# numpy.random.vonmises

random.vonmises(mu, kappa, size=None)

  Draw samples from a von Mises distribution.

  Samples are drawn from a von Mises distribution with specified mode (mu) and dispersion (kappa), on the interval [-pi, pi].

  The von Mises distribution (also known as the circular normal distribution) is a continuous probability distribution on the unit circle. It may be thought of as the circular analogue of the normal distribution.

  Note

  New code should use the `vonmises` method of a `default_rng()` instance instead; please see the Quick Start.

  Parameters
    mu float or array_like of floats

      Mode (“center”) of the distribution.

    kappa float or array_like of floats

      Dispersion of the distribution, has to be >=0.

    size int or tuple of ints, optional

      Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `mu` and `kappa` are both scalars. Otherwise, `np.broadcast(mu, kappa).size` samples are drawn.

  Returns
    out ndarray or scalar

      Drawn samples from the parameterized von Mises distribution.

  See also

  `scipy.stats.vonmises`

    probability density function, distribution, or cumulative density function, etc.

  `random.Generator.vonmises`

    which should be used for new code.

  Notes

  The probability density for the von Mises distribution is

  \[p(x) = \frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)},\]

  where \(\mu\) is the mode and \(\kappa\) the dispersion, and \(I_0(\kappa)\) is the modified Bessel function of order 0.

  The von Mises is named for Richard Edler von Mises, who was born in Austria-Hungary, in what is now the Ukraine. He fled to the United States in 1939 and became a professor at Harvard. He worked in probability theory, aerodynamics, fluid mechanics, and philosophy of science.

  References

  1

    Abramowitz, M. and Stegun, I. A. (Eds.). “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing,” New York: Dover, 1972.

  2

    von Mises, R., “Mathematical Theory of Probability and Statistics”, New York: Academic Press, 1964.

  Examples

  Draw samples from the distribution:

      >>> mu, kappa = 0.0, 4.0 # mean and dispersion
      >>> s = np.random.vonmises(mu, kappa, 1000)

  Display the histogram of the samples, along with the probability density function:

      >>> import matplotlib.pyplot as plt
      >>> from scipy.special import i0
      >>> plt.hist(s, 50, density=True)
      >>> x = np.linspace(-np.pi, np.pi, num=51)
      >>> y = np.exp(kappa*np.cos(x-mu))/(2*np.pi*i0(kappa))
      >>> plt.plot(x, y, linewidth=2, color='r')
      >>> plt.show()

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