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Series Question (from a probability question)

  1. Sep 17, 2007 #1
    [SOLVED] Series Question (from a probability question)

    EDIT: Found it, never mind.


    Here is my question:

    How do I find what [tex]\sum_{k=1}^{\infty}k(1/2)^k[/tex] is?

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    Here is the original question and my work.
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    A couple decides to continue to have children until a daughter is born. What is the expected number of children of this couple?
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    So suppose P(having a daughter) = 1/2.

    Define the random variable X = number of children until a daughter is born.

    Then [itex]f_X(x) = (1-1/2)^{x-1}(1/2)[/itex]

    So,
    [tex]E(X) = \sum_{x = 1}^{\infty}x f_X(x) = \sum_{k=1}^{\infty}k(1/2)(1/2)^{k-1}[/tex]

    or as I wrote it above.

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    Now I believe everything I have done up this point is correct, however I can't remember how to do series. I have been looking at my calculus book, but I have yet to find what I need. Any hints would be greatly appreciated, thanks!


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    EDIT: Found it online, seems to be a common geometric series, not sure why I couldn't find it in my book.
     
    Last edited: Sep 17, 2007
  2. jcsd
  3. Sep 17, 2007 #2

    Dick

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    You know how to find f(x)=sum(x^n), right? f'(x)=sum(n*x^(n-1)). Is that enough of a hint?
     
  4. Sep 17, 2007 #3

    Dick

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    It's good to know how it's derived as well.
     
  5. Sep 17, 2007 #4
    Thanks, indeed it is good to know how to derive it.
     
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