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Math Help - [SOLVED] 4th central moment

  1. #1
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    [SOLVED] 4th central moment

    Hi,

    I need to find  \mu_4 for an iid sample from an Exp (1) distribution. I am given the hint to "think" Gamma functions. I would appreciate any directional advice! (I know it's the 4th central moment, but not sure how to get toward Gamma functions...)

    Thanks!
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by Statistik View Post
    Hi,

    I need to find  \mu_4 for an iid sample from an Exp (1) distribution. I am given the hint to "think" Gamma functions. I would appreciate any directional advice! (I know it's the 4th central moment, but not sure how to get toward Gamma functions...)

    Thanks!
    The forth central moment is given by

    \mathbb{E}[X^4]=\int_{0}^{\infty}x^4f(x)dx

    The density function for an exponential is

    f(x)=\lambda e^{-\lambda x}

    So putting the two facts together gives

    \mathbb{E}[X^4]=\lambda\int_{0}^{\infty}x^4e^{-\lambda x}dx

    From here make the u=-\lambda x \implies du =-\lambda dx

    This should look alot like a gamma function.
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  3. #3
    MHF Contributor matheagle's Avatar
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    You can also obtain the fourth moment of a gamma by differentiating the MGF four times.

    M(t)=(1-\beta t)^{-\alpha}
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