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Probability Theory: Poisson Distribution

  1. Sep 8, 2013 #1
    1. The problem statement, all variables and given/known data

    A random variable has a Poisson distribution with parameter λ = 2. Compute the following probabilities, giving an exact answer and a decimal approximation.

    P(X ≥ 4)


    2. Relevant equations

    P(X = k) = λke/k!

    3. The attempt at a solution

    P(X ≥ 4) = Ʃk = 4 λke/k!

    = λ4e/4! + λ5e/5! + λ6e/6! + [itex]\cdots[/itex]

    = e4/4! + λ5/5! + λ6/6! + [itex]\cdots[/itex]]

    = e Ʃk = 4 λk/k!

    Let n = k - 4

    =e Ʃn = 0 λn+4/(n+4)!

    plug in λ = 2

    = e-2 Ʃn = 0 2n+4/(n+4)!

    = e-2 Ʃn = 0 2n24/(n+4)!

    = 16e-2 Ʃn = 0 2n/(n+4)!

    This is as far as I have gotten, but I'm not sure I'm on the correct track. I used wolfram alpha to reduce the summation term, Ʃn = 0 2n/(n+4)!, to [1/48(3e2-19)], but I'm at a loss as to how to get there myself.

    Anyone have any suggestions? It would be greatly appreciated!
     
  2. jcsd
  3. Sep 8, 2013 #2

    Ray Vickson

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    ##P(X \geq 4) = 1-P(X \leq 3).##
     
  4. Sep 8, 2013 #3

    BruceW

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    Ray's 'clue' is what you must use for this problem. generally, this would be a difficult problem. But luckily, 4 is not a large number.
     
  5. Sep 8, 2013 #4
    Doh! That little fact completely slipped my mind. Thanks guys!
     
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