Arcsine Distribution


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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 [1].

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:

arcsine distribution PDF
Arcsine distribution PDF.

The PDF is supported on (0 < x < 1), otherwise the density is 0.

The arcsin PDF is U-shaped [2].

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 includes arcsin (inverse sine), which is where the distribution gets its name.

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.

References

[1] Rasnick, Rebecca, “Generalizations of the Arcsine Distribution” (2019). Electronic Theses and Dissertations. Paper 3565. https://dc.etsu.edu/etd/3565

[2] Graphed with Desmos: https://www.desmos.com/calculator