are there any density function family with the following properties?

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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

Answers (1)

John D'Errico
John D'Errico on 16 Apr 2015
A beta distribution is the logical choice.
  2 Comments
Mr M.
Mr M. on 16 Apr 2015
Hm? Beta is typically asymmetric, and in the symmetric case it has nonzero derivative at least one of the edges
John D'Errico
John D'Errico on 18 Apr 2015
Edited: John D'Errico on 18 Apr 2015
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.

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