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I am wondering whether a family of probability distributions with the following form of a density function has a name:

$$f(x)=C*\operatorname{Exp}(-B|x|^A)$$

where $A$, $B$ and $C$ are positive constants, with $B$ being a "scaling" constant and $C$ selected so that $f(x)$ integrates to 1.

When $A=1$ this corresponds to a Laplace distribution (with $B=1/b$ and $C=1/2b$ to obtain variance is $2b^2$) and $A=2$ yields the Gaussian distribution (for variance $\sigma^2$ we set $B=1/2\sigma^2$ and $C=1/\sqrt{2\pi\sigma^2}$).

I realize that this belongs to the exponential family. I am just wondering if this specific form has been studied and whether it has any interesting properties (in addition to the properties of the exponential family)... If there is a good reference discussing this, please post.

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Generalized normal distribution –  S.B. Sep 13 '12 at 20:02
    
Thank you, @sbahmani! If you submit your one-liner as an answer, I'll accept it. –  M.B.M. Sep 13 '12 at 20:11

1 Answer 1

up vote 2 down vote accepted

Generalized normal distribution!

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