Use the Arcsine Distribution Calculator to analyze data following an arcsine distribution. Understand the fundamentals of the arcsine distribution and perform accurate calculations with ease.”

Arcsine Distribution Calculator

The Arcsine distribution is a special case of the Beta distribution with parameters $ \frac{1}{2} $ and $ \frac{1}{2} $.

Its PDF is given by: $$ f(x)=\frac{1}{\pi\sqrt{x(1-x)}},\quad 0 < x < 1, $$ and its CDF by: $$ F(x)=\frac{2}{\pi}\arcsin(\sqrt{x}),\quad 0 \le x \le 1. $$

The expected value is $0.5$ and the variance is $0.125$.

Step 1: Enter Value

$ x $ (evaluation point, $0 \le x \le 1$):

Enter a number between 0 and 1 (e.g., 0.5)

Arcsine Distribution Calculator (In-Depth Explanation)

The Arcsine distribution is a special case of the Beta distribution with parameters $ \alpha = \frac{1}{2} $ and $ \beta = \frac{1}{2} $. Known for its U-shaped probability density function, the standard arcsine distribution is defined on the interval (0,1) and exhibits unique properties – such as symmetry about 0.5 and divergence at the boundaries. This guide provides an in-depth explanation of both the standard and the general (location-scale) arcsine distributions, including their probability density functions, distribution functions, moments, and applications.

Overview
Key Concepts
Distribution Functions
Step-by-Step Calculation Process
Practical Examples
Moments and Properties
General Arcsine Distribution
Common Applications
Conclusion

1. Overview

The standard arcsine distribution is a continuous probability distribution on the interval (0,1) with the probability density function:

$ g(x) = \frac{1}{\pi \sqrt{x(1-x)}} $, for $ x \in (0,1) $

The name arises from the appearance of the arcsine function in its cumulative distribution function and quantile function. Notably, the standard arcsine distribution is U-shaped, symmetric about $ x = \frac{1}{2} $, decreases then increases with a minimum at $ \frac{1}{2} $, and diverges as $ x $ approaches 0 or 1.

2. Key Concepts

To understand the arcsine distribution, consider these key points:

Special Case of Beta Distribution: It is the Beta distribution with $ \alpha = \frac{1}{2} $ and $ \beta = \frac{1}{2} $.
Probability Density Function (PDF): Emphasizes the boundaries, leading to a U-shaped curve.
Cumulative Distribution Function (CDF): Has a simple form in terms of the arcsine function.
Quantile Function: Easily computed using the sine function.
Moments: The mean is $ \frac{1}{2} $ and the variance is $ \frac{1}{8} $; higher moments are given by product formulas.

3. Distribution Functions

For the standard arcsine distribution defined on $ (0,1) $:

Probability Density Function (PDF):

$ g(x) = \frac{1}{\pi \sqrt{x(1-x)}} $

Cumulative Distribution Function (CDF):

$ G(x) = \frac{2}{\pi} \arcsin(\sqrt{x}) $, for $ x \in [0,1] $

Quantile Function:

$ G^{-1}(p) = \sin^2\left(\frac{\pi}{2}p\right) $, for $ p \in [0,1] $

Proofs: The appearance of the arcsine function in the CDF and quantile function justifies the name of the distribution. For instance, the quartiles are:

$ q_1 = \sin^2\left(\frac{\pi}{8}\right) \approx 0.1464 $
$ q_2 = \frac{1}{2} $ (median)
$ q_3 = \sin^2\left(\frac{3\pi}{8}\right) \approx 0.8536 $

4. Step-by-Step Calculation Process

Input a Value:
Choose a value $ x $ in the interval (0,1) to compute the PDF or CDF.
Calculate the PDF:
Substitute $ x $ into the PDF:

$ g(x) = \frac{1}{\pi \sqrt{x(1-x)}} $
Calculate the CDF:
Substitute $ x $ into the CDF:

$ G(x) = \frac{2}{\pi} \arcsin(\sqrt{x}) $
Determine Quantiles:
For a given probability $ p $, compute the quantile using:

$ G^{-1}(p) = \sin^2\left(\frac{\pi}{2}p\right) $

5. Practical Examples

Example: Evaluating at $ x = 0.25 $

Scenario: Compute the PDF and CDF at $ x = 0.25 $.

PDF Calculation:

$ g(0.25) = \frac{1}{\pi \sqrt{0.25 \times 0.75}} $

CDF Calculation:

$ G(0.25) = \frac{2}{\pi} \arcsin(\sqrt{0.25}) = \frac{2}{\pi} \arcsin(0.5) $

Since $ \arcsin(0.5) = \frac{\pi}{6} $, it follows that:

$ G(0.25) = \frac{2}{\pi} \times \frac{\pi}{6} = \frac{1}{3} $

Thus, at $ x = 0.25 $, the PDF is $ \frac{1}{\pi \sqrt{0.1875}} $ and the CDF is approximately $ 0.3333 $.

6. Moments and Properties

For a random variable $ Z $ following the standard arcsine distribution:

Mean: $ E(Z) = \frac{1}{2} $
Variance: $ \mathrm{var}(Z) = \frac{1}{8} $
General Moments: For $ n \in \mathbb{N} $,
$ E(Z^n) = \prod_{j=0}^{n-1} \frac{2j+1}{2j+2} $
Moment Generating Function (MGF):
$ m(t)=E(e^{tZ})=\sum_{n=0}^{\infty}\left(\prod_{j=0}^{n-1}\frac{2j+1}{2j+2}\right)\frac{t^n}{n!}, \quad t\in\mathbb{R} $
Skewness and Kurtosis: $ \text{skew}(Z)=0 $ and $ \text{kurt}(Z)=\frac{3}{2} $ (as provided in the literature, note that some sources may express kurtosis differently).

Moreover, the standard arcsine distribution is intimately connected with the Beta distribution and can be simulated via the random quantile method. For example, if $ U $ is a standard uniform random variable, then:

$ X=\sin^2\left(\frac{\pi}{2}U\right) $

has the standard arcsine distribution.

7. General Arcsine Distribution

The standard arcsine distribution can be generalized by introducing a location parameter $ a $ and a scale parameter $ w>0 $. If $ Z $ follows the standard arcsine distribution, then:

$ X = a + wZ $

has the general arcsine distribution on the interval $ (a, a+w) $ with:

Probability Density Function (PDF):

$ f(x)=\frac{1}{\pi \sqrt{(x-a)(a+w-x)}}, \quad x\in(a,a+w) $

Cumulative Distribution Function (CDF):

$ F(x)=\frac{2}{\pi}\arcsin\left(\sqrt{\frac{x-a}{w}}\right), \quad x\in[a,a+w] $

Quantile Function:

$ F^{-1}(p)=a+w\sin^2\left(\frac{\pi}{2}p\right), \quad p\in[0,1] $

The moments transform accordingly:

Mean: $ E(X)=a+\frac{w}{2} $
Variance: $ \mathrm{var}(X)=\frac{w^2}{8} $
Skewness and Kurtosis: Remain unchanged, i.e., $ \text{skew}(X)=0 $ and $ \text{kurt}(X)=\frac{3}{2} $.

8. Common Applications

Random Processes: Modeling the proportion of time a process spends in a given state.
Finance: Analyzing asset return distributions and market trends.
Physics: Studying diffusion processes and particle motion, including connections with Brownian motion (e.g., the last zero of a Brownian motion process follows an arcsine distribution).
Simulation: Generating random variables using the quantile method due to the closed-form expression of the quantile function.

9. Conclusion

The Arcsine Distribution Calculator provides a comprehensive framework for understanding and computing the properties of both the standard and the generalized arcsine distributions. By leveraging the elegant forms of the probability density function,

$ g(x) = \frac{1}{\pi \sqrt{x(1-x)}} $

and the cumulative distribution function,

$ G(x) = \frac{2}{\pi} \arcsin(\sqrt{x}) $

along with their quantile functions and moment properties, users can explore the unique U-shaped behavior and its wide range of applications in statistical analysis, simulation, and various scientific fields.