\( \newcommand{\R}{\mathbb{R}} \) \( \newcommand{\N}{\mathbb{N}} \) \( \newcommand{\Z}{\mathbb{Z}} \) \( \newcommand{ \E}{\mathbb{E}} \) \( \newcommand{\P}{\mathbb{P}} \) \( \newcommand{\var}{\text{var}} \) \( \newcommand{\sd}{\text{sd}} \) \( \newcommand{\bs}{\boldsymbol} \) \( \newcommand{\sgn}{\text{sgn}} \) \( \newcommand{\skw}{\text{skew}} \) \( \newcommand{\kur}{\text{kurt}} \)
  1. Random
  2. 4. Special Distributions
  3. The Semicircle Distribution

The Semicircle Distribution

The semicircle distribution plays a very important role in the study of random matrices. It is also known as the Wigner distribution in honor of the physicist Eugene Wigner, who did pioneering work on random matrices.

The Standard Semicircle Distribution

Definition

Random variable \( X \) has the standard semicircle distribution if \(X\) has a continuous distribution on \( [-1, 1] \) with probability density function \( g \) given by \[ g(x) = \frac{2}{\pi} \sqrt{1 - x^2}, \quad x \in [-1, 1] \]

The graph of \( g \) is the upper half of the circle of radius 1 centered at the origin, hence the name.

Distribution Functions

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

  1. \( g \) is symmetric about \( x = 0 \).
  2. \( g \) increases and then decreases with mode at \( x = 0 \).
  3. \( g \) is concave downward.
Proof:

As noted earlier, the graph of \( g \) is the upper half of the circle of radius 1 centered at the origin. The area under this semicircle is \( \pi / 2 \) so it follows immediately that \( g \) is a valid probability density function—the constant \( 2 / \pi \) in \( g \) is the normalizing constant.

Open special distribution simulator and select the semicircle distribution. With the default parameter value, note the shape of the probability density function. Run the simulation 1000 times and compare the empirical density function to the probability density function.

The distribution function \( G \) of \( X \) is \[ G(x) = \frac{1}{2} + \frac{1}{\pi} x \sqrt{1 - x^2} + \frac{1}{\pi} \arcsin(x), \quad x \in [-1, 1] \]

Proof:

Of course \( G(x) = \int_{-1}^x g(t) \, dt \) for \( -1 \le x \le 1 \). The integral is evaluated by using the trigonometric substitution \( t = \sin(\theta) \).

We cannot give the quantile function \( G^{-1} \) in closed form, but values of this function can be approximated. Clearly by symmetry, \( G^{-1}\left(\frac{1}{2} - p\right) = -G^{-1}\left(\frac{1}{2} + p\right) \) for \( 0 \le p \le \frac{1}{2} \). In particular, the median is 0.

Open the special distribution simulator and select the semicircle distribution. With the default parameter value, note the shape of the distribution function. Compute the first and third quartiles.

Moments

By symmetry, the odd order moments of \( X \) are 0. The even order moments can be computed explicitly.

For \( n \in \N \)

\begin{align} \E\left(X^{2n}\right) & = \left(\frac{1}{2}\right)^{2n} \frac{1}{n + 1} \binom{2n}{n} \\ \E\left(X^{2n+1}\right) & = 0 \end{align}
Proof:

Clearly \( X \) has moments of all orders since the PDF \( g \) is bounded and the support interval is bounded. So by symmetry, the odd order moments are 0, and we just need to prove the result for the even order moments. Note that \[ \E\left(X^{2n}\right) = \int_{-1}^1 x^{2n} \frac{2}{\pi} \sqrt{1 - x^2} \, dx \] We use the substitution \( x = \sin(\theta) \) to get \[ \E\left(X^{2n}\right) = \int_{-\pi/2}^{\pi/2} \frac{2}{\pi} \sin^{2n}(\theta) \cos^2(\theta) \, d\theta \] This integral can be evaluated by standard calculus methods to give the result above.

The numbers \( C_n = \frac{1}{n+1} \binom{2n}{n}\) for \( n \in \N \) are known as the Catalan numbers, and are named for the Belgian mathematician Eugene Catalan. In particular, we can compute the mean, variance, skewness, and kurtosis.

The mean and variance of \( X \) are

  1. \( \E(X) = 0 \)
  2. \( \var(X) = \frac{1}{4} \)

Open the special distribution simulator and select the semicircle distribution. With the default parameter value, note the size and location of the mean \(\pm\) standard deviation bar. Run the simulation 1000 times and compare the empirical mean and standard deviation to the true mean and standard deviation.

The skewness and kurtosis of \( X \) are

  1. \( \skw(X) = 0 \)
  2. \( \kur(X) = 2 \)
Proof:

The standard score of \( X \) is \( 2 X \). Hence \( \skw(X) = E\left(2^3 X^3\right) = 0 \). Of course, this is also clear from the symmetry of the distribution of \( X \). Similarly, by the results on moments above \[ \kur(X) = \E\left(2^4 X^4\right) = 2^4 \left(\frac{1}{2}\right)^4 \frac{1}{3}\binom{4}{2} = 2 \]

It follows that the excess kurtosis is \(\kur(X) - 3 = -1\).

Related Distributions

The semicircle distribution has simple connections to the continuous uniform distribution.

If \((X, Y)\) is uniformly distributed on the circular region in \(\R^2\) centered at the orgin with radius 1, then \(X\) and \(Y\) each have the standard semicircular distribution.

Proof:

\((X, Y)\) has joint PDF \((x, y) \mapsto 1/\pi\) on \(C = \{(x, y) \in \R^2: x^2 + y^2 \le 1\}\). Hence \(X\) has PDF \[ g(x) = \int_{-\sqrt{1 - x^2}}^{\sqrt{1 - x^2}} \frac{1}{\pi} \, dy = \frac{2}{\pi} \sqrt{1 - x^2}, \quad x \in [-1, 1] \]

It's easy to simulate a random point that is uniformly distributed on circular region in the previous theorem, and this provides a way of simulating a standard semicircle distribution. This is important since we can't use the random quantile method of simulation.

Suppose that \(U\), \(V\), and \(W\) are independent random variables, each with the standard uniform distribution (random numbers). Let \(R = \max\{U, V\}\) and \(\Theta = 2 \pi W\), and then let \(X = R \cos(\Theta)\), \(Y = R \sin(\Theta)\). Then \((X, Y)\) is uniformly distributed on the circular region of radius 1 centered at the origin, and hence \(X\) and \(Y\) each have the standard semicircle distribution.

Proof:

\(U\) and \(V\) have CDF \(u \mapsto u\) for \(u \in [0, 1]\) and therefore \(R\) has CDF \(r \mapsto r^2\) for \(r \in [0, 1]\). Hence \(R\) has PDF \(r \mapsto 2 r\) for \(r \in [0, 1]\). On the other hand, \(\Theta\) is uniformly distributed on \([0, 2 \pi)\) and hence has density \(\theta \mapsto 1 / 2 \pi\) on \([0, 2 \pi)\). By independence, the Joint PDF of \((R, \Theta)\) is \((r, \theta) \mapsto (2 r)(1 / 2 \pi) = r / \pi \) on \(\{(r, \theta): 0 \le r \le 1, 0 \le \theta \le 2 \pi\}\). For the polar coordinate transformation \((x, y) \mapsto (r \cos(\theta), r \sin(\theta))\), the Jacobian is \(r\). Hence by the change of variables theorem, \((X, Y)\) has PDF \[ (x, y) \mapsto \frac{r}{\pi} \frac{1}{r} = \frac{1}{\pi} \text{ on } \{(x, y) \in \R^2: x^2 + y^2 \le 1\} \]

Of course, note that \(X\) and \(Y\) in the previous theorem are not independent. Another method of simulation is to use the rejection method. This method works well since the semicircle distribution has a bounded support interval and a bounded probability density function.

Open the rejection method app and select the semicircle distribution. Keep the default parameters to get the standard semicirle distribution. Run the simulation 1000 times and note the points in the scatterplot. Compare the empirical density function, mean, and standard deviation to their distributional counterparts.

The General Semicircle Distribution

Like so many standard distributions, the standard semicircle distribution is usually generalized by adding location and scale parameters.

Definition

Suppose that \(Z\) has the standard semicircle distribution. For \(a \in \R\) and \(r \in (0, \infty)\), \(X = a + r Z\) has the semicircle distribution with center (location parameter) \(a\) and radius (scale parameter) \(r\).

Distribution Functions

\(X\) has probability density function \(f\) given by \[ f(x) = \frac{2}{\pi r^2} \sqrt{r^2 - (x - a)^2}, \quad x \in [a - r, a + r] \]

Proof:

This follows from a standard result for location-scale families. Recall that \[ f(x) = \frac{1}{r} g\left(\frac{x - a}{r}\right), \quad \frac{x - a}{r} \in [-1, 1] \] where \( g \) is the standard semicircle PDF given above.

The graph of \( f \) is the upper half of the circle of radius \( r \) centered at \(a\). The area under this semicircle is \( \pi r^2 / 2 \) so as a check on our work, we see that \( f \) is a valid probability density function.

\( f \) satisfies the following properties:

  1. \( f \) is symmetric about \( x = a \).
  2. \( f \) increases and then decreases with mode at \( x = a \).
  3. \( f \) is concave downward.

Open special distribution simulator and select the semicircle distribution. Vary the center \(a\) and the radius \( r \), and note the shape of the probability density function. For selected values of \(a\) and \( r \), run the simulation 1000 times and compare the empirical density function to the probability density function.

The distribution function \( F \) of \( X \) is \[ F(x) = \frac{1}{2} + \frac{x - a}{\pi r^2} \sqrt{r^2 - (x - a)^2} + \frac{1}{\pi} \arcsin\left(\frac{x - a}{r}\right), \quad x \in [a - r, a + r] \]

Proof:

This follows from a standard result for location-scale families: \[ F(x) = G\left(\frac{x - a}{r}\right), \quad \frac{x - a}{r} \in [-1, 1]\] where \( G \) is the standard semicircle CDF given above.

As in the standard case, we cannot give the quantile function \( F^{-1} \) in closed form, but values of this function can be approximated. Recall that \(F^{-1}(p) = a + r G^{-1}(p)\) where \( G^{-1} \) is the standard semicircle quantile function. In particular, \( F^{-1}\left(\frac{1}{2} - p\right) = 2 a - F^{-1}\left(\frac{1}{2} + p\right) \) for \( 0 \le p \le \frac{1}{2} \). The median is \(a\).

Open the special distribution simulator and select the semicircle distribution. Vary the center \(a\) and the radius \( r \), and note the shape of the distribution function. For selected values of \(a\) and \( r \), compute the first and third quartiles.

Moments

Of course, the moments of \(X\) can be computed from the moments of the standard variable \(Z\) given above. Using the binomial theorem and the linearity of expected value we have \[ \E\left(X^n\right) = \sum_{k=0}^n \binom{n}{k} r^k a^{n-k} \E\left(Z^k\right), \quad n \in \N \]

In particular,

The mean and variance of \( X \) are

  1. \( \E(X) = a \)
  2. \( \var(X) = r^2 / 4 \)

When the center is 0, the general moments have a simple form:

Suppose that \( a = 0 \). For \( n \in \N \), \begin{align} \E\left(X^{2n}\right) & = \left(\frac{r}{2}\right)^{2n} \frac{1}{n + 1} \binom{2n}{n} \\ \E\left(X^{2n+1}\right) & = 0 \end{align}

Proof:

This follows from the moment results for the standard variable \(Z\) given above since \(X^m = r^m Z^m\) for \(m \in \N\).

Open the special distribution simulator and select the semicircle distribution. Vary the center \(a\) and the radius \( r \), and note the size and location of the mean \( \pm \) standard deviation bar. For selected values of \(a\) and \( r \), run the simulation 1000 times and compare the empirical mean and standard deviation to the distribution mean and standard deviation.

The skewness and kurtosis of \( X \) are

  1. \( \skw(X) = 0 \)
  2. \( \kur(X) = 2 \)
Proof:

These results follow immediately from the corresponding results for the standard distribution, given above. Recall that skewness and kurtosis are defined in terms of the standard score, which is independent of the location and scale parameters..

Once again, the excess kurtosis is \( \kur(X) - 3 = -1 \).

Related Distributions

Since the semicircle distribution is a location-scale family, it's invariant under location-scale transformations.

Suppose that \( X \) has the semicircle distribution with center \(a\) and radius \( r \). If \(b \in \R\) and \( c \in (0, \infty) \) then \(b + c X \) has the semicircle distribution with center \(b + c a\) and radius \( c r \).

Proof:

This follows immediately from the representation \(X = a + r Z\) where \(Z\) has the standard semicircle distribution.

One member of the beta family of distributions is a semicircle distribution:

The beta distribution with left parameter \(3/2 \) and right parameter \(3/2 \) is the semicircle distribution with center \(1/2\) and radius \(1/2\).

Proof:

By definition, the beta distribution with left and right parameters \(3/2\) has PDF \[ f(x) = \frac{1}{B(3/2, 3/2)}x^{1/2}(1 - x)^{1/2}, \quad x \in [0, 1] \] But \(B(3/2, 3/2) = \pi/8\) and \(x^{1/2}(1 - x)^{1/2} = \sqrt{x - x^2}\). Completing the square gives \[ f(x) = \frac{8}{\pi} \sqrt{\frac{1}{4} - \left(x - \frac{1}{2}\right)^2}, \quad x \in [0, 1] \] which is the PDF of the semicircle distribution with center \(1/2\) and radius \(1/2\).

Since we can simulate a variable \(Z\) with the standard semicircle distribution by the method in given above, we can simulate a variable with the semicircle distribution with center \(a\) and radius \(r\) by our very definition: \(X = a + r Z\). Once again, the rejection method also works well since the support and probability density fucntion of \(X\) are bounded.

Open the rejection method app and select the semicircle distribution. For selected values of \(a\) and \(r\), run the simulation 1000 times and note the points in the scatterplot. Compare the empirical density function, mean and standard deviation to their distributional counterparts.