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Math Help - Moment-generating Function

  1. #1
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    Moment-generating Function

    Show that if a random variable has the probability density
    f(x) = {\frac {1}{2}e^{-|x|}} for -∞ < x < ∞
    its moment-generating function is given by
    Mx(t) = {\frac {1}{1-t{^2}}}

    I know that I need to find the integral, but I am getting confused. Could you please help?!
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  2. #2
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    Quote Originally Posted by TheHolly View Post
    Show that if a random variable has the probability density
    f(x) = {\frac {1}{2}e^{-|x|}} for -∞ < x < ∞
    its moment-generating function is given by
    Mx(t) = {\frac {1}{1-t{^2}}}

    I know that I need to find the integral, but I am getting confused. Could you please help?!
    By definition:


    M_{X}(t) = \frac{1}{2} \int_{-\infty}^{+\infty} e^{tx} \, e^{-|x|} \, dx


    Substitute |x| = x for x > 0 and |x| = -x for x < 0:


    = \frac{1}{2} \left( \int_{-\infty}^{0} e^{tx} \, e^{x} \, dx + \int^{\infty}_{0} e^{tx} \, e^{-x} \, dx \right)


    = \frac{1}{2} \left( \int_{-\infty}^{0} e^{(1+t)x} \, dx + \int^{\infty}_{0} e^{-(1-t) x} \, dx \right)


    = \frac{1}{2} \left( \frac{1}{1 + t} \left[ e^{(1+t)x} \right]_{-\infty}^{0} - \frac{1}{1 - t} \left[ e^{-(1-t)x}\right]_{0}^{+\infty} \right)


    = \frac{1}{2} \left( \frac{1}{1 + t} + \frac{1}{1 - t} \right)


    and you should be able to finish it off from here.
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  3. #3
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    Thank you! Really, after I saw how to break that integral up into two integrals, I was able to solve the rest myself.
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