First attempt at coding up the semicircle distribution

Author: jiahao

src/GaussianDensities.jl

@@ -0,0 +1,66 @@
+using Distributions
+
+immutable Semicircle <: ContinuousUnivariateDistribution
+    mean::Float64
+    radius::Float64
+    Semicircle(mu, r) = r > 0 ? new(float64(mu), float64(r)) : error("radius r must be positive")
+end
+
+#Standard (normalized) - note this is NOT the standard used in to RMT (r = 2)
+Semicircle(mu) = Semicircle(mu, 1.0)
+Semicircle() = Semicircle(0.0, 1.0)
+
+# Properties
+# For convenience, this is ordered like the Wikipedia infobox template ;)
+
+#Probability density function
+function pdf(d::Semicircle, x::Real)
+    r, a = d.mean, d.radius
+    return 2/(pi*r^2) * sqrt(r^2 - (x-a)^2)
+end
+
+#Cumulative density function
+function cdf(d::Semicircle, x::Real)
+    r, a = d.mean, d.radius
+    if x < -a
+        return 0.0
+    elseif x > a
+        return 1.0
+    else
+        return 0.5 + (x-a)/(pi*r^2) * sqrt(r^2 - (x-a)^2) + 1/pi * asin((x-a)/r)
+    end
+end
+
+function logpdf(d::Laplace, x::Real)
+    log(pdf(d, x))
+end
+function quantile(d::Laplace, p::Real)
+    0.0
+end
+
+mean(d::Semicircle) = d.mean
+median(d::Semicircle) = d.mean
+mode(d::Semicircle) = d.mean
+
+std(d::Semicircle) = (d.radius/2)
+var(d::Semicircle) = (d.radius/2)^2
+skewness(d::Semicircle) = 0.0
+kurtosis(d::Semicircle) = 2.0
+entropy(d::Semicircle) = log(pi*R) - 0.5
+
+## Moment generating function
+function mgf(d::Semicircle, t::Real)
+    r = t * d.mean
+    2 * besseli(1, r)/r
+end
+
+# Characteristic function
+function cf(d::Semicircle, t::Real)
+    r = t * d.mean
+    2 * besselj(1, r)/r
+end
+
+## Common methods
+#XXX Use the fact that Semicircle(0.5, 0.5) = Beta(1.5, 1.5)
+rand(d::Semicircle) = 2*(rand(Beta(1.5, 1.5) - 0.5))
+insupport(d::Normal, x::Number) = real_valued(x) & (abs(x) <= d.radius)