I am having issues trying to do something that is fairly simple. Given the interval [0,1] I need to find a function from [0,1] to [0,1] that concentrates the point density around $x=0.5$.
So for example: [0,0.25,0.5,0.75,1] would become [0, 0.4, 0.5, 0.6, 1]
But I am not finding a function with this property that follows a normal distribution centered at 0.5.
Map $[0,1]$ linearly to $[-1,1]$, apply $t \mapsto t^{2n+1}$, which pushes numbers closer to $0$, the center point, and then map back to $[0,1]$, again linearly.
Explicitly $$x \mapsto \frac12((2x-1)^{2n+1} + 1)$$
As you make $n$ larger the values become more tightly concentrated around $0$.