Do anyone know what is Inverse Cumulative Distribution Function of Normal Distribution formula or equation? I am looking at Google but did not find any good answer.


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:

graph showing inverse cdf of standard normal

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 $\text{erf}^{-1}$ as a special function ($\text{erf}$ is often regarded as a special function), then it could be written as a function of that, but one could as well call $\Phi^{-1}$ a special function.


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