Generalized Normal Distribution
X ~ GenNormal(μ, α, β)
Unifying / Tail-Shape Family • Published July 22, 2026
Also known as the Exponential Power distribution -- the cleanest demonstration on this site that three already-published distributions are one parameter apart: β=1 gives the already-published Laplace exactly, β=2 gives the already-published Normal exactly, and β→∞ converges toward the already-published Uniform (all three ties confirmed numerically). The excess-kurtosis formula evaluates to exactly 0 at β=2 and exactly 3 at β=1, matching those sheets' own reported values. Maximum likelihood of the shape parameter has no closed form.
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About This Reference Sheet
Also known as the Exponential Power distribution -- the cleanest demonstration on this site that three already-published distributions are one parameter apart: β=1 gives the already-published Laplace exactly, β=2 gives the already-published Normal exactly, and β→∞ converges toward the already-published Uniform (all three ties confirmed numerically). The excess-kurtosis formula evaluates to exactly 0 at β=2 and exactly 3 at β=1, matching those sheets' own reported values. Maximum likelihood of the shape parameter has no closed form.
Support
x ∈ (-∞, ∞)
Parameters
μ ∈ ℝ (location), α > 0 (scale), β > 0 (shape)