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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 sample mean for the circular uniform distribution will be concentrated about zero, becoming more concentrated as N increases. The distribution of the sample mean for the uniform distribution is given by:[2]
where consists of intervals of in the variables, subject to the constraint that and are constant, or, alternatively, that and are constant. The distribution of the angle is uniform
where 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 or . It follows that they will have zero mean and a variance of 1/2. By the central limit theorem, in the limit of large N, and , 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 two normally distributed variables, will be Chi-distributed with two degrees of freedom (i.e.Rayleigh-distributed) and variance :