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The **arcsine distribution** is a symmetric probability distribution with a minimum at x = 1⁄2 and vertical asymptotes at x = 0 and x = 1. Its name comes from the fact that the cumulative distribution function (CDF) involves the arcsine (inverse sine); “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:

- Renewal theory (the branch of probability theory that generalizes the Poisson process for arbitrary holding times.).
- 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).

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 Properties

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).

The standard arcsine distribution is a special case of the beta distribution with *α* = *β* = ½. This only holds true for support on (0, 1), otherwise tapered tails appear at the top and the bottom of the arcsine distribution, which violates the properties of the beta distribution [3].

The arcsine distribution is also a special case of the Pearson type I distribution.

## CDF

The cumulative distribution function 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.

## Other properties for the arcsine distribution:

- Mean = ½
- Median = ½
- Mode = the standard arcsin distribution is U shaped and has no mode
- Variance = ⅛
- Skewness = 0
- Kurtosis = -3/2

The standard arcsine probability density function satisfies:

- Symmetry about x = ½.
- Decreasing function to a minimum value of
*x*= ½, then increasing. - Concave up
*f(x*) → ∞ as x ↓ 0 and as x ↑ 1.

## Arcsine exponentiated-X (ASE-X_

He et al. [4] proposed a new family of distributions called the Arcsine exponentiated-X distributions, with CDF

Where

- G(t; ξ)= the CDF of the baseline distribution
- ξ = parameter vector of the baseline distribution
- λ = an additional shape parameter.

## 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

[3] Chave, A. (2017). Computational Statistics in the Earth Sciences With Applications in MATLAB. Cambridge University Press.

[4] He, W. et al. (2020). The Arcsine Exponentiated-*X* Family: Validation and Insurance Application. Hindawi. Volume 2020 | Article ID 8394815 | https://doi.org/10.1155/2020/8394815