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Basics of moment generating functions

  1. Apr 9, 2013 #1

    Mentallic

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    I missed my class that introduced us to moment generating functions, and my notes are missing some pretty essential parts to helping me understand them, so here it is:


    1. The problem statement, all variables and given/known data

    Find the moment generating function of [itex]f_X(x)=e^{-x}, x>0[/itex]



    2. Relevant equations

    [tex]E(X^R)=\int_{-\infty}^{\infty}X^Rf_X(x)dx[/tex]

    R must be a natural number I believe.

    [tex]m_X(u) = E(e^{uX})[/tex]



    3. The attempt at a solution

    I'm unsure if the expectation is applied in the same way, in other words, would this be correct?

    [tex]m_X(u) = E(e^{ux})[/tex]

    [tex]= \int_{-\infty}^{\infty}e^{uX}f_X(x)dx[/tex]

    [tex]= \int_{0}^{\infty}e^{ux}e^{-x}dx[/tex]

    [tex]= \int_{0}^{\infty}e^{x(u-1)}dx[/tex]

    And I'm not quite sure what the u in this case is either.
     
  2. jcsd
  3. Apr 9, 2013 #2

    vela

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  4. Apr 9, 2013 #3

    Mentallic

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    Oh awesome, thanks vela.

    I can also see that the expectation can be found quite easily after deriving the moment generating function, which is good because I'll be needing it :smile:
     
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