Editing Normal distribution (section)

==== Using the Taylor series and Newton's method for the inverse function ====
An application for the above Taylor series expansion is to use [[Newton's method]] to reverse the computation.  That is, if we have a value for the [[cumulative distribution function]], <math display=inline>\Phi(x)</math>, but do not know the x needed to obtain the <math display=inline>\Phi(x)</math>, we can use Newton's method to find x, and use the Taylor series expansion above to minimize the number of computations.  Newton's method is ideal to solve this problem because the first derivative of <math display=inline>\Phi(x)</math>, which is an integral of the normal standard distribution, is the normal standard distribution, and is readily available to use in the Newton's method solution.

To solve, select a known approximate solution, <math display=inline>x_0</math>, to the desired {{tmath|\Phi(x)}}.  <math display=inline>x_0</math> may be a value from a distribution table, or an intelligent estimate followed by a computation of <math display=inline>\Phi(x_0)</math> using any desired means to compute. Use this value of <math display=inline>x_0</math> and the Taylor series expansion above to minimize computations.

Repeat the following process until the difference between the computed  <math display=inline>\Phi(x_{n})</math> and the desired {{tmath|\Phi}}, which we will call <math display=inline>\Phi(\text{desired})</math>, is below a chosen acceptably small error, such as 10<sup>−5</sup>, 10<sup>−15</sup>, etc.:

<math display=block>x_{n+1} = x_n - \frac{\Phi(x_n,x_0,\Phi(x_0))-\Phi(\text{desired})}{\Phi'(x_n)}\,,</math>

where

: <math display=inline>\Phi(x,x_0,\Phi(x_0))</math> is the <math display=inline>\Phi(x)</math> from a Taylor series solution using <math display=inline>x_0</math> and <math display=inline>\Phi(x_0)</math>

<math display=block>\Phi'(x_n)=\frac{1}{\sqrt{2\pi}}e^{-x_n^2/2}\,.</math>

When the repeated computations converge to an error below the chosen acceptably small value, ''x'' will be the value needed to obtain a <math display=inline>\Phi(x)</math> of the desired value, {{tmath|\Phi(\text{desired})}}.