Is there a probability distribution with a CDF shaped like the sinh or logit function? I am looking for a probability distribution with the shape of the CDF similar to the sinh or logit function. I need it to be centered at 0 and symmetric, meet the usual CDF restrictions. I understand that most likely it will need to be defined on a specific interval, that's okay. The important property is that F'(x) < 1, F''(x) < 0 close to the left of the mean, and F'(x)<1 , F''(x)>0 close to the right. For more farther away values, however defined, I need F'(x) > 1, F''(x) > 0. Needless to say, the easier the formula, the better!
I tried to play around with the sinh function but I couldn't get it to work. Any ideas? Here is an example of the cdf I am looking for:

 A: Given the desire for simplicity, here are two suggestions:
U-quadratic pdf $\quad f(x) = \frac32 x^2 \quad \text {defined on } (-1,1)$

with cdf:

The cdf is:  $F(x) = \frac12 (1 + x^3)$ on (-1,1)

Inverse Triangular pdf $\quad f(x) = |x| \quad \text {defined on } (-1,1)$

with cdf:

The cdf is:
$$F(x) = \left\{
\begin{array}{cc}
 1 & \text{if } x\geq 1 \\
 \frac{1}{2} \left(1+x^2\right) & \text{if } 0<x<1 \\
 \frac{1}{2} \left(1-x^2\right) & \text{if } -1<x\leq 0 \\
 0 & \text{otherwise}  \\
\end{array}
\right. $$
A: There is a natural (and important) distribution on $[0,1]$, the arcsine distribution (http://www.randomservices.org/random/special/Arcsine.html). Its cdf is 
$$F(x)=\begin{cases}0 & \text{for} & x \leq 0 \\ \frac{2}{\pi}\arcsin(\sqrt{x})& \text{for} & 0<x<1 \\ 1 & \text{for} & x \geq 1 \end{cases}$$
Remarks:


*

*"$\arcsin$" is now a less common notation - though more correct - than "$\sin^{-1}$".

*The "arcsine" distribution is a particular case of a "beta" distribution.

*Edit: cdf $F$ should be replaced by $F_a$ defined on $-\frac{1}{2a}<x<\frac{1}{2a}$ by 
$$F_a(x):=\frac{2}{\pi}\arcsin\left(\sqrt{ax+\frac12}\right)$$
with $a$ arbitrary small in order that its curve is symmetrical with respect to point $(0,1/2)$, and that its derivative is arbitrarily small in the vicinity of $0$.
The graphical representation of $F$ is as follows:

