Indefinite Integration of the Gamma Integral
and Related Statistical Applications

Proof of Equation (4.13)

\begin{equation*}
{{M}_{x}}\left( t \right)=\frac{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}{2D}\exp \left( \mu t +\frac{{{\sigma }^{2}}{{t}^{2}}}{2}-\frac{{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}} \right)h_{\tfrac{-1}{2}}^{\frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}}}{\Bigr|}_{a}^{b}
\end{equation*}Proof:
The moment-generating function ${{M}_{x}}\left( t \right)$ is the Laplace Transform of $f\left( x \right)$
\begin{align*}
E\left( {{e}^{tx}} \right)= &\int_{a}^{b}{{{e}^{tx}}}f\left( x \right)dx \\
& =\int_{a}^{b}{\exp \left( tx \right)}\frac{\exp \left( \frac{-{{\left( x-\mu \right)}^{2}}}{2{{\sigma }^{2}}} \right)}{D}dx \\
& =\frac{1}{D}\int_{a}^{b}{\exp \left( \frac{-{{\left( x-\mu \right)}^{2}}}{2{{\sigma }^{2}}}+tx \right)dx} \\
& =\frac{\exp \left( \mu t+\frac{{{\sigma }^{2}}{{t}^{2}}}{2} \right)}{D}\int_{a}^{b}{\exp \left( \frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}} \right)dx}
\end{align*}
We evaluate $\int_{a}^{b}{\exp \left( \frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}} \right)dx}$ first.
Let $u=\frac{x-\left( \mu +{{\sigma }^{2}}t \right)}{\sqrt{2}\sigma }$ and $du=\frac{dx}{\sqrt{2}\sigma }$.
\begin{align*}
&\int_{a}^{b}{\exp \left( \frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}} \right)dx} \\
=&\sqrt{2}\sigma \int_{\frac{a-\left( \mu +{{\sigma }^{2}}t \right)}{\sqrt{2}\sigma }}^{\frac{b-\left( \mu +{{\sigma}^{2}}t \right)}{\sqrt{2}\sigma }}{\exp \left( -{{u}^{2}} \right)du} \\
=&\sqrt{2}\sigma \left\{ \frac{x-\left( \mu +{{\sigma }^{2}}t \right)}{2\sqrt{2}\sigma }\exp \left( \frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}} \right)h_{\tfrac{-1}{2}}^{\frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}}}{\Bigr|}_{a}^{b} \right\} \\
= &\frac{x-\left( \mu +{{\sigma }^{2}}t \right)}{2}\exp \left( \frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}} \right)h_{\tfrac{-1}{2}}^{\frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}} {2{{\sigma }^{2}}}}{\Bigr|}_{a}^{b} \\
\end{align*}
Therefore, we derive the moment-generating function
\begin{equation*}
{{M}_{x}}\left( t \right)=\frac{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}{2D}\exp \left( \mu t+\frac{{{\sigma }^{2}}{{t}^{2}}}{2}-\frac{{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}} \right)h_{\tfrac{-1}{2}}^{\frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}}}{\Bigr|}_{a}^{b}
\end{equation*}
$\square$

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