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I am looking for a (commonly used) probability density function, which would look like a normal distribution flipped upside down. It would look like a uniform distribution with a dent in the middle.

Just to be clear, I am dealing with a continuous random variable within some range, say [-1,1].

Sometimes I have data which is concentrated around zero, but other times I have data which is concentrated toward 1 and -1, while the region around zero is relatively underpopulated. Is there a kind of pdf which (depending on some parameter) can represent these two cases?

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Beta distribution? You'd have to rescale your [-1,1] into [0,1].

enter image description here

Don't know if this will get the level of weight you want into the corners. Decreasing the shape parameters forces more weight to them though.

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  • $\begingroup$ Actually, the shape parameters are the exact parameters I need to adjust to fit both cases I mentioned. It looks like this is exactly what I was looking for! $\endgroup$ – Angelorf Feb 10 '14 at 10:32
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    $\begingroup$ The beta distribution illustrated above has an antimode at $x = 0.5$. This kind of minor terminology polarises people all the way from "yet another term we don't really need" to "just the term I was looking for". $\endgroup$ – Nick Cox Feb 10 '14 at 11:36
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    $\begingroup$ Well, the term is certainly better than 'abyss'! $\endgroup$ – Angelorf Feb 10 '14 at 17:20

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