List of integrals of Gaussian functions

In the expressions in this article,

\phi (x)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}x^{2}}

is the standard normal probability density function,

\Phi (x)=\int _{-\infty }^{x}\phi (t)\,dt={\frac {1}{2}}\left(1+\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)\right)

is the corresponding cumulative distribution function (where erf is the error function), and

T(h,a)=\phi (h)\int _{0}^{a}{\frac {\phi (hx)}{1+x^{2}}}\,dx

is Owen's T function.

Owen[1] has an extensive list of Gaussian-type integrals; only a subset is given below.

Indefinite integrals

$\int \phi (x)\,dx=\Phi (x)+C$
$\int x\phi (x)\,dx=-\phi (x)+C$
$\int x^{2}\phi (x)\,dx=\Phi (x)-x\phi (x)+C$
$\int x^{2k+1}\phi (x)\,dx=-\phi (x)\sum _{j=0}^{k}{\frac {(2k)!!}{(2j)!!}}x^{2j}+C$ [2]
$\int x^{2k+2}\phi (x)\,dx=-\phi (x)\sum _{j=0}^{k}{\frac {(2k+1)!!}{(2j+1)!!}}x^{2j+1}+(2k+1)!!\,\Phi (x)+C$

In the previous two integrals, $n!!$ is the double factorial: for even $n$ it is equal to the product of all even numbers from 2 to $n$ , and for odd $n$ it is the product of all odd numbers from 1 to $n$ ; additionally it is assumed that $0!! = (-1)!! = 1$ .

$\int \phi (x)^{2}\,dx={\frac {1}{2{\sqrt {\pi }}}}\Phi \left(x{\sqrt {2}}\right)+C$
$\int \phi (x)\phi (a+bx)\,dx={\frac {1}{t}}\phi \left({\frac {a}{t}}\right)\Phi \left(tx+{\frac {ab}{t}}\right)+C,\qquad t={\sqrt {1+b^{2}}}$ [3]
$\int x\phi (a+bx)\,dx=-{\frac {1}{b^{2}}}\left(\phi (a+bx)+a\Phi (a+bx)\right)+C$
$\int x^{2}\phi (a+bx)\,dx={\frac {1}{b^{3}}}\left((a^{2}+1)\Phi (a+bx)+(a-bx)\phi (a+bx)\right)+C$
$\int \phi (a+bx)^{n}\,dx={\frac {1}{b{\sqrt {n(2\pi )^{n-1}}}}}\Phi \left({\sqrt {n}}(a+bx)\right)+C$
$\int \Phi (a+bx)\,dx={\frac {1}{b}}\left((a+bx)\Phi (a+bx)+\phi (a+bx)\right)+C$
$\int x\Phi (a+bx)\,dx={\frac {1}{2b^{2}}}\left((b^{2}x^{2}-a^{2}-1)\Phi (a+bx)+(bx-a)\phi (a+bx)\right)+C$
$\int x^{2}\Phi (a+bx)\,dx={\frac {1}{3b^{3}}}\left((b^{3}x^{3}+a^{3}+3a)\Phi (a+bx)+(b^{2}x^{2}-abx+a^{2}+2)\phi (a+bx)\right)+C$
$\int x^{n}\Phi (x)\,dx={\frac {1}{n+1}}\left(\left(x^{n+1}-nx^{n-1}\right)\Phi (x)+x^{n}\phi (x)+n(n-1)\int x^{n-2}\Phi (x)\,dx\right)+C$
$\int x\phi (x)\Phi (a+bx)\,dx={\frac {b}{t}}\phi \left({\frac {a}{t}}\right)\Phi \left(xt+{\frac {ab}{t}}\right)-\phi (x)\Phi (a+bx)+C,\qquad t={\sqrt {1+b^{2}}}$
$\int \Phi (x)^{2}\,dx=x\Phi (x)^{2}+2\Phi (x)\phi (x)-{\frac {1}{\sqrt {\pi }}}\Phi \left(x{\sqrt {2}}\right)+C$
$\int e^{cx}\phi (bx)^{n}\,dx={\frac {e^{\frac {c^{2}}{2nb^{2}}}}{b{\sqrt {n(2\pi )^{n-1}}}}}\Phi \left({\frac {b^{2}xn-c}{b{\sqrt {n}}}}\right)+C,\qquad b\neq 0,n>0$

Definite integrals

$\int _{-\infty }^{\infty }x^{2}\phi (x)^{n}\,dx={\frac {1}{\sqrt {n^{3}(2\pi )^{n-1}}}}$
$\int _{-\infty }^{0}\phi (ax)\Phi (bx)\,dx={\frac {1}{2\pi |a|}}\left({\frac {\pi }{2}}-\arctan \left({\frac {b}{|a|}}\right)\right)$
$\int _{0}^{\infty }\phi (ax)\Phi (bx)\,dx={\frac {1}{2\pi |a|}}\left({\frac {\pi }{2}}+\arctan \left({\frac {b}{|a|}}\right)\right)$
$\int _{0}^{\infty }x\phi (x)\Phi (bx)\,dx={\frac {1}{2{\sqrt {2\pi }}}}\left(1+{\frac {b}{\sqrt {1+b^{2}}}}\right)$
$\int _{0}^{\infty }x^{2}\phi (x)\Phi (bx)\,dx={\frac {1}{4}}+{\frac {1}{2\pi }}\left({\frac {b}{1+b^{2}}}+\arctan(b)\right)$
$\int _{-\infty }^{\infty }x\phi (x)^{2}\Phi (x)\,dx={\frac {1}{4\pi {\sqrt {3}}}}$
$\int _{0}^{\infty }\Phi (bx)^{2}\phi (x)\,dx={\frac {1}{2\pi }}\left(\arctan(b)+\arctan {\sqrt {1+2b^{2}}}\right)$
$\int _{-\infty }^{\infty }\Phi (a+bx)^{2}\phi (x)\,dx=\Phi \left({\frac {a}{\sqrt {1+b^{2}}}}\right)-2T\left({\frac {a}{\sqrt {1+b^{2}}}},{\frac {1}{\sqrt {1+2b^{2}}}}\right)$
$\int _{-\infty }^{\infty }x\Phi (a+bx)^{2}\phi (x)\,dx={\frac {2b}{\sqrt {1+b^{2}}}}\phi \left({\frac {a}{t}}\right)\Phi \left({\frac {a}{{\sqrt {1+b^{2}}}{\sqrt {1+2b^{2}}}}}\right)$ [4]
$\int _{-\infty }^{\infty }\Phi (bx)^{2}\phi (x)\,dx={\frac {1}{\pi }}\arctan {\sqrt {1+2b^{2}}}$
$\int _{-\infty }^{\infty }x\phi (x)\Phi (bx)\,dx=\int _{-\infty }^{\infty }x\phi (x)\Phi (bx)^{2}\,dx={\frac {b}{\sqrt {2\pi (1+b^{2})}}}$
$\int _{-\infty }^{\infty }\Phi (a+bx)\phi (x)\,dx=\Phi \left({\frac {a}{\sqrt {1+b^{2}}}}\right)$
$\int _{-\infty }^{\infty }x\Phi (a+bx)\phi (x)\,dx={\frac {b}{t}}\phi \left({\frac {a}{t}}\right),\qquad t={\sqrt {1+b^{2}}}$
$\int _{0}^{\infty }x\Phi (a+bx)\phi (x)\,dx={\frac {b}{t}}\phi \left({\frac {a}{t}}\right)\Phi \left(-{\frac {ab}{t}}\right)+{\frac {1}{\sqrt {2\pi }}}\Phi (a),\qquad t={\sqrt {1+b^{2}}}$
$\int _{-\infty }^{\infty }\ln(x^{2}){\frac {1}{\sigma }}\phi \left({\frac {x}{\sigma }}\right)\,dx=\ln(\sigma ^{2})-\gamma -\ln 2\approx \ln(\sigma ^{2})-1.27036$

References

Owen 1980.
Patel & Read (1996) lists this integral above without the minus sign, which is an error. See calculation by WolframAlpha.
Patel & Read (1996) report this integral with error, see WolframAlpha.
Patel & Read (1996) report this integral incorrectly by omitting x from the integrand.

Owen, D. (1980). "A table of normal integrals". Communications in Statistics: Simulation and Computation. B9 (4): 389–419. doi:10.1080/03610918008812164.
Patel, Jagdish K.; Read, Campbell B. (1996). Handbook of the normal distribution (2nd ed.). CRC Press. ISBN 0-8247-9342-0.

Lists of integrals
Rational functions Irrational functions Trigonometric functions Inverse trigonometric functions Hyperbolic functions Inverse hyperbolic functions Exponential functions Logarithmic functions Gaussian functions Definite integrals

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.