Continuous and differentiable bell-shaped distribution on $[a, b]$ Is there a distribution with these three properties?

*

*Supported on $[a, b]$ for any $a, b\in\mathbb{R}$ and $a < b$

*Continuous and Differentiable (that is $\nabla_x p(x)$ can be computed)

*Bell-curved (or at least not as flat as the Uniform(a, b))


One option is to transform a beta distribution.
$Beta(3,3)$ has your desired properties on $[0,1]$.
Now subtract $1/2$ to center the distribution.
Next, multiply to stretch or compress the distribution.
Finally, add your desired mean.
