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Moments from characteristic function geometric distribution

  1. Jan 6, 2017 #1
    1. The problem statement, all variables and given/known data

    Hi,

    I have the probabilty density: ##p_{n}=(1-p)^{n}p , n=0,1,2... ##

    and I am asked to find the characteristic function: ##p(k)= <e^{ikn}> ## and then use this to determine the mean and variance of the distribution.

    2. Relevant equations

    I have the general expression for the characteristic function : ##\sum\limits^{\infty}_{n=0} \frac{(-ik)^m}{m!} <x^{m}> ## * , from which can equate coefficients of ##k## to find the moments.

    3. The attempt at a solution

    So I have ## <e^{-ikn}>=\sum\limits^{\infty}_{n-0} (1-p)^{n}p e^{-ikn} ##

    I understand the solution given in my notes which is that this is equal to, after some rearranging etc, expanding out using taylor :

    ## 1 + \frac{(1-p)}{ p} (-k + 1/2 (-ik)^{2} + O(k^3) ) + \frac{(1-p)^{2}} { p^2 } ( (-ik)^{2} + O(k^3))##
    and then equating coefficients according to *

    However my method was to do the following , and I'm unsure why it is wrong:

    ## <e^{-ikn}>=\sum\limits^{\infty}_{n=0} (1-p)^{n} p e^{-ikn} = \sum\limits^{\infty}_{n=0} (1-p)^{n}p \frac{(-ik)^n}{n!} ##

    And so comparing to * ## \implies ##

    ## \sum\limits^{\infty}_{n=0} (1-p)^{n}p = \sum\limits^{\infty}_{n=0} <x^{n}> ##

    Anyone tell me what I've done wrong? thank you, greatly appreciated.
     
  2. jcsd
  3. Jan 6, 2017 #2

    Ray Vickson

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    Simplify:
    $$p (1-p)^n e^{-ikn} = p x^n, \; \text{where} \; x = (1-p)e^{-ik}$$
    Now just sum the geometric series ##\sum x^n##.
     
  4. Jan 8, 2017 #3
    Yeah that's fine, and in the end you get the 'solution given' which I said I understand. That's what's been done before expanding out.

    My problem was wondering what is wrong with the method I post after that...
     
    Last edited: Jan 8, 2017
  5. Jan 8, 2017 #4

    Ray Vickson

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    You seem to be saying that ##p(1-p)^n e^{-ikn}## is the same as ##p(1-p)^n (-ik)^n/n!##, but this is obviously false.
     
    Last edited: Jan 8, 2017
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