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.


  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

   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",