This is an artcle of my recent acquaintance. It is about various integrals involving the standard normal distribution $$\Phi(x)$$ and its derivative. Besides it is a handy reference, it is interesting to see how the author organize the hundreds of integrals in a manner that is easy to lookup.

The paper considers integrals in the general form

$\int f(x) \Phi(a+bx)^c \Phi(bx)^d \phi(a+bx)^g \phi(bx)^h x^i dx$

and 14 different functions $$f(x)$$ are considered. They are encoded by a single (hexadecimal?) digit, such as $$f(x)=e^{ax}$$ is 1 and $$f(x)=\ln\Phi(x)$$ is 6. Then, together with $$(c,d,g,h,i)$$, a 6-digit number can be used to represent the integral. For example, number 001000 is

$\int\Phi(bx)dx = x\Phi(bx)+\frac1b\phi(bx)$

and number 100010 is

$\int e^{cx}\phi(x)dx = e^{c^2/2}\Phi(x-c)$

There is a different paper1 (based on error function erf() instead of $$\Phi(x)$$) of similar nature but a bit more difficult to use as the integrals are categorized in broad sections, and no such handy indexing is provided.

PDF

1. Edward W. Ng and Murray Geller. A table of integrals of the error functions. J. Research of the National Bureau of Standards — B. Mathematical Sciences, 73B(1), 1969. PDF

## Bibliographic data

@article{
title = "A table of normal integrals",
author = "D. B. Owen",
journal = "Communication in Statistics --- Simulation and Computation",
volume = "9",
number = "4",
year = "1980",
pages = "389--419",
doi = "10.1080/03610918008812164",
}