# scipy.stats.multivariate_t

scipy.stats.multivariate_t = <scipy.stats._multivariate.multivariate_t_gen object>

  A multivariate t-distributed random variable.

  The loc parameter specifies the location. The shape parameter specifies the positive semidefinite shape matrix. The df parameter specifies the degrees of freedom.

  In addition to calling the methods below, the object itself may be called as a function to fix the location, shape matrix, and degrees of freedom parameters, returning a “frozen” multivariate t-distribution random.

  Parameters
    x array_like

      Quantiles, with the last axis of x denoting the components.

    loc array_like, optional

      Location of the distribution. (default `0`)

    shape array_like, optional

      Positive semidefinite matrix of the distribution. (default `1`)

    df float, optional

      Degrees of freedom of the distribution; must be greater than zero. If `np.inf` then results are multivariate normal. The default is `1`.

    allow_singular bool, optional

      Whether to allow a singular matrix. (default `False`)

    random_state{None, int, `numpy.random.Generator`,

      > `numpy.random.RandomState`}, optional

      If seed is None (or np.random), the `numpy.random.RandomState` singleton is used. If seed is an int, a new `RandomState` instance is used, seeded with seed. If seed is already a `Generator` or `RandomState` instance then that instance is used.

  Notes

  Setting the parameter loc to `None` is equivalent to having loc be the zero-vector. The parameter shape can be a scalar, in which case the shape matrix is the identity times that value, a vector of diagonal entries for the shape matrix, or a two-dimensional array_like. The matrix shape must be a (symmetric) positive semidefinite matrix. The determinant and inverse of shape are computed as the pseudo-determinant and pseudo-inverse, respectively, so that shape does not need to have full rank.

  The probability density function for `multivariate_t` is

  \[f(x) = \frac{\Gamma(\nu + p)/2}{\Gamma(\nu/2)\nu^{p/2}\pi^{p/2}|\Sigma|^{1/2}} \exp\left[1 + \frac{1}{\nu} (\mathbf{x} - \boldsymbol{\mu})^{\top} \boldsymbol{\Sigma}^{-1} (\mathbf{x} - \boldsymbol{\mu}) \right]^{-(\nu + p)/2},\]

  where \(p\) is the dimension of \(\mathbf{x}\), \(\boldsymbol{\mu}\) is the \(p\)-dimensional location, \(\boldsymbol{\Sigma}\) the \(p \times p\)-dimensional shape matrix, and \(\nu\) is the degrees of freedom.

  New in version 1.6.0.

  Examples

      >>> import matplotlib.pyplot as plt
      >>> from scipy.stats import multivariate_t
      >>> x, y = np.mgrid[-1:3:.01, -2:1.5:.01]
      >>> pos = np.dstack((x, y))
      >>> rv = multivariate_t([1.0, -0.5], [[2.1, 0.3], [0.3, 1.5]], df=2)
      >>> fig, ax = plt.subplots(1, 1)
      >>> ax.set_aspect('equal')
      >>> plt.contourf(x, y, rv.pdf(pos))

  Methods

  ``pdf(x, loc=None, shape=1, df=1, allow_singular=False)`` | Probability density function.
  ``logpdf(x, loc=None, shape=1, df=1, allow_singular=False)`` | Log of the probability density function.
  ``rvs(loc=None, shape=1, df=1, size=1, random_state=None)`` | Draw random samples from a multivariate t-distribution.

scipy.stats.random_correlation scipy.stats.multivariate_hypergeom