1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    Staff Emeritus
    Science Advisor

    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

    User Avatar
    Science Advisor
    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Probability Generating Function / Geometric
Loading...