Arcsine Distribution

The Arcsine distribution is a probability distribution for a random variable whose probability density function (PDF) is proportional to the reciprocal of the square root of the product of the variable and one minus the variable.

The Arcsine distribution is a special case of the Beta distribution where both shape parameters equal 0.5. For the standard form defined on [0, 1], its Cumulative Distribution Function (CDF) is:

And its Probability Density Function (PDF) is:

While the standard form is defined on [0, 1], the distribution can be parameterized with arbitrary lower and upper bounds to shift and scale the interval.

The Arcsine distribution is used in various fields such as physics, finance, and engineering. For example, it can model the distribution of the fraction of time a Brownian motion spends above the origin.

Another application is in reliability engineering, where it can describe the distribution of the time to failure of certain systems.

The Arcsine distribution is U-shaped with probability mass concentrated at the boundaries of its support interval.

Notable properties include:

• The distribution is symmetric about $\frac{1}{2}$.

• The distribution has infinite density at x = 0 and x = 1. Even so, all moments of the distribution exist despite the infinite densities at the boundaries.

• The distribution has a unique U-shaped curve that is steepest near the boundaries.

The Arcsine distribution has several relationships to other probability distributions:

• It is a special case of the Beta distribution with parameters α = β = 1/2.

• If X follows a uniform distribution on [0, 2π] then sin²(X) follows an Arcsine distribution on [0, 1].

• If X follows an Arcsine distribution on [0, 1], then 1-X also follows an Arcsine distribution on [0, 1].

The arcsine distribution is implemented by the ArcsineDistribution class. The implementation allows for distributions over a non-standard interval. It has two constructors. The first constructor takes no arguments and returns the standard arcsine distribution over [0, 1]. The second constructor takes two arguments: the lower and upper bounds of the distribution.

var arcsine1 = new ArcsineDistribution(-1, 1);

Dim arcsine1 = New ArcsineDistribution(-1.0, 1.0)

No code example is currently available or this language may not be supported.

let arcsine1 = ArcsineDistribution(-1.0, 1.0)

A singleton instance of the standard distribution is available as the Standard property. The code below draws a random sample from the standard distribution:

var random = new Pcg32();
double sample = ArcsineDistribution.Standard.Sample(random);

Dim random = New Pcg32()
Dim sample = ArcsineDistribution.Standard.Sample(random)

No code example is currently available or this language may not be supported.

let random = Pcg32()
let sample = ArcsineDistribution.Standard.Sample(random)

The arcsine distribution is a special form of the Beta distribution. You can refer to the following sources:

• Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). "Continuous Univariate Distributions, Volume 2", Chapter 25. Wiley Series in Probability and Statistics.

• Forbes, C., Evans, M., Hastings, N., & Peacock, B. (2011). "Statistical Distributions", Chapter 4. Wiley Series in Probability and Statistics.

Other Resources