# Why the CDF for the Normal Distribution can not be expressed as a closed form function?

I am working my way through Think Stats, where the author states that

"there is no closed form expression for the normal cumulative density function"

but does not provide any further details as to why this is the case, simply saying that the alternative is to write it in terms of the error function.

Is there some way to intuit why the Normal Distribution can not be expressed as a closed form function?

• This was originally proved by Liouville in the early 19th century. Here are some resources to start with: (1) B. Conrad, Integration in elementary terms, (2) M. Rosenlicht, Integration in finite terms, Amer. Math. Monthly 79 (1972), 963–972, and (3) The Risch algorithm. – cardinal Nov 12 '11 at 16:01
• – cardinal Nov 12 '11 at 16:02
• Probably a little heavy going for me, was hoping for an intuitive explanation! – Joel Nov 12 '11 at 16:11
• Yes, it is a little heavy going, I admit. Conrad's writeup is really nice, but I think even he is a little overoptimistic in his assessment that the intended audience is "talented high-school students"! – cardinal Nov 12 '11 at 16:13
• The basic idea is that an antiderivative that can be written in terms of a finite number of "elementary" functions has to have a pretty particular form. When you take $e^{-x^2}$ as the function for which you desire to find an antiderivative, you then show that the particular form needed cannot actually be found. – cardinal Nov 12 '11 at 16:19

## 1 Answer

You first have to think about the definition of "closed form". The obvious kind of "closed form" is polynomials; Having only addition and multiplication, their values can actually be computed directly and not approximated using tables.
Does the log function have a "closed form"? Yes- by the common convention (see comments below). And no- in the sense it cannot be computed directly and its values are taken from tables. The gamma function is another such example. There is indeed an historic convention of calling some functions "closed form". However, once you note that most functions are actually computed using tables, or approximated using polynomials, then the CDF is not much different than a log.

• $\log$ is considered an elementary function, so I think your explanation leaves a little bit to be desired. In fact, the form that the antiderivative solution of $e^{-x^2}$ must take is a sum of functions that are compositions of $\log$ with other elementary functions! – cardinal Nov 12 '11 at 17:13
• I think you may be conflating computability with having a closed form. Moreover, a closed form is much broader than a polynomial in elementary functions; for instance, it can include compositions of such functions. It is also interesting that the logarithm (and its inverse, the exponential) can be easily and exactly computed for a large number of values, whereas the error function cannot; and that derivatives and integrals of the logarithm can be exactly computed symbolically, whereas (at least for integrals) that is not the case for the error function. – whuber Nov 12 '11 at 17:17
• @cardinal; My point is that what is considered as "closed" is just a matter of definition; Quoting Wikipedia: "...an expression is said to be a closed-form expression if it can be expressed analytically in terms of a bounded number of certain "well-known" functions". Define the error function as "well-known" and you have yourself a "closed form". Following that rationale, many "elementary functions" might not have a "closed form" either. – JohnRos Nov 12 '11 at 17:27
• @JohnRos: The term elementary function has a very precise definition which, as far as I know, actually arose as part of the work that Liouville originally did in this area. In my experience, when people speak of "closed form" expressions they almost always implicitly mean in terms of these elementary functions and that is certainly the case in this instance. :) – cardinal Nov 12 '11 at 17:47
• I am very annoyed by the fact that there does not appear to be a simple way to make either an en-dash or em-dash in comments, which is where most of my participation happens. :) (You can make them in answers, however, by using the appropriate HTML codes.) – cardinal Nov 12 '11 at 18:49