Probability Distributions > Arcsine distribution

The **arcsine distribution** and the *generalized arcsine distribution *are found in the study of random walk fluctuation, renewal theory, and as the Jeffrey’s prior for successes in a Bernoulli trial. The generalized **arcsine distribution**, used in mathematical statistics, is a special case of the beta I distribution, when a = b = ½.

It’s called the arcsine distribution because “arcsin x” is the arc that has a sine of x.

Although the distribution is used in some fairly weighty mathematical areas, a simple example of what the distribution can be used for is as the fraction of time a player can win a coin toss game, assuming the coins are fairly weighted (Rasnick, 2019).

## PDF for the Arcsine Distribution

The probability density function (PDF) of the **standard arcsine distribution** is defined as:

The PDF is supported for an interval of between 0 and 1 (0 < 1). Outside of this boundary, the density is zero. The distribution can be generalized to include any bounded support between two values *a* and *b*, or by using scale parameters and location parameters.

The PDF of the arcsine distribution is U-shaped:

## CDF for the Arcsine Distribution

The cumulative distribution function (CDF) for the arcsin function is:

The CDF is valid for x-values from 0 to 1 (0 < x < 1); The CDF is concentrated near the boundary values 0 and 1. As the distribution gets very close to 1, it tends to infinity. It includes the inverse sin (arcsin), which is where the distribution gets its name.

## References

Agrawal, N. et al. Arcsin Distribution. Retrieved January 21, 2020 from: https://www.boost.org/doc/libs/1_58_0/libs/math/doc/html/math_toolkit/dist_ref/dists/arcine_dist.html

Balakrishnan, N. & Nevzorov, V. (2004). A Primer on Statistical Distributions. John Wiley & Sons.

Hazewinkel, M. (1994). Encyclopaedia of Mathematics (set). Springer Science & Business Media.

Mörters, P. & Peres, Y. (2010). Brownian Motion. Cambridge University Press.

R-forge distributions Core Team. (2009-2010). Handbook on probability distributions. Retrieved January 21, 2020 from: http://www.stat.rice.edu/~dobelman/textfiles/distributions.r-forge.pdf

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