product term in parentheses is just the distribution of the mean for a circular uniform distribution. This means that the distribution of the mean direction of a von Mises distribution is a von Mises distribution , or, equivalently, . Entropy By definition, the information entropy of the von Mises distribution is where is any interval of length . The logarithm of the density of the Von Mises distribution is straightforward: The characteristic function representation for the Von Mises distribution is: where . Substituting these expressions into the entropy integral, exchanging the order of integration and summation, and using the orthogonality of the cosines, the entropy may be written: For , the von Mises distribution becomes the circular uniform distribution and the entropy attains its maximum value of . Notice that the Von Mises distribution maximizes the entropy when the real and imaginary parts of the first circular moment are specified or, equivalently, the circular mean and circular variance are specified. See also Bivariate von Mises distribution Directional statistics Von Mises–Fisher distribution Kent distribution References Further reading Abramowitz, M. and Stegun, I. A. (ed.), Handbook of Mathematical Functions, National Bureau of Standards, 1964; reprinted Dover Publications, 1965. "Algorithm AS 86: The von Mises Distribution Function", Mardia, Applied Statistics, 24, 1975 (pp. 268–272). "Algorithm 518, Incomplete Bessel Function I0: The von Mises Distribution", Hill, ACM Transactions on Mathematical Software, Vol. 3, No. 3, September 1977, Pages 279–284. Best, D. and Fisher, N. (1979). Efficient simulation of the von Mises distribution. Applied Statistics, 28, 152–157. Evans, M., Hastings, N., and Peacock, B., "von Mises Distribution". Ch. 41 in Statistical Distributions, 3rd ed. New York. Wiley 2000. Fisher, Nicholas I., Statistical Analysis of Circular Data. New York. Cambridge 1993. "Statistical Distributions", 2nd. Edition, Evans, Hastings, and Peacock, John Wiley and Sons, 1993, (chapter 39). Continuous distributions Directional statistics Exponential family distributions