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Any idea on how to integrate the following?

  1. May 28, 2014 #1
    1. The problem statement, all variables and given/known data

    The c.d.f. of a random variable X is given by the following formula

    F(x) = 1 - e-x2 x≥0

    Find M(t). Moment generating function.

    3. The attempt at a solution

    I've found the pdf as

    f(x) = F'(x) = 2xe-x2 x≥0

    To determine M(t) I need to do the following

    M(t) = E(etX) = ∫etx 2xe-x2

    Then I'm not sure how to go about integrating. Any ideas or clues to point me in the right direction will be greatly appreciated. Thanks.
     
  2. jcsd
  3. May 28, 2014 #2

    jbunniii

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    Try completing the square: ##x^2 - tx = x^2 - tx + (t/2)^2 - (t/2)^2 = (x - t/2)^2 - (t/2)^2##, so
    $$\int e^{tx}2xe^{-x^2}dx = e^{(t/2)^2} \int 2x e^{-(x-t/2)^2} dx$$
    which should be manageable with a bit more work.
     
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