Normal Distribution in Bounded Domain with Zero Endpoints Is there a known probability distribution $p(x)$ that approximates a normal distribution near $x=0$, but is bounded within a domain $x \in [a,b]$ such that $0 \in [a,b]$ and $p(a)=p'(a)=p(b)=p'(b)=0$?

One nice approximation satisfying those conditions has the pdf
$$p(x)=6\left(\frac{x^2}{25}-\frac14\right)^{\!2} \text{ on }\left[-\frac{5}{2},\frac{5}{2}\right]$$
with $p(x)=0$ outside that range. This is shown below in orange, and is a transformed version of the Beta(3,3) distribution.

