Is there a parametrized family of diverse continuous functions whose input and output are reals in [0.0, 1.0]?
The closest I can think of is the Beta distribution:
1) CDF works, but all the functions are monotonically increasing, so not super interesting. I can add a "flip" parameter to also get the monotonically decreasing versions.
2) PDF output is not [0.0, 1.0], but I could divide by max. Again would need to add a "flip" parameter to get the horizontally flipped versions.
I could add another flag to pick between these two.
Thanks!
The set of all continuous functions from $[0,1]$ to $[0,1]$ is a semigroup $S$ with respect to a composition. The group $H(1)$ of its invertible elements is the group of homeomorphisms of $[0,1]$. In particular, $H(1)$ contains a parametrized semigroup $P=\{x^t:t>0\}$. If you want a parametrized family consisting of non-monotonic map, you can pick any non-monotonic map $g\in S$ (for instance, $g(x)=4x(1-x)$ or $g(x)=\sin(\pi x)$) and consider a parametrized family $Pg$ or $gP$). Moreover, for any parametrized family $F$ of functions from $[0,1]$ to $\Bbb R$ and a function $f$ from $\Bbb R$ to $[0,1]$ (for instance, $f(x)=\sin^2 x$ or $f(x)=\frac {2}{\pi}\arctan (x)$), $f(F)$ is a parametrized family of functions from $[0,1]$ to $[0,1]$.