Arcsine Distribution
X ~ Arcsine(a, b)
Bounded / U-Shaped Family • Published July 22, 2026
One of Lévy's random-walk arcsine laws, and exactly the already-published Beta(1/2,1/2) on (0,1): PDF/CDF verified against scipy.stats.arcsine, the classic cos(Θ) circular-motion construction confirmed via 3-million-draw simulation. Has no formal MLE -- the likelihood is unbounded as the endpoints approach the sample min/max -- yet the sample min/max converge to the true endpoints at a verified O(1/n²) rate, a full order of n faster than the already-published Uniform's own endpoint estimator.
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About This Reference Sheet
One of Lévy's random-walk arcsine laws, and exactly the already-published Beta(1/2,1/2) on (0,1): PDF/CDF verified against scipy.stats.arcsine, the classic cos(Θ) circular-motion construction confirmed via 3-million-draw simulation. Has no formal MLE -- the likelihood is unbounded as the endpoints approach the sample min/max -- yet the sample min/max converge to the true endpoints at a verified O(1/n²) rate, a full order of n faster than the already-published Uniform's own endpoint estimator.
Support
x ∈ (a, b)
Parameters
a < b (endpoints)