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Math Help - Finding the Moment Generating Function of the Geometric Distribution

  1. #1
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    Finding the Moment Generating Function of the Geometric Distribution

    Hi
    Not too sure if this is the correct area, but hopefully someone can help!
    I've got a problem that has arisen from another problem:
    <br />
\begin{aligned}\sum_{k=0}^\infty e^{tk} p(1-p)^k <br />
&=pe^t \sum_{k=0}^\infty (e^t(1-p))^k \end{aligned}

    but I can't see how to go from the last line to:

    \begin{array}{cc}p(1-p)^{k-1}&\mbox{k=1,2,3...}<br />
 \end{array}\\
    Last edited by mr fantastic; August 7th 2009 at 10:57 PM. Reason: Re-titled the post
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  2. #2
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    Quote Originally Posted by bigdoggy View Post
    Not too sure if this is the correct area, but hopefully someone can help!
    I've got a problem that has arisen from another problem:
    \begin{aligned}\sum_{k=0}^\infty e^{tk} p(1-p)^k <br />
&=pe^t \sum_{k=0}^\infty (e^t(1-p))^k \end{aligned}
    but I can't see how to go from the last line to:
    \begin{array}{cc}p(1-p)^{k-1}&\mbox{k=1,2,3...}<br />
 \end{array}\\
    Frankly I see multiple problems with this question.
    Maybe you should post the original question.

    This bit of mathematical factoring is impossible.
    \begin{aligned}\sum_{k=0}^\infty e^{tk} p(1-p)^k <br />
&=\color{red}pe^t \sum_{k=0}^\infty (e^t(1-p))^k \end{aligned}
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  3. #3
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    Hi

    Yes you're right!

    \begin{aligned}\sum_{k=1}^\infty e^{tk} p(1-p)^{k-1 }<br />
&= \frac{p}{1-p} \sum_{k=0}^\infty (e^t(1-p))^k \end{aligned}

    How would I solve this to produce:

    \frac{pe^t}{1-(1-p)e^t}
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  4. #4
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    Quote Originally Posted by bigdoggy View Post
    Hi

    Yes you're right!

    \begin{aligned}\sum_{k=1}^\infty e^{tk} p(1-p)^{k-1 }<br />
&= \frac{p}{1-p} \sum_{k=0}^\infty (e^t(1-p))^k \end{aligned}

    How would I solve this to produce:

    \frac{pe^t}{1-(1-p)e^t}
    It would help if you posted the original question (as already requested) which I assume was finding the moment generating function of the geometric distribution.

    You will find that that particular question has been solved at least once in these forums (use the Search tool).
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  5. #5
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    I wonder what's going on with MGF... there have been several these last 2 days

    And this question is part of this : http://www.mathhelpforum.com/math-he...97251-mgf.html

    If you didn't understand some steps, you could've just asked them there, I would have answered...
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  6. #6
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    Quote Originally Posted by Moo View Post
    I wonder what's going on with MGF... there have been several these last 2 days

    And this question is part of this : http://www.mathhelpforum.com/math-he...97251-mgf.html

    If you didn't understand some steps, you could've just asked them there, I would have answered...
    Therefore ..... thread closed.
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