Which function should I use to output a parabola-sigmoid-like of maximum $n$, $0 \le n \le 1$, with input of $0$ to $1$ It could be a simple question but I can't find the answer.
I have an input which is a decimal value between $0$ and $1$.
I need to find a function, sort of a parabolic sigmoid, which returns $1$ at its maximum, $n$ between $0$ and $1$, and progressively tend to $0$ when input tends to $0$ or $1$.
Thank you for your help.
 A: You might be able to do it with a cubic $f(x) = ax^3 +bx^2 + cx$. That function is $0$ at $0$.
Since $f(1) = 0$, 
$$
a + b + c = 0 .
$$
Since $f(n) = 1$, 
$$
an^3 + bn^2 + cn= 1 .
$$
Since $n$ is the maximum the derivative is $0$ there so
$$
3an^2 + 2bn + c = 0 .
$$
Since $n$ is known, you have three linear equations to solve for the three coefficients.
Warning: check that the function is positive on the unit interval - it might not be.
Another idea is to start with the parabola
$$
 x^2 - x
$$
and then find a function $g$ mapping the unit interval to itself that distorts the values on the $x$-axis to move the maximum at $1/2$ to $n$ where you want it. Your answer would be
$$
f(x) = g(x)^2 1 g(x).
$$
PS. "sigmoid" is not the shape you are looking for. 
A: After some researches, I found the function I wanted. It is a probability density function which solves exactly my problem.
$\displaystyle f(x, \mu, s) = \frac{ { 1 } }{s(e ^ \frac{ { x - \mu } } {{2s} } + e ^ \frac{ { -x - \mu } } {{2s} })^2}$
(The function can also be seen here)
