von Mises distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ von Mises(κ,μ) is used to indicate that the random variable X has the von Mises distribution with shape parameter κ and location parameter μ. A von Mises random variable X with parameters κ and μ has probability density function

f(x) = \frac{e^{κ cos(x−μ)}}{2π I_{0}(κ)} 0 < x < 2π

for all κ and for 0 < μ < 2π. The Modified Bessel function of the first kind of order 0 is used for the von Mises random variable and is defined as

I_{0}(κ) = ∑_{i=0}^{∞} \frac{κ^{2i}}{2^{2i}(i!)^{2}}

for all κ. The probability density function for μ = π and two different values of κ is illustrated below.

PSfrag replacements

κ = 1
κ = 4

0 π 2π x f(x)

The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = \frac{∫_{0}^{x}e^{κ cos(t−μ)}dt}{2π I_{0}(κ)} 0 < x < 2π.

The survivor function, hazard function, inverse distribution function, moment generating function, and characteristic functions are all mathematically intractable. The median of X is

μ.

The population mean of X is

E[X] = μ.

The population variance, skewness, and kurtosis of X are mathematically intractable.