This article presents an exact solution to the two-dimensional normal integral as follows:

\begin{multline*}
\text{     }\frac{1}{\left( 2\pi  \right)\left| \Sigma  \right|}\int_{{{A}_{1}}}^{{{B}_{1}}}{\int_{{{A}_{2}}}^{{{B}_{2}}}{\exp \left\{ -\tfrac{1}{2}(\mathbf{x}-\mu {)}'{{\Sigma }^{-1}}(\mathbf{x}-\mu ) \right\}d{{x}_{1}}d{{x}_{2}}}}\\\shoveleft =\frac{1}{8\pi}{{t}_{1}}h_{\tfrac{-1}{2}}^{\tfrac{-1}{2}{{{t}_{1}}^{2}}}\exp \left\{ \frac{-{{t}_{1}}^{2}}{2} \right\} \left( \frac{{{t}_{2}}}{\sqrt{1-{{\rho }^{2}}}}-\frac{\rho {{t}_{1}}}{\sqrt{1-{{\rho }^{2}}}} \right) \cdot \\
\qquad \qquad h_{\tfrac{-1}{2}}^{\tfrac{-1}{2}{{\left( \tfrac{{{t}_{2}}}{\sqrt{1-{{\rho }^{2}}}}-\tfrac{\rho {{t}_{1}}}{\sqrt{1-{{\rho }^{2}}}} \right)}^{2}}} \exp \left\{ \frac{-1}{2}{{\left( \frac{{{t}_{2}}}{\sqrt{1-{{\rho }^{2}}}}-\frac{\rho {{t}_{1}}}{\sqrt{1-{{\rho }^{2}}}} \right)}^{2}} \right\} \\
\shoveleft +\frac{1}{4\pi}\rho \sqrt{1-{{\rho }^{2}}} \exp \left\{ \frac{-{{t}_{1}}^{2}}{2} \right\} \exp \left\{ \frac{-1}{2}{{\left( \frac{{{t}_{2}}}{\sqrt{1-{{\rho }^{2}}}}-\frac{\rho {{t}_{1}}}{\sqrt{1-{{\rho }^{2}}}} \right)}^{2}} \right\} \\
\qquad \cdot \sum\limits_{i=0}^{\infty }{\sum\limits_{j=0}^{i}{\Bigg\{C_{i-j}^{i}{{\left( -{{\rho }^{2}} \right)}^{i-j}}{{\left( {{{t}_{1}}^{2}} \right)}^{j}}\cdot \frac{{{\Delta }^{i-j}}{{h}^{\tfrac{-1}{2}{{\left( \tfrac{{{t}_{1}}}{\sqrt{1-{{\rho }^{2}}}}-\tfrac{\rho {{t}_{2}}}{\sqrt{1-{{\rho }^{2}}}} \right)}^{2}}}}\left[ \tfrac{-1}{2}+j \right]}{\left( -1+2j \right)!!}\Bigg\}}} \\
\shoveleft +\frac{1}{4\pi}{{t}_{1}}\left( \frac{{{t}_{1}}}{\sqrt{1-{{\rho }^{2}}}}-\frac{\rho {{t}_{2}}}{\sqrt{1-{{\rho }^{2}}}} \right)\rho \exp \left\{ \frac{-{{t}_{1}}^{2}}{2} \right\} \exp \left\{ \frac{-1}{2}{{\left( \frac{{{t}_{2}}}{\sqrt{1-{{\rho }^{2}}}}-\frac{\rho {{t}_{1}}}{\sqrt{1-{{\rho }^{2}}}} \right)}^{2}} \right\} \\
\qquad \quad \cdot \sum\limits_{i=0}^{\infty }{\sum\limits_{j=0}^{i}{\Bigg\{C_{i-j}^{i}{{\left( -{{\rho }^{2}} \right)}^{i-j}}{{\left( {{{t}_{1}}^{2}} \right)}^{j}}\cdot \frac{{{\Delta }^{i-j}}{{h}^{\tfrac{-1}{2}{{\left( \tfrac{{{t}_{1}}}{\sqrt{1-{{\rho }^{2}}}}-\tfrac{\rho {{t}_{2}}}{\sqrt{1-{{\rho }^{2}}}} \right)}^{2}}}}\left[ \tfrac{1}{2}+j \right]}{\left( 1+2j \right)!!}\Bigg\}}}\Bigg|_{{{t}_{1}}=\frac{{{A}_{1}}-{{\mu }_{1}}}{{{\sigma }_{1}}}}^{{{t}_{1}}=\frac{{{B}_{1}}-{{\mu }_{1}}}{{{\sigma }_{1}}}}\Bigg|_{{{t}_{2}}=\frac{{{A}_{2}}-{{\mu }_{2}}}{{{\sigma }_{2}}}}^{{{t}_{2}}=\frac{{{B}_{2}}-{{\mu }_{2}}}{{{\sigma }_{2}}}} \\
\end{multline*}


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