An unbounded probability density function does not imply an atom. HereThe last equivalence follows from . The endpoint interval has much more probability than the same interval under a uniform distribution, whose value is , but its probability still tends to zero. This is an integrable singularity.
For example, the first 1% of the unit interval contains about of the probability; the last 1% contains the same amount because the arcsine distribution is a symmetric probability distribution. Each exact endpoint still has zero probability. Density and interval probability answer different questions.
Endpoint density diverges, but every finite bin has finite probability
. Original exact curves and bin probabilities. Left: the arcsine cumulative distribution function compared with the uniform one. Right: ten equal-width bins have probabilities , not heights sampled from the divergent density. Their total is one. Details.
A numerical integration routine must respect these endpoint integrable singularities. The substitution givesIt turns a difficult-looking density integral into a constant measure in angle. Integrating finite bins is also a faithful way to draw the distribution near its endpoints; sampling to make a bar height is not.
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