The arcsine distribution is a probability distribution with a cumulative distribution function (CDF) involving the arcsine. It is so-named because “arcsin x” is the arc with a sine of x – where sine is a trigonometric function that gives one of three possible ratios of the lengths of a triangle’s sides.
The distribution is used in several areas including:
- Jeffrey’s prior for Bernoulli trial successes.
- The Erdős arcsine law (which states that the prime divisors of a number have a distribution related to the arcsine distribution).
- Random walk fluctuations (such as those seen in stock price fluctuations).
- Renewal theory (the branch of probability theory that generalizes the Poisson process for arbitrary holding times.).
One practical use case is in determining the fraction of time a player can win a coin toss game, assuming fair coins .
The standard arcsine distribution is a special case of the beta distribution when a = b = ½; If a random variable X is arcsine distributed, then X ~ Beta(1⁄2,1⁄2). It is also a special case of the Pearson type I distribution.
Arcsine Distribution PDF
The standard arcsine distribution probability density function (PDF) is defined as:
The PDF is supported on (0 < x < 1), otherwise the density is 0.
The distribution can be generalized to include bounded support between any values a and b, or by using scale and location parameters. Specifically, using the transformation:
for a a ≤ x ≤ b, with PDF
on the interval (a, b).
Arcsine Distribution CDF
The cumulative distribution function (CDF) is:
The CDF is valid on the interval (0 < x < 1) and is concentrated near boundary values 0 and 1, tending to infinity at the endpoints.
 Rasnick, Rebecca, “Generalizations of the Arcsine Distribution” (2019). Electronic Theses and Dissertations. Paper 3565. https://dc.etsu.edu/etd/3565
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