Circular uniform distribution

In probability theory and directional statistics, a circular uniform distribution is a probability distribution on the unit circle whose density is uniform for all angles.

The probability density function (pdf) of the circular uniform distribution, e.g. with

at base angle

In these terms, the circular moments of the circular uniform distribution are all zero, except for

is the Kronecker delta symbol.

Here the mean angle is undefined, and the length of the mean resultant is zero.

The sample mean of a set of N measurements

drawn from a circular uniform distribution is defined as: where the average sine and cosine are:[1] and the average resultant length is: and the mean angle is: The sample mean for the circular uniform distribution will be concentrated about zero, becoming more concentrated as N increases.

in the variables, subject to the constraint that

is uniform and the distribution of

is the Bessel function of order zero.

There is no known general analytic solution for the above integral, and it is difficult to evaluate due to the large number of oscillations in the integrand.

A 10,000 point Monte Carlo simulation of the distribution of the mean for N=3 is shown in the figure.

For certain special cases, the above integral can be evaluated: For large N, the distribution of the mean can be determined from the central limit theorem for directional statistics.

Since the angles are uniformly distributed, the individual sines and cosines of the angles will be distributed as: where

By the central limit theorem, in the limit of large N,

, being the sum of a large number of i.i.d's, will be normally distributed with mean zero and variance

The mean resultant length

, being the square root of the sum of squares of two normally distributed independent variables, will be Chi-distributed with two degrees of freedom (i.e.Rayleigh-distributed) and variance

: The differential information entropy of the uniform distribution is simply where

This is the maximum entropy any circular distribution may have.