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Math Help - Moment Generating Function Gamma Distrib

  1. #1
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    Moment Generating Function Gamma Distrib

    \displaystyle\frac{1}{\sqrt{7}\Gamma\left(\frac{1}  {2}\right)}\lim_{b\to\infty}\int_0^b\left(x^{-1/2}\exp\left[-x\left(\frac{1}{7}-t\right)\right]\right) \ dx

    \displaystyle\frac{1}{\sqrt{7}\Gamma\left(\frac{1}  {2}\right)}\lim_{b\to\infty}\int_0^b\left(x^{-1/2}\exp\left[\frac{-x}{\frac{1}{\left(\frac{1}{7}-t\right)}}\right]\right)  \ dx

    \displaystyle\alpha=\frac{1}{2} \ \beta=\frac{1}{\frac{1}{7}-t}

    \displaystyle =\frac{1}{\sqrt{7}\Gamma\left(\frac{1}{2}\right)}\  left(\frac{1}{\frac{1}{7}-t}\right)^{1/2}\Gamma\left(\frac{1}{2}\right)

    Correct?
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  2. #2
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    Quote Originally Posted by dwsmith View Post
    \displaystyle\frac{1}{\sqrt{7}\Gamma\left(\frac{1}  {2}\right)}\lim_{b\to\infty}\int_0^b\left(x^{-1/2}\exp\left[-x\left(\frac{1}{7}-t\right)\right]\right) \ dx

    \displaystyle\frac{1}{\sqrt{7}\Gamma\left(\frac{1}  {2}\right)}\lim_{b\to\infty}\int_0^b\left(x^{-1/2}\exp\left[\frac{-x}{\frac{1}{\left(\frac{1}{7}-t\right)}}\right]\right) \ dx

    \displaystyle\alpha=\frac{1}{2} \ \beta=\frac{1}{\frac{1}{7}-t}

    \displaystyle =\frac{1}{\sqrt{7}\Gamma\left(\frac{1}{2}\right)}\  left(\frac{1}{\frac{1}{7}-t}\right)^{1/2}\Gamma\left(\frac{1}{2}\right)

    Correct?
    After simplification, does it agree with the general result given here: Gamma distribution - Wikipedia, the free encyclopedia ?
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  3. #3
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    It does.

    \displaystyle\frac{1}{(1-7t)^{1/2}}

    But when I take the second derivative as t = 0 and subtract that from the mean squared, I don't obtain the correct variance.
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  4. #4
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    Quote Originally Posted by dwsmith View Post
    It does.

    \displaystyle\frac{1}{(1-7t)^{1/2}}

    But when I take the second derivative as t = 0 and subtract that from the mean squared, I don't obtain the correct variance.
    I do You should try again.
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