Probability Distribution List >

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:

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.

## 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