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  1. Random
  2. 4. Special Distributions
  3. The Arcsine Distribution

The Arcsine Distribution

The arcsine distribution is important in the study of Brownian motion and prime numbers, among other applications.

The Standard Arcsine Distribution

Definition

Random variable \(Z\) has the standard arcsine distribution if \(Z\) has probability density function \(g\) given by \[g(x) = \frac{1}{\pi \sqrt{x (1 - x)}}, \quad x \in (0, 1)\]

Distribution Functions

The function \(g\) is a valid probability density function and satisfies the following properties:

  1. \(g\) is symmetric about \(x = \frac{1}{2}\).
  2. \(g\) decreases and then increases with minimum value at \( x = \frac{1}{2} \).
  3. \( g \) is concave upward
  4. \(g(x) \to \infty\) as \(x \downarrow 0\) and as \(x \uparrow 1\).
Proof:

The substitution \(u = \sqrt{x}\), \(x = u^2\), \(dx = 2 u \, du\) gives \[\int_0^1 \frac{1}{\pi \sqrt{x (1 - x)}} dx = \int_0^1 \frac{2}{\pi \sqrt{1 - u^2}} du = \frac{2}{\pi} \arcsin(u) \biggm\vert_0^1 = \frac{2}{\pi} \left(\frac{\pi}{2} - 0\right) = 1\]

  1. Note that \(g\) is a function of \(x\) only through \(x (1 - x)\).
  2. This follows from standard calculus: \[g^\prime(x) = \frac{2 x - 1}{2 \pi [x (1 - x)]^{3/2}}\]
  3. This also follows from standard calculus: \[ g^{\prime\prime}(x) = \frac{3 - 8 x + 8 x^2}{4 \pi [x (1 - x)]^{5/2}} \]
  4. The limits are clear.

In particular, the standard arcsine distribution is U-shaped and has no mode.

Open the Special Distribution Simulator and select the arcsine distribution. Keep the default parameter values and note the shape of the probability density function. Run the simulation 1000 times and compare the emprical density function to the probability density function.

The proof that \( g \) is a valid probability density function explains the name arcsine distribution, as does the form of the distribution function given next.

\(Z\) has distribution function \(G\) given by \(G(x) = \frac{2}{\pi} \arcsin\left(\sqrt{x}\right)\) for \(x \in [0, 1]\).

Proof:

Again, using the substitution \(u = \sqrt{t}\), \(t = u^2\), \(dt = 2 u \, du\): \[G(x) = \int_0^x \frac{1}{\pi \sqrt{t (1 - t)}} dt = \int_0^{\sqrt{x}} \frac{2}{\pi \sqrt{1 - u^2}} du = \frac{2}{\pi}\arcsin(t) \biggm\vert_0^{\sqrt{x}} = \frac{2}{\pi} \arcsin\left(\sqrt{x}\right) \]

\(Z\) has quantile function \(G^{-1}\) given by \(G^{-1}(p) = \sin^2\left(\frac{\pi}{2} p\right)\) for \(p \in [0, 1]\). In particular, the quartiles of \(Z\) are

  1. \(q_1 = \sin^2\left(\frac{\pi}{8}\right) = \frac{1}{4}(2 - \sqrt{2}) \approx 0.1464\), the first quartile
  2. \(q_2 = \frac{1}{2}\), the median
  3. \(q_3 = \sin^2\left(\frac{3 \pi}{8}\right) = \frac{1}{4}(2 + \sqrt{2}) \approx 0.8536\), the third quartile
Proof:

The formula for the quantile function follows from the distribution function by solving \(p = G(x)\) for \(x\) in terms of \(p \in [0, 1]\).

Open the Special Distribution Calculator and select the arcsine distribution. Keep the default parameter values and note the shape of the distribution function. Compute selected values of the distribution function and the quantile function.

Moments

The mean, variance, and standard deviation of \(Z\) are

  1. \(\E(Z) = \frac{1}{2}\)
  2. \(\var(Z) = \frac{1}{8}\)
  3. \(\sd(Z) = \frac{1}{2 \sqrt{2}} \approx 0.3536\)
Proof:
  1. The mean is \(\frac{1}{2}\) by symmetry.
  2. Using the usual substitution \(u = \sqrt{x}\), \(x = u^2\) \(dx = 2 u \, du\) and then the substitution \(u = sin(\theta)\), \(du = \cos(\theta) \, d\theta\) gives \[\E\left(Z^2\right) = \int_0^1 \frac{1}{\pi \sqrt{x (1 - x)}} dx = \int_0^1 \frac{2 u^4}{\pi \sqrt{1 - u^2}} = \int_0^{\pi/2} \frac{2}{\pi} \sin^4(\theta) d\theta = \frac{2}{\pi} \frac{3 \pi}{16} = \frac{3}{8}\]

Open the Special Distribution Simulator and select the arcsine distribution. Keep the default parameter values. Run the simulation 1000 times and compare the empirical mean and stadard deviation to the true mean and standard deviation.

The moments of \(Z\) (about 0) are \[\E\left(Z^n\right) = \prod_{j=0}^{n-1} \frac{2 j + 1}{2 j + 2}, \quad n \in \N\]

Proof:

The same integral substitutions as before gives \[\E(Z^n) = \int_0^{\pi/2} \frac{2}{\pi} \sin^{2 n}(\theta) d\theta = \prod_{j=0}^{n-1} \frac{2 j + 1}{2 j + 2}\]

Of course, the moments can be used to give a formula for the moment generating function, but this formula is not particularly helpful since it is not in closed form.

\(Z\) has moment generating function \(m\) given by \[m(t) = \E\left(e^{t Z}\right) = \sum_{n=0}^\infty \left(\prod_{j=0}^{n-1} \frac{2 j + 1}{2 j + 2}\right) \frac{t^n}{n!}, \quad t \in \R\]

The skewness and kurtosis of \(Z\) are

  1. \(\skw(Z) = 0\)
  2. \(\kur(Z) = \frac{3}{2}\)
Proof:
  1. The skewness is 0 by the symmetry of the distribution.
  2. The result for the kurtosis follows from the standard formula for kurtosis in terms of the moments: \(\E(Z) = \frac{1}{2}\), \(\E\left(Z^2\right) = \frac{3}{8}\), \(\E\left(Z^3\right) = \frac{5}{16}\), and \(\E\left(Z^4\right) = \frac{35}{128}\).

Related Distributions

The standard arcsine distribution is a special case of the beta distribution.

The standard arcsine distribution is the beta distribution with left parameter \(\frac{1}{2}\) and right parameter \(\frac{1}{2}\).

Proof:

The beta distribution with parameters \(a = b = \frac{1}{2}\) has PDF \[x \mapsto \frac{1}{B(1/2, 1/2)} x^{-1/2}(1 - x)^{-1/2}, \quad x \in (0, 1)\] But \(B\left(\frac{1}{2}, \frac{1}{2}\right) = \pi\), so this is the standard arcsine PDF.

Since the quantile function above is in closed form, the standard arcsine distribution can be simulated by the random quantile method.

Connections with the standard uniform distribution.

  1. If \(U\) has the standard uniform distribution (a random number) then \(X = \sin^2\left(\frac{\pi}{2} U\right)\) has the standard arcsine distribution.
  2. If \(X\) has the standard arcsine distribution then \(U = \frac{2}{\pi} \arcsin\left(\sqrt{X}\right)\) has the standard uniform distribution.

Open the random quantile simulator and select the arcsine distribution. Keep the default parameters. Run the experiment 1000 times and compare the empirical probability density function, mean, and standard deviation to their distributional counterparts. Note how the random quantiles simulate the distribution.

The following exercise illustrates the connection between the Brownian motion process and the standard arcsine distribution.

Open the Brownian motion simulator. Keep the default time parameter and select the last zero random variable. Note that this random variable has the standard arcsine distribution. Run the experiment 1000 time and compare the empirical probability density function, mean, and standard deviation to their distributional counterparts. Note how the last zero simulates the distribution.

The General Arcsine Distribution

The standard arcsine distribution is generalized by adding location and scale parameters.

Definition

If \(Z\) has the standard arcsine distribution, and if \(a \in \R\) and \(w \in (0, \infty)\), then \(X = a + w Z\) has the arcsine distribution with location parameter \(a\) and scale parameter \(w\).

Distribution Functions

Suppose that \(X\) has the arcsine distribution with location parameter \(a\) and scale parameter \(w\).

\(X\) has a continuous distribution on \( (a, a + w) \) with probability density function \(f\) given by \[f(x) = \frac{1}{\pi \sqrt{(x - a)(a + w - x)}}, \quad x \in (a, a + w)\]

  1. \(f\) is symmetric about \(a + \frac{1}{2} w\).
  2. \(f\) decreases and then increases with minimum value at \( x = a + \frac{1}{2} w \).
  3. \( f \) is concave upward.
  4. \(f(x) \to \infty\) as \(x \downarrow a\) and as \(x \uparrow a + w\).
Proof:

Recall that \(f(x) = \frac{1}{w} g\left(\frac{x - a}{w}\right)\) where \(g\) is the PDF of the standard arcsine distribution, given above.

An alternate parameterization of the general arcsine distribution is by the endpoints of the support interval: the left endpoint (location parameter) \(a\) and the right endpoint \(b = a + w\).

Open the Special Distribution Simulator and select the arcsine distribution. Vary the location and scale parameters and note the shape and location of the probability density function. For selected values of the parameters, run the simulation 1000 times and compare the emprical density function to the probability density function.

\(X\) has distribution function \(F\) given by \[F(x) = \frac{2}{\pi} \arcsin\left(\sqrt{\frac{x - a}{w}}\right), \quad x \in [a, a + w]\]

Proof:

Recall that \(F(x) = G[(x - a) / w)\) where \(G\) is the CDF of the standard arcsine distribution given above.

\(X\) has quantile function \(F^{-1}\) given by \(F^{-1}(p) = a + w \sin^2\left(\frac{\pi}{2} p\right)\) for \(p \in [0, 1]\) In particular, the quartiles of \(X\) are

  1. \(q_1 = a + w \sin^2\left(\frac{\pi}{8}\right) = a + \frac{1}{4}\left(2 - \sqrt{2}\right) w\), the first quartile
  2. \(q_2 = a + \frac{1}{2} w\), the median
  3. \(q_3 = a + w \sin^2\left(\frac{3 \pi}{8}\right) = a + \frac{1}{4}\left(2 + \sqrt{2}\right) w\), the third quartile
Proof:

Recall that \(F^{-1}(p) =a + w G^{-1}(p)\) where \(G^{-1}\) is the quantile function of the standard arcsine distribution given in (4).

Open the Special Distribution Calculator and select the arcsine distribution. Vary the parameters and note the shape and location of the distribution function. For various values of the parameters, compute selected values of the distribution function and the quantile function.

Moments

Again, we assume that \(X\) has the arcsine distribution with location parameter \(a\) and scale parameter \(w\).

The mean, variance, and standard deviation of \(X\) are

  1. \(\E(X) = a + \frac{1}{2} w\)
  2. \(\var(X) = \frac{1}{8} w^2\)
  3. \(\sd(Z) = \frac{1}{2 \sqrt{2}} w\)
Proof:

These results from the representation \(X = a + w Z\) and the corresponding results for the standard arcsine distribution above.

Open the Special Distribution Simulator and select the arcsine distribution. Vary the parameters and note the size and location of the mean\(\pm\)standard deviation bar. For various values of the parameters, run the simulation 1000 times and compare the empirical mean and stadard deviation to the true mean and standar deviation.

The moments of \(X\) can be obtained from the moments of \(Z\), but the results are messy, except when the location parameter is 0.

Suppose the location parameter \(a = 0\). The moments of \(X\) (about 0) are \[\E(X^n) = w^n \prod_{j=0}^{n-1} \frac{2 j + 1}{2 j + 2}, \quad n \in \N\]

Proof:

This follows from the representation \(X = w Z\) and the corresponding result for the standard arcsine distribution above.

\(X\) has moment generating function \(M\) given by \[M(t) = \E\left(e^{t X}\right) = e^{a t} \sum_{n=0}^\infty \left(\prod_{j=0}^{n-1} \frac{2 j + 1}{2 j + 2}\right) \frac{w^n t^n}{n!}, \quad t \in \R\]

Proof:

Recall that \(M(t) = e^{a t} m(w t)\) where \(m\) is the moment generating function of the standard arcsine distribution given above.

The skewness and kurtosis of \(X\) are

  1. \(\skw(X) = 0\)
  2. \(\kur(X) = \frac{3}{2}\)
Proof:

Recall that the skewness and kurtosis are defined in terms of the standard score of \(X\) and hence are invariant under a location-scale transformation.

Related Distributions

By construction, the general arcsine distribution is a location-scale family, and so is closed under location-scale transformations.

If \(X\) has the arcsine distribution with location parameter \(a\) and scale parameter \(w\) and if \(c \in \R\) and \(d \in (0, \infty)\) then \(c + d X\) has the arcsine distribution with location parameter \(c + a d\) scale parameter \(d w\).

Since the quantile function above is in closed form, the arcsine distribution can be simulated by the random quantile method.

Connections with the standard uniform distribution.

  1. If \(U\) has the standard uniform distribution (a random number) and \(a \in \R\), \(b \in (0, \infty)\), then \(X = a + w \sin^2\left(\frac{\pi}{2} U\right)\) has the arcsine distribution with location parameter \(a\) and scale parameter \(b\).
  2. If \(X\) has the arcsine distribution with location parameter \(a\) and scale parameter \(b\) then \(U = \frac{2}{\pi} \arcsin\left(\sqrt{\frac{X - a}{w}}\right)\) has the standard uniform distribution.

Open the random quantile simulator and select the arcsine distribution. Vary the parameters and note the location and shape of the probability density function. For selected parameter values, run the experiment 1000 times and compare the empirical probability density function, mean, and standard deviation to their distributional counterparts. Note how the random quantiles simulate the distribution.

The following exercise illustrates the connection between the Brownian motion process and the arcsine distribution.

Open the Brownian motion simulator and select the last zero random variable. Vary the time parameter \( t \) and note that the last zero has the arcsine distribution on the interval \( (0, t) \). Run the experiment 1000 time and compare the empirical probability density function, mean, and standard deviation to their distributional counterparts. Note how the last zero simulates the distribution.