Appendix: Some Useful Integral Identities and Symmetry Properties of Normal Random Variables

Throughout all the formulas below, a, b, c, A, B, C are any real constants and X is a normal random variable with mean µ ∊ ℝ and standard deviation σ > 0, i.e., X ~ Norm(µ, σ2) with PDF ${\varphi }_{\mu ,\sigma }\left(x\right)\equiv \frac{{\text{e}}^{-{\left(x-\mu \right)}^2/2{\sigma }^2}}{\sigma \sqrt{2\pi }},-\infty <x<\infty$. The functions $\mathcal{N}$(x) and $\mathcal{N}$2(x, y; ρ) are the standard normal univariate and bivariate cumulative distribution functions, respectively.

$\text{E}\left[{\text{e}}^{BX}{\mathbb{I}}_{\left\{X>A\right\}}\right]\equiv {\displaystyle {\int }_A^{\infty }{\text{e}}^{Bx}{\varphi }_{\mu ,\sigma }\left(x\right)\text{d}x={\text{e}}^{\mu B+\frac{1}{2}{\sigma }^2{B}^2}\mathcal{N}}\left(\sigma B+\frac{\mu -A}{\sigma }\right)\text{ (A.1)}$

$\text{E}\left[{\text{e}}^{BX}{\mathbb{I}}_{\left\{X<A\right\}}\right]\equiv {\displaystyle {\int }_{-\infty }^A{\text{e}}^{Bx}{\varphi }_{\mu ,\sigma }\left(x\right)\text{d}x={\text{e}}^{\mu B+\frac{1}{2}{\sigma }^2{B}^2}\mathcal{N}}\left(-\sigma B+\frac{A-\mu }{\sigma }\right)\text{ (A.2)}$

$\text{E}\left[\mathcal{N}\left(AX+C\right)\right]\equiv {\displaystyle {\int }_{-\infty }^{\infty }\mathcal{N}\left(Ax+C\right){\varphi }_{\mu ,\sigma }\left(x\right)\text{d}x=\mathcal{N}}\left(\frac{\mu A+C}{\sqrt{1+{\sigma }^2{A}^2}}\right)\text{ (A.3)}$

$\begin{array}{c}\text{E}\left[{\text{e}}^{BX}\mathcal{N}\left(AX+C\right)\right]\equiv {\displaystyle {\int }_{-\infty }^{\infty }{\text{e}}^{Bx}\mathcal{N}\left(Ax+C\right){\varphi }_{\mu ,\sigma }\left(x\right)}\text{d}x\\ ={\text{e}}^{\mu B+\frac{1}{2}{\sigma }^2{B}^2}\mathcal{N}\left(\frac{\mu A+C+{\sigma }^{2}AB}{\sqrt{1+{\sigma }^2{A}^2}}\right)\end{array}\text{ (A.A)}$

${\displaystyle {\int }_0^{\infty }\mathcal{N}\left(Ax+B\right){\text{e}}^x\text{d}x=-\mathcal{N}\left(B\right)+{\text{e}}^{\left(1-2AB\right)/2A^2}\mathcal{N}\left(\frac{1-AB}{|A|}\right)}\left(\text{for}A<0\right)\text{ (A.5)}$

${\displaystyle {\int }_0^{\infty }\mathcal{N}\left(Ax+B\right){\text{e}}^{-x}\text{d}x=\mathcal{N}\left(B\right)+\mathrm{sgn} \left(A\right){\text{e}}^{\left(1+2AB\right)/2A^2}\mathcal{N}\left(-\frac{1+AB}{|A|}\right)}\text{ (A.6)}$

$\begin{array}{c}{\displaystyle {\int }_{-\infty }^{\infty }\mathcal{N}\left(Ax+B\right){\text{e}}^x\text{d}x}={\text{e}}^{\left(1-2AB\right)/2A^2}\mathcal{N}\left(\frac{1-AB}{|A|}\right)\\ -{\text{e}}^{\left(1+2AB\right)/2A^2}\mathcal{N}\left(-\frac{1+AB}{|A|}\right)\left(\text{for}A<0\right)\end{array}\text{ (A.7)}$

${\displaystyle {\int }_0^{\infty }\mathcal{N}\left(Ax+B\right)\text{d}x=\frac{1}{|A|}\left[B\mathcal{N}\left(B\right)+\frac{{\text{e}}^{-B^2/2}}{\sqrt{2\pi }}\right]}\left(\text{for}A<0\right)\text{ (A.8)}$

$\begin{array}{c}\text{E}\left[\mathcal{N}\left(AX+C\right){\mathbb{I}}_{\left\{X<B\right\}}\right]\equiv {\displaystyle {\int }_{-\infty }^B\mathcal{N}\left(Ax+C\right){\varphi }_{\mu ,\sigma }\left(x\right)\text{d}x}\\ ={\mathcal{N}}_2\left(\frac{B-\mu }{\sigma },\frac{\mu A+C}{\sqrt{1+{\sigma }^2{A}^2}};\frac{-\sigma A}{\sqrt{1+{\sigma }^2{A}^2}}\right)\end{array}\text{ (A.9)}$

$\begin{array}{c}\text{E}\left[\mathcal{N}\left(AX+C\right){\mathbb{I}}_{\left\{X>B\right\}}\right]\equiv {\displaystyle {\int }_B^{\infty }\mathcal{N}\left(Ax+C\right){\varphi }_{\mu ,\sigma }\left(x\right)\text{d}x}\\ ={\mathcal{N}}_2\left(\frac{\mu -B}{\sigma },\frac{\mu A+C}{\sqrt{1+{\sigma }^2{A}^2}};\frac{\sigma A}{\sqrt{1+{\sigma }^2{A}^2}}\right)\end{array}\text{ (A.10)}$

Note that formulas (A.1)–(A.10) simplify in the special case when $X^{\underset{¯}{\underset{¯}{d}}}Z~Norm\left(0,1\right)$ by setting µ = 0, σ = 1.

The bivariate normal CDF also has useful symmetry relations such as

${\mathcal{N}}_2\left(a,b;\rho \right)={\mathcal{N}}_2\left(b,a;\rho \right),\text{ (A.11)}$

${\mathcal{N}}_2\left(a,b;\rho \right)+{\mathcal{N}}_2\left(a,-b;-\rho \right)=\mathcal{N}\left(a\right).\text{ (A.12)}$

In the expectation formulas below, Z1,Z2 are i.i.d. standard normal random variables with covariance Cov(Z1,Z2)= ρ, |ρ| < 1, i.e., with joint PDF $n_2\left(x,y;\rho \right)\equiv \frac{{\partial }^2}{\partial x\partial y}{\mathcal{N}}_2\left(x,y;\rho \right)=\frac{1}{2\pi \sqrt{1-{\rho }^2}}{\text{e}}^{-\left(x^2+y^2-2\rho xy\right)/2\left(1-{\rho }^2\right)}$.

$\begin{array}{c}\text{E}\left[{\text{e}}^{-B{Z}_2}{\mathbb{I}}_{\left\{Z_1<a,Z_2<b\right\}}\right]\equiv {\displaystyle \int {\displaystyle {\int }_{{ℝ}^2}{\text{e}}^{-By}}{n}_2\left(x,y;\rho \right){\mathbb{I}}_{\left\{x<a,y<b\right\}}\text{d}x\text{d}y}\\ ={\text{e}}^{\frac{1}{2}{B}^2}{\mathcal{N}}_2\left(a+\rho B,b+B;\rho \right)\end{array}\text{ (A.13)}$

$\text{E}\left[{\text{e}}^{-B{Z}_2}{\mathbb{I}}_{\left\{Z_1>a,Z_2<b\right\}}\right]={\text{e}}^{\frac{1}{2}{B}^2}{\mathcal{N}}_2\left(-\left(a+\rho B\right),b+B;-\rho \right)\text{ (A.14)}$

$\text{E}\left[{\text{e}}^{-B{Z}_2}{\mathbb{I}}_{\left\{Z_1<a,Z_2>b\right\}}\right]={\text{e}}^{\frac{1}{2}{B}^2}{\mathcal{N}}_2\left(a+\rho B,-\left(b+B\right);-\rho \right)\text{ (A.15)}$

$\text{E}\left[{\text{e}}^{-B{Z}_2}{\mathbb{I}}_{\left\{Z_1>a,Z_2>b\right\}}\right]={\text{e}}^{\frac{1}{2}{B}^2}{\mathcal{N}}_2\left(-\left(a+\rho B\right),-\left(b+B\right);\rho \right)\text{ (A.16)}$

Note: interchanging a ↔ b in (A.13)–(A.16) gives respectively equivalent formulas for the expectations $\text{E}\left[{\text{e}}^{-B{Z}_1}{\mathbb{I}}_{\left\{Z_1<a,Z_2<b\right\}}\right]$, $\text{E}\left[{\text{e}}^{-B{Z}_1}{\mathbb{I}}_{\left\{Z_1<a,Z_2>b\right\}}\right]$, $\text{E}\left[{\text{e}}^{-B{Z}_1}{\mathbb{I}}_{\left\{Z_1>a,Z_2<b\right\}}\right]$, and $\text{E}\left[{\text{e}}^{-B{Z}_1}{\mathbb{I}}_{\left\{Z_1>a,Z_2>b\right\}}\right]$.