cumulative distribution functionnormal distribution
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?
There's no closed form expression for the inverse cdf of a normal (a.k.a. the quantile function of a normal). It looks like this:
There are various ways to express the function (e.g. as an infinite series or as a continued fraction), and numerous approximations (which is how computers are able to "calculate" it).
Reasonably accurate approximations are tedious to write and not especially enlightening (except in so far as the general forms convey a little insight into common ways of approximating functions you can't easily obtain in closed form).
If you regard $\text{erf}^{-1}$ or $\text{erf}$ itself as a special function, then it could be written as a function of one of those, but one could as well call $\Phi^{-1}$ a special function and be done in one step.
You're making the substitution $x = z - u$ to transform the integral. The differential of this is:
$$ dx = 0 - du = - du $$
So the calculation finishes up like this:
$$=\int_{0}^{1}f_X(z-u)du = - \int_{z}^{z-1}f_X(x)dx = \int_{z-1}^{z}f_X(x)dx$$
Best 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.