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Homework Help: Probability Generating Function / Geometric

  1. Feb 15, 2012 #1
    1. The problem statement, all variables and given/known data

    a) [itex]P(X=x)=pq^x,\,x\geq 0[/itex]

    Find the PGF.

    b) [itex]P(X=x)=pq^{|x|},\,x\,\epsilon\,\text{Z}[/itex]

    Find the PGF.

    2. The attempt at a solution

    a) [itex]G_X(s)=E(s^X)=\displaystyle\sum_{x\geq 0}pq^x s^x=p\displaystyle\sum_{x\geq 0}(qs)^x=\frac{p}{1-qs}[/itex]

    b) Not sure about this one... Is it: as above for [itex]x\geq0[/itex]. And for [itex]x<0[/itex]:

    [itex]G_X(s)=E(s^X)=\displaystyle\sum_{x>0}pq^{-x} s^{-x}=p\displaystyle\sum_{x\geq 0}(qs)^{-x}=\ldots[/itex]
     
    Last edited: Feb 15, 2012
  2. jcsd
  3. Feb 15, 2012 #2

    HallsofIvy

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    Since |-x|= |x| the negative values of x just double the value.
     
  4. Feb 15, 2012 #3
    Oh, okay I see. I got it totally mixed up.

    Is it: [itex]p+p\displaystyle\sum_{x\geq0}(qs)^{2x}=\ldots[/itex]
     
  5. Feb 15, 2012 #4

    Ray Vickson

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    Homework Helper

    In (b), try writing out a few terms:
    [tex] G = p + pq^1 s^1 + pq^{|-1|} s^{|-1|} + pq^2 s^2 + pq^{|-2|} s^{|-2|} + \cdots . [/tex]

    RGV
     
  6. Feb 15, 2012 #5
    [itex]p+2p\displaystyle\sum_{x>0}(qs)^{x}=p+\frac{2(p+qs-1)}{1-qs}[/itex]

    C'est correct?
     
    Last edited: Feb 15, 2012
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