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Probability: Discrete Random Variable

  1. Jul 12, 2012 #1
    1. The problem statement, all variables and given/known data
    Suppose X is a discrete random variable whose probability generating function is
    G(z) = z^2 * exp(4z-4)


    2. Relevant equations
    No idea


    3. The attempt at a solution
    I'm thinking that due to the exponent on the z term, that the exp(4z-4) would be the
    P[X=3] = exp(4z-4), but i'm not even sure of this.

    I honestly have no idea where to even start on a problem like this. Any sort of guidance would be great.
     
  2. jcsd
  3. Jul 12, 2012 #2

    Simon Bridge

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  4. Jul 12, 2012 #3
    All the info given in [1] is what was given for the problem, I forgot to say that I am suppose to find the expected value, var[x], and the distribution on x.

    I honestly have no idea what I am doing. Any hints would be great.
     
  5. Jul 12, 2012 #4

    Simon Bridge

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    Well, that link has the information. Have a look, have a go, and then show us where you get stuck.

    Start with [tex]m_X(t) = \sum_{k=0}^n {P(X=k) e^{kt}}[/tex]... to get the distribution, then use the definitions for expectation and variance.
     
  6. Jul 12, 2012 #5

    Ray Vickson

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    What you have written is the "moment-generating function", rather that the probability generating function. For a discrete random variable [itex]X \in \{0,1,2,\ldots \}[/itex] the probability generating function is
    [tex] G(z) \equiv E z^X = \sum_{k=0}^{\infty} P(X=k) z^k.[/tex]
    See, eg., http://en.wikipedia.org/wiki/Probability-generating_function .


    RGV
     
  7. Jul 13, 2012 #6

    Simon Bridge

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    Why yes I did, and it is indeed - please see post #2, and the link from that post, for the reasoning behind that :)
     
  8. Jul 13, 2012 #7

    Ray Vickson

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    Of course one can use the moment-generating function for discrete, integer-valued random variables, but it is not very convenient; the moment-generating function (or Laplace transform) works better for continuous random variables. In the OP's example, the mgf would be
    [tex] M_X(t) = G(e^t) = e^{2t - 4 + 4e^t},[/tex]
    which is not particularly nice to work with.

    RGV
     
  9. Jul 13, 2012 #8

    Simon Bridge

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    Well... either way OP has a place to start.
     
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