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 $p(a)=p'(a)=p(b)=p'(b)=0$?

Another potential option is a scaled and translated logit-normal distribution with pdf
$$p(x)=\frac{(b - a)\exp \left(-\frac{\left(\log \left(\frac{x-a}{b-x}\right)-\mu \right)^2}{2 \sigma ^2}\right)}{\sqrt{2 \pi } \sigma  (x-a) (b-x)}$$
with parameters $-\infty<\mu<\infty$ and $\sigma>0$.
Here's an example:

