Von Mises Distribution Calculator

Compute the probability density, cumulative probability, and upper-tail probability of the circular von Mises distribution from a mean direction and concentration.

Input

Compute probabilities for the circular von Mises distribution (circular normal). Enter the angle x, mean direction mu, and concentration kappa.

Angle unit

Angle x

The angle at which to evaluate the probability.

Mean direction mu

The central direction of the distribution.

Concentration kappa

A value of 0 or more. Larger values concentrate around mu.

Result

Probability density f(x) at x = 1

0.2057145

Cumulative F(x)

0.88957774

Upper tail 1 − F(x)

0.11042226

Mean direction mu

Circular variance

0.30222534

Probability density (x-axis: Radians)

Cumulative distribution (x-axis: Radians)

How it works

The probability density is f(x) = exp(kappa cos(x − mu)) / (2 pi I0(kappa)), where I0 is the first-kind modified Bessel function, implemented here from its series expansion.
The cumulative distribution F(x) is obtained by numerically integrating the density from −pi to x with the composite Simpson rule. The upper-tail probability is 1 − F(x).
The mean direction equals mu and the circular variance is 1 − I1(kappa)/I0(kappa). A larger concentration kappa makes the distribution tighter around mu.
Angles may be entered in degrees or radians and are normalized internally to the range from −pi to pi.

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