Half circle distribution

Probability distribution
name =Half-Circle
type =density
pdf_

parameters =$r:~r\; in\; (-infty,infty)$
support =$-r\; le\; x\; le\; r\; !$
pdf =$f(x)=\{2sqrt\{r^2\; -\; x^2\}over\; pi\; r^2\; \},\; forall\; x\; in\; [-r\; ,\; r]$
cdf =$F(x)=0.5\; +\; \{arcsin(x/r)\; over\; pi\}\; +\; \{xsqrt\{1\; -\; \{x^2\; over\; r^2$ over pi imes r}, xin [-r , r]
mean =$0$
median =$0$
mode =$0$
variance =$r^2/4$
skewness =$0$
kurtosis =TBD
entropy =TBD
mgf =TBD
char =TBD

In probability theory and statistics, the half-circle distribution is a continuous probability distribution defined by the unique half-circle that goes between [-r , r] , where r = radius, whose area is one. Note that the half-circle is effectively a "half-ellipse", in general, as its area is bound to 1. The half-circle distribution is circular for radius=1 and elliptical for radius $ot=$ 1.

: $f(x|r)=\; \{2sqrt\{r^2\; -\; x^2\}over\; pi\; r^2\; \},\; quad\; ext\{for\; \}\; x\; in\; [-r,\; r]\; .$

Parameter relations

The Half-circle distribution has effectively only 1 parameter "radius", when it is centered in the origin. If shifts are allowed, then another (center) parameter will be necessary to completely describe the distribution. The radius is half of the range of the distribution.

Applications

This distribution is a useful educational illustration of a compactly supported continuous distribution.

Interactive demonstrations

The SOCR tools allow interactive manipulations and computations of the Half-circle distributions, among other continuous and discrete distributions. Go to [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Distributions] and select the Circle Distribution from the drop-down list of distributions in this Java applet.