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Probability generating function (binomial distribution)

  1. Oct 12, 2009 #1
    1. The problem statement, all variables and given/known data
    The probabilty generating funtion G is definied for random varibles whos range are [tex]\subset[/tex] {0,1,2,3,......}. If Y is such a random variable we will call it a counting random varible. Its probabiltiy generating function is [tex]G(s) = E(s^{y}) [/tex] for those s's such that [tex]E(|s|^{y})[/tex]) < [tex]\infty[/tex].

    2. Relevant equations

    binomial distribution = [tex]\left(\stackrel{n}{y}\right)[/tex][tex]p^{y}[/tex][tex]q^{n-y}[/tex] , y = 0,1,2,3,....n and 0 [tex]\leq[/tex] p [tex]\leq[/tex] 1

    3. The attempt at a solution

    What i have so far is...

    [tex]G(s) = E(s^{y}) [/tex] = [tex]\sum[/tex] [tex]s^{y}[/tex][tex]\left(\stackrel{n}{y}\right)[/tex][tex]p^{y}[/tex][tex]q^{n-y}[/tex]

    [tex]G(s) = E(s^{y}) [/tex] = [tex]\sum[/tex] [tex]\left(\stackrel{n}{y}\right)[/tex][tex](sp)^{y}[/tex][tex]q^{n-y}[/tex]

    not sure where to go from that. i managed to do it for the geometric random variable distribution b/c there was no "n choose y". Thanks to wiki, I know what the answer should be. The answer is G(s) = [tex][(1-p) + ps]^{n}[/tex]. I can't see how they went from what i have above to that.
     
  2. jcsd
  3. Oct 12, 2009 #2
    well i just realized that [tex]G(s) = E(s^{y}) [/tex] = [tex]\sum[/tex] [tex]\left(\stackrel{n}{y}\right)[/tex][tex](sp)^{y}[/tex][tex]q^{n-y}[/tex]

    is the same thing as [tex](q + sp)^{n}[/tex] .

    Also by definition [tex] p + q = 1 \Rightarrow q = 1-p [/tex] which means...

    [tex]G(s) = E(s^{y}) = [(1-p) + ps]^{n} [/tex]
     
    Last edited: Oct 12, 2009
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