Closed Form of Normal Distribution

Question

What does closed form in following sentence mean and why we need tables of c.d.f.?

Normal distributions's p.d.f. cannot be integrated in closed form, and hence tables of the c.d.f. or computer programs are necessary in order to compute probabilities and quantiles for normal distributions.

Who still uses tables these days? O_o Most of the computing environments I know can easily evaluate the CDF of the normal distribution (as well as the inverse), and even if they weren't implemented, the algorithms for numerically evaluating them are not too complicated. — J. M. ain't a mathematician, Jul 08 '12 at 16:46

Robert Israel · Answer 1 · 2012-07-08T16:48:23.833

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"Closed form" is not really a well-defined concept, and in fact I would call $\dfrac12+\dfrac12 {{\rm erf}\left(x\sqrt {2}/2\right)}$ a closed form. Basically the question is whether it can be expressed as a finite expression using "well-known" functions. I would say erf is a well-known function, but others might disagree.

What is true is that the normal cdf is not an elementary function. An elementary function is built up from constants and the variable $x$ using a finite number of the following steps:

the arithmetic operations +-*/
exponentials
logarithms
root of a polynomial whose coefficients are elementary functions

Note that the usual trig and inverse trig functions are included, since they can be expressed by (complex) exponentials and logarithms.

edited Jul 08 '12 at 16:48

answered Jul 08 '12 at 16:42

Robert Israel

448,999

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See here. – Did Jul 08 '12 at 16:46
By the way, that Wikipedia entry is wrong: it's not just $n$'th roots, it's roots of polynomials. For example, $z$ satisfying $z^5 + x z + 1$ is an elementary function, although it can't be expressed in terms of radicals. – Robert Israel Jul 08 '12 at 16:52
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I don't know; I wouldn't consider the Bring radical as elementary... – J. M. ain't a mathematician Jul 08 '12 at 16:57
@RobertIsrael: You might want to indicate a source giving a definition you agree with. – Did Jul 08 '12 at 17:11
For example, see Risch, "The problem of integration in finite terms", Trans. Amer. Math. Soc. 139 (1969), 167-189: http://dx.doi.org/10.2307%2F1995313 – Robert Israel Jul 08 '12 at 19:07
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@RobertIsrael Thanks (but please use the @ symbol). – Did Jul 08 '12 at 19:41

Closed Form of Normal Distribution

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