Indefinite Integration of the Gamma Integral
and Related Statistical Applications

Proof of Equation (4.15)

$$\begin{equation*} {{m}_{2}}={{\mu }^{2}}+{{\sigma }^{2}}-\frac{{{\sigma }^{2}}}{D}\exp \left( \frac{-{{\left[ x-\mu \right]}^{2}}}{2{{\sigma }^{2}}} \right)\left( x+\mu \right){\Bigr|}_{a}^{b}. \end{equation*}$$ Proof:
$$\begin{align*} & \frac{{{\partial }^{2}}{{M}_{x}}\left( t \right)}{\partial {{t}^{2}}} \\ & =\frac{-{{\sigma }^{2}}}{D}\frac{\partial }{\partial t}\left\{ \exp \left( \frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}} \right){\Bigr|}_{a}^{b} \exp \left( \mu t+\frac{{{\sigma }^{2}}{{t}^{2}}}{2} \right) \right\} \\ & +\frac{1}{D}\frac{\partial }{\partial t}\left\{ \frac{x-\left( \mu +{{\sigma }^{2}}t \right)}{2}\exp \left( \frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}} \right)h_{\tfrac{-1}{2}}^{\frac{-{{\left[x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}}}{\Bigr|}_{a}^{b} \right\}\exp \left( \mu t+\frac{{{\sigma }^{2}}{{t}^{2}}}{2} \right)\left( \mu +t{{\sigma }^{2}} \right) \\ & +\frac{1}{D}\left\{ \frac{x-\left( \mu +{{\sigma }^{2}}t \right)}{2}\exp \left( \frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}} \right)h_{\tfrac{-1}{2}}^{\frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}}}{\Bigr|}_{a}^{b} \right\}\frac{\partial }{\partial t}\left[ \exp \left( \mu t+\frac{{{\sigma }^{2}}{{t}^{2}}}{2} \right) \right]\left( \mu +t{{\sigma }^{2}} \right) \\ & +\frac{1}{D}\left\{ \frac{x-\left( \mu +{{\sigma }^{2}}t \right)}{2}\exp \left( \frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}} \right)h_{\tfrac{-1}{2}}^{\frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}}}{\Bigr|}_{a}^{b} \right\}\left[ \exp \left( \mu t+\frac{{{\sigma }^{2}}{{t}^{2}}}{2} \right) \right]\frac{\partial }{\partial t}\left( \mu +t{{\sigma }^{2}} \right). \\ \end{align*}$$
With a few steps of operations,
$$\begin{align*} & \frac{{{\partial }^{2}}{{M}_{x}}\left( t \right)}{\partial {{t}^{2}}}\\ &=\frac{-{{\sigma }^{2}}x}{D}\exp \left( \frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}} \right)\exp \left( \mu t+\frac{{{\sigma }^{2}}{{t}^{2}}}{2} \right){\Bigr|}_{a}^{b} \\ & -\frac{{{\sigma }^{2}}}{D}\left( \mu +t{{\sigma }^{2}} \right)\exp \left( \mu t+\frac{{{\sigma }^{2}}{{t}^{2}}}{2} \right)\exp \left( \frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}} \right){\Bigr|}_{a}^{b} \\ & +\frac{x-\left( \mu +{{\sigma }^{2}}t \right)}{2D}{\left( \mu +{{\sigma }^{2}}t \right)^{2}}\exp {\left( \mu t+\frac{{{\sigma }^{2}}{{t}^{2}}}{2} \right)}\exp \left( \frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}} \right)h_{\tfrac{-1}{2}}^{\frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}}}{\Bigr|}_{a}^{b} \\ & +\frac{x-\left( \mu +{{\sigma }^{2}}t \right)}{2D}\left( {{\sigma }^{2}} \right)\exp \left( \mu t+\frac{{{\sigma }^{2}}{{t}^{2}}}{2} \right)\exp \left( \frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}} \right)h_{\tfrac{-1}{2}}^{\frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}}}{\Bigr|}_{a}^{b} . \end{align*}$$
Let $t=0$ and we derive
$$\begin{equation*} {{m}_{2}}=\frac{{{\partial }^{2}}{{M}_{x}}\left( t \right)}{\partial {{t}^{2}}}{{|}_{t=0}}={{\mu }^{2}}+{{\sigma }^{2}}-\frac{{{\sigma }^{2}}}{D}\exp \left( \frac{-{{\left[ x-\mu \right]}^{2}}}{2{{\sigma }^{2}}} \right)\left( x+\mu \right){\Bigr|}_{a}^{b}. \end{equation*}$$
$\square$

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