the book the following case and questions are stated:

    - A doctor refers 4.8 patients/week to the MRI department
    - We can expect the weekly demand for MRI-slots is POISSON distributed, the standard deviation (2.2) seems to indicate just that.
    - we can calculate the probability that more than 5 MRI-slots are needed (35%)
    - However the number of unused slots also increases (on average 0.97 when 5 slots are reserved; 1.62 when 6 slots are reserved)

    Exercises:

    1) recreate the histogram that is in the book showing the Poisson distribution with Mean 4.8
    2)Calculate the probability that there is a shortage in MRI-slots given different numbers of reserved slots
    3) Reproduce the average number of unused slots and the average shortage.

    1) I have managed to do exercise 1, by generating probabilities with =POISSON(slot demand; 0.48; False) - please find my worksheet attached - and created a histogram from that data
    2)I used a SUMIF function to calculate this; the numbers check out with what's in the book
    3)
    - I haven't got the slightest clue how to get from the given data to an average number of unused slots.
    - When they talk about the "average shortage" does this mean for any given number of reserved slots(0-15), subtract the mean (4.8) and then average those results? What kind of information would you get from that number (2.7 in this case)?

    I would be very grateful if someone were to point me in the right direction on that last question.

    best regards,


    Marc
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    Re: POISSON distribution question

    From the given information, I guess we have the following:

    Let N \sim \text{Poisson}(\lambda) be the demand.
    Let s be the number of MRI slots provided.

    Number of unused slots = \max\{s - N, 0\}
    Number of shortage = \max\{N - s, 0\}

    Then the expected value
    E[\max\{s - N, 0\}]

    = \sum_{n = 0}^{s - 1} (s - n) e^{-\lambda} \frac {\lambda^n} {n!} + 0

    = s^2 - \sum_{n = 1}^{s - 1} e^{-\lambda} \frac {\lambda^n} {(n-1)!}

    which you can compute it by excel. The expected number of shortage is left to you as exercise.

    The information of this is that when you run this system for a very long time, the average number of unused slots/shortage across different periods will be very close to the theoretical values. It gives you an idea about the efficiency of the system, and try to find out the what number of slots is most efficient (in terms of something).

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    Re: POISSON distribution question


    First off all, thank you BGM for your reply.

    It has been at least 15 years since I have had any statistics course (haven't really used it since, at least not in this way) so I am kind of digging through my old notes as I am going along.

    The good news is, I managed to reproduce the number of unused slots with the first of your formulae (http://www.talkstats.com/~talkmath/t...c14a6c8d-1.gif). If I try to use the 2nd formula, which I understand should give the same results as the first, the outcomes are rather more exotic. I assume I am making a mistake in the maths...

    I am also struggling with the notation you use (i.e. max{s-N,0}). I understood it to mean "the maximum value from a sample set containing 2 numbers (s-N and 0)", so won't it always equal s-N?

    I am already extremely grateful for a practically usable formula (at least I can continue my work). I can identify parts of it (such as the the part that computes the POISSON distribution), but I don't fully understand the rest of it. A bit out of my league I am afraid....

    best regards,

    Marc

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