Logit Transformation

$\begin{figure}\begin{center}\BoxedEPSF{LogitTransform.epsf scaled 700}\end{center}\end{figure}$

The function

$\begin{displaymath} z=f(x)=\ln\left({x\over 1-x}\right). \end{displaymath}$

This function has an inflection point at

, where

$\begin{displaymath} f''(x)={2x-1\over x^2(x-1)^2}=0. \end{displaymath}$

Applying the logit transformation to values obtained by iterating the Logistic Equation generates a sequence of Random Numbers having distribution

$\begin{displaymath} P_z={1\over \pi(e^{x/2}+e^{-x/2})}, \end{displaymath}$

which is very close to a Gaussian Distribution.

References

Collins, J.; Mancilulli, M.; Hohlfeld, R.; Finch, D.; Sandri, G.; and Shtatland, E. ``A Random Number Generator Based on the Logit Transform of the Logistic Variable.'' Computers in Physics 6, 630-632, 1992.

Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 244-245, 1995.