are there any density function family with the following properties?

I would like probability density function family with the following properties: - defined on a finite range - symmetric, one maximum at the center - smooth, with finite tails and zero derivative at the edge of the range - goes to a thin Gaussian as variance parameter goes to zero - goes to a smooth rectangle-function as variance parameter goes upwards

Answer 1

[John D'Errico]

A beta distribution is the logical choice.

[Mr M.]

Hm? Beta is typically asymmetric, and in the symmetric case it has nonzero derivative at least one of the edges

[John D'Errico]

Then I think you don't know the beta distribution at all well.

ezplot(@(x) betapdf(x,3,3),[0, 1])

As long as the pair of beta parameters are equal, the beta pdf will be completely symmetric.

And as long as they are sufficiently large, (assuming I recall the beta pdf correctly, any parameters of 2 or larger should be sufficient) the end point first derivative will be zero.

The beta distribution will be flat (reduces to a uniform distribution) when the parameters are both 1.

It is true that when the parameters are in the interval [1,2], then the end point first derivatives will be non-zero.

ezplot(@(x) betapdf(x,1.01,1.01),[0, 1])

So IF you absolutely need a zero end point derivative for the case where the distribution tends to a uniform, you could hack together something. But it won't be any common distribution with a name on it. In fact, essentially the only bounded distribution that has properties much as you want is the beta.