Indefinite Integration of the Gamma Integral
and Related Statistical Applications

Proof of Equation (4.14)

$$\begin{equation*} {{m}_{1}}=\mu-\frac{{{\sigma}^{2}}}{D}\exp\left(\frac{-{{\left[x-\mu \right]}^{2}}}{2{{\sigma }^{2}}} \right){\Bigr|}_{a}^{b} \end{equation*}$$ Proof:
Taking the first partial derivative on $t$, we derive
$$\begin{align*} &\frac{\partial {{M}_{x}}\left( t \right)}{\partial t}\\ &=\frac{1}{D}\frac{\partial }{\partial t}\left\{ \frac{x-\left( \mu +{{\sigma }^{2}}t \right)}{2}\exp \left( \frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}} \right)h_{\tfrac{-1}{2}}^{\frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}}}{\Bigr|}_{a}^{b} \right\}\exp \left( \mu t+\frac{{{\sigma }^{2}}{{t}^{2}}}{2} \right) \\ & +\frac{1}{D}\left\{ \frac{x-\left( \mu +{{\sigma }^{2}}t \right)}{2}\exp \left( \frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}} \right)h_{\tfrac{-1}{2}}^{\frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}}}{\Bigr|}_{a}^{b} \right\}\frac{\partial }{\partial t}\exp \left( \mu t+\frac{{{\sigma }^{2}}{{t}^{2}}}{2} \right), \end{align*}$$
where
$$\begin{align*} & \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\} \\ & =\frac{\partial }{\partial t}\left[ \frac{x-\left( \mu +{{\sigma }^{2}}t \right)}{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} \\ & +\left[ \frac{x-\left( \mu +{{\sigma }^{2}}t \right)}{2} \right]\frac{\partial }{\partial t}\left\{ \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{-{{\sigma }^{2}}}{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} \\ & +\frac{{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{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} \\ & =-{{\sigma }^{2}}\exp \left( \frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}} \right)\left\{ \frac{1}{2}h_{\tfrac{-1}{2}}^{\frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}}}-\frac{{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}}h_{\tfrac{1}{2}}^{\frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}}} \right\}{\Bigr|}_{a}^{b} \\ & =-{{\sigma }^{2}}\exp \left( \frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}} \right){\Bigr|}_{a}^{b}, \end{align*}$$
and
$$\begin{equation*} \frac{\partial }{\partial t}\exp \left( \mu t+\frac{{{\sigma }^{2}}{{t}^{2}}}{2} \right)=\exp \left( \mu t+\frac{{{\sigma }^{2}}{{t}^{2}}}{2} \right)\left( \mu +t{{\sigma }^{2}} \right). \end{equation*}$$
Therefore,
$$\begin{multline*} \frac{\partial {{M}_{x}}\left( t \right)}{\partial t}=\frac{-{{\sigma }^{2}}}{D}\exp \left( \frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right)\right]}^{2}}}{2{{\sigma }^{2}}} \right){\Bigr|}_{a}^{b} \exp \left( \mu t+\frac{{{\sigma }^{2}}{{t}^{2}}}{2} \right) \\ +\frac{1}{D}\left\{ \frac{x-\left( \mu +{{\sigma }^{2}}t \right)}{2}\exp \left( \frac{-{{\left[ x-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}} \right)h_{\tfrac{-1}{2}}^{\frac{-{{\left[ u-\left( \mu +{{\sigma }^{2}}t \right) \right]}^{2}}}{2{{\sigma }^{2}}}}{\Bigr|}_{a}^{b} \right\}\\ \cdot \exp \left( \mu t+\frac{{{\sigma }^{2}}{{t}^{2}}}{2} \right)\left( \mu +t{{\sigma }^{2}} \right). \end{multline*}$$
Let $t=0$ and we derive
$$\begin{equation*} {{m}_{1}}=\frac{\partial {{M}_{x}}\left( t \right)}{\partial t}{{|}_{t=0}}=\mu -\frac{{{\sigma }^{2}}}{D}\exp \left( \frac{-{{\left[ x-\mu \right]}^{2}}}{2{{\sigma }^{2}}} \right){\Bigr|}_{a}^{b} \end{equation*}$$
$\square$

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