Von Mises Distribution - Distribution of The Mean

Distribution of The Mean

The distribution of the sample mean for the von Mises distribution is given by:

P(\bar{R},\bar{\theta})\,d\bar{R}\,d\bar{\theta}=\frac{1}{ (2\pi I_0(k))^N}\int_\Gamma \prod_{n=1}^N \left( e^{\kappa\cos(\theta_n-\mu)} d\theta_n\right) = \frac{e^{\kappa N\bar{R}\cos(\bar{\theta}-\mu)}}{I_0(\kappa)^N}\left(\frac{1}{(2\pi)^N}\int_\Gamma \prod_{n=1}^N d\theta_n\right)

where N is the number of measurements and consists of intervals of in the variables, subject to the constraint that and are constant, where is the mean resultant:

\bar{R}^2=|\bar{z}|^2= \left(\frac{1}{N}\sum_{n=1}^N \cos(\theta_n) \right)^2 + \left(\frac{1}{N}\sum_{n=1}^N \sin(\theta_n) \right)^2

and is the mean angle:

\overline{\theta}=\mathrm{Arg}(\overline{z}). \,

Note that product term in parentheses is just the distribution of the mean for a circular uniform distribution.

Read more about this topic:  Von Mises Distribution

Famous quotes containing the words distribution of and/or distribution:

    “In this distribution of functions, the scholar is the delegated intellect. In the right state, he is, Man Thinking. In the degenerate state, when the victim of society, he tends to become a mere thinker, or, still worse, the parrot of other men’s thinking.”
    —Ralph Waldo Emerson (1803–1882)

    “In this distribution of functions, the scholar is the delegated intellect. In the right state, he is, Man Thinking. In the degenerate state, when the victim of society, he tends to become a mere thinker, or, still worse, the parrot of other men’s thinking.”
    —Ralph Waldo Emerson (1803–1882)