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Thread: Moments Generating Function

  1. #1
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    Moments Generating Function

    Moments Generating Function-topic10-1.jpg
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  2. #2
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    Re: Moments Generating Function

    $V(X) = E[X^2] - \left(E[X]^2\right)$

    $E[X^k] = \left . \dfrac{d^k}{dt^k}M_X(t) \right|_{t=0}$

    $M_X(t) = \dfrac{1}{1-3t}$

    $E[X] = \left . \dfrac{d}{dt}M_X(t)\right|_{t=0} = \left .\dfrac{3}{(1-3 t)^2} \right|_{t=0} = 3$

    Similarly

    $E[X^2] = \left . \dfrac{18}{(1-3 t)^3} \right|_{t=0} = 18$

    $V[X] = 18 - 3^3 = 9$
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