Half circle distribution

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Half-Circle
Probability density function
Plot of the Half-Circle Density Function
Cumulative distribution function
{{{cdf_image}}}
Parameters r:~r \in (-\infty,\infty)
Support -r \le x \le r \!
Probability density function (pdf) f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r]
Cumulative distribution function (cdf) F(x)=0.5 + {\arcsin(x/r) \over \pi} + {x\sqrt{1 - {x^2 \over r^2}} \over \pi \times r},\ x\in [-r , r]
Mean 0
Median 0
Mode 0
Variance r2 / 4
Skewness 0
Excess kurtosis TBD
Entropy TBD
Moment-generating function (mgf) TBD
Characteristic function 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 \not= 1.

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

[edit] 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.

[edit] Applications

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

[edit] Interactive demonstrations

The SOCR tools allow interactive manipulations and computations of the Half-circle distributions, among other continuous and discrete distributions. Go to SOCR Distributions and select the Circle Distribution from the drop-down list of distributions in this Java applet.


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