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Conditional distribution

  1. May 11, 2014 #1
    let X1 and X2 be independent Poisson variables with respective parameters μ1 and μ2. Let S = X1 + X2. Is X1 given S=s a binomial dsitribution? What is the parameters?


    I just can show that S is a Poisson with mean μ1 + μ2. But im not confirm X1 given S is a binomial or not? Someone please help to prove it.
     
  2. jcsd
  3. May 11, 2014 #2

    mathman

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    S cannot be binomial. X2 ranges over all integers. Therefore S will also, while binomial range is finite.
     
  4. May 11, 2014 #3

    statdad

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    If the conditional distribution is intended you want to know about X1 when S = X1 + X2 = s, a fixed value. In this setting X1 cannot range over all integer values.
     
  5. May 12, 2014 #4

    mathman

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    You're right. I misread the question.

    It looks binomial:

    P(X1=x|S=s) = (μ1xμ2(s-x))/{x!(s-x)!(μ1+μ2)s}
     
    Last edited: May 12, 2014
  6. May 13, 2014 #5

    mathman

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    I was sloppy. There should be a coefficient of s!
     
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