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Proving the poisson distribution is normalized

  1. Oct 3, 2011 #1
    SOLVED




    1. The problem statement, all variables and given/known data

    I am trying to prove that the poisson distribution is normalized, I think I've got an ok start but just having trouble with the next step.


    2. Relevant equations

    A counting experiment where the probability of observing n events (0≤n<∞) is given by:


    P(n) = (μ^n)/n! * e^(-μ)

    Where μ is a real number.


    3. The attempt at a solution

    Background (possibly incorrect)

    So it's discrete, as n will take integer values; I need a sum not an integral.

    Ʃ P(n) from n=0 to ∞ is just given by:

    Ʃ (μ^n)/n! * e^(-μ)

    And e^(-μ) does not vary with n, so:

    Ʃ P(n) = e^(-μ) * Ʃ (μ^n)/n!

    Important bit

    Now as I am trying to prove it is normalized, i need to get Ʃ P(n) = 1, so I assume my problem is getting from:

    (μ^n) / n!

    to

    e^μ


    Any tips or help would be much appreciated, thanks in advance.
     
    Last edited: Oct 3, 2011
  2. jcsd
  3. Oct 3, 2011 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    You are doing probability applications but have never seen this material before? Oh, well: see, eg., http://www.mcs.sdsmt.edu/tkowalsk/notes/Common-Taylor-series.pdf [Broken] .

    RGV
     
    Last edited by a moderator: May 5, 2017
  4. Oct 3, 2011 #3
    Yeah it's come up in my 2nd year Quantum module. Thanks for the link, I think I'd just forgotten that, curse of the double gap year.

    Makes sense now, thanks again.
     
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