# Logistic function

The lo­gis­tic func­tion is a sig­moid func­tion that maps the real num­bers to the unit in­ter­val $$(0, 1)$$ us­ing the for­mula $$\displaystyle f(x) = \frac{1}{1 + e^{-x}}$$.

More gen­er­ally, there ex­ists a fam­ily of lo­gis­tic func­tions that can be writ­ten as $$\displaystyle f(x) = \frac{M}{1 + \alpha^{c(x_0 - x)}}$$, where:

• $$M$$ is the up­per bound of the func­tion (in which case the func­tion maps to the in­ter­val $$(0, M)$$ in­stead). When $$M = 1$$, the lo­gis­tic func­tion is usu­ally be­ing used to calcu­late some prob­a­bil­ity or pro­por­tion of a to­tal.

• $$x_0$$ is the in­flec­tion point of the curve, or the value of $$x$$ where the func­tion’s growth stops speed­ing up and starts slow­ing down.

• $$\alpha$$ is a vari­able con­trol­ling the steep­ness of the curve.

• $$c$$ is a scal­ing fac­tor for the dis­tance.

## Applications

• The lo­gis­tic func­tion is used to model growth rates of pop­u­la­tions in an ecosys­tem with a limited car­ry­ing ca­pac­ity.

• The in­verse lo­gis­tic func­tion (with $$\alpha = 2$$) is used to con­vert a prob­a­bil­ity to log-odds form for use in Bayes’ rule.

• The lo­gis­tic func­tion (with $$\alpha = 10$$ and $$c = 1/400$$) is used to calcu­late the ex­pected prob­a­bil­ity of a player win­ning given a spe­cific differ­ence in rat­ing in the Elo rat­ing sys­tem.

Parents:

• Mathematics

Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.