# Family of functions ranging from zero to one with parameter influencing rate of approach to one

What is a family of functions which goes from 0 to 1 and has one or more parameters that alters the rate at which a value of 1 is approached?

• @user333 Could you work on improving your question, please? As it stands, it is difficult to determine exactly what question is being asked, and what would be the purpose of such a function in a statistical context (ok I guess it might be related to your previous questions, e.g. on random walk).
– chl
Apr 7 '11 at 11:19

How do you like the Sigmoid Function ? By adding a parameter k one gets

$f_k(x)=\frac{1}{1+exp(-k*x)}$.

By setting k you can control how steep the function is and hence how fast it approaches 1.

• Any idea how to allow for "finer" tuning via two parameters? Where to put second parameter? Apr 19 '11 at 15:19
• @user333 yes. What about $f_{k,c}(x)=\frac{1}{1+exp(-k*x+c)}$ ? More generalized: $f(x)=\frac{1}{1+exp(g(x))}$ where g(x) is any function you like. Apr 19 '11 at 16:04

There's an incredible amount of those functions... and it really depends on what you want to do with it.

In general any function $f(x)$ with no infinite bounds can be limited to [0;1] by simply using $\frac{f(x)-min}{max-min}$ (where min and max are the minimum and maximum of the function).

One commonly used function is the sigmoid.