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Best way to integrate a moment generating function?

  1. Oct 14, 2011 #1
    1. The problem statement, all variables and given/known data

    ∫etxx2e-x

    2. Relevant equations

    M(t) = etx f(x) dx

    3. The attempt at a solution

    I know the solution is -1/(t-3)3, however I'm having difficulty integrating the function. UV - ∫ V DU is extremely long and challenging, I'm wondering if there is a shortcut (i.e. quotient rule?)

    Also, there is a process used here but I'm unable to understand it:

    Untitled-1.jpg
     
  2. jcsd
  3. Oct 14, 2011 #2

    LCKurtz

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    Unfortunately, integration by parts is the antiderivative method generated by the product or quotient rules. I don't know of any shortcut. Let u = x2 the first round than u = x the second round and you should be able to integrate the result directly.
     
  4. Oct 14, 2011 #3

    lurflurf

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    It looks like that example integrated by parts multiple times. It is easy if you practice. Another method could be differentiation by the parameter.
    Let In(t) be integrals like yours where n is the power of x

    Then notice
    In(t)=DnI0(t)=Dn(-1/(t-1))

    where
    D is differentiation with respect to t
    I0(t)=(-1/(t-1))
     
  5. Oct 14, 2011 #4
    I did the integration by parts, and I ended up with 1/(1-t)3...is that incorrect?

    The mean and variance were still the same as the book's answers (μ = 3, σ2 = 3)
     
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