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# scipy.stats.multivariate_normal

`scipy.stats.``multivariate_normal` = <scipy.stats._multivariate.multivariate_normal_gen object>

  A multivariate normal random variable.

  The mean keyword specifies the mean. The cov keyword specifies the covariance matrix.

  Parameters:

  x : array_like

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

  mean : array_like, optional

  > Mean of the distribution (default zero)

  cov : array_like, optional

  > Covariance matrix of the distribution (default one)

  allow_singular : bool, optional

  > Whether to allow a singular covariance matrix. (Default: False)

  random_state : None or int or np.random.RandomState instance, optional

  > If int or RandomState, use it for drawing the random variates. If None (or np.random), the global np.random state is used. Default is None.

  Alternatively, the object may be called (as a function) to fix the mean

  and covariance parameters, returning a “frozen” multivariate normal

  random variable:

  rv = multivariate_normal(mean=None, cov=1, allow_singular=False)

  > - Frozen object with the same methods but holding the given mean and covariance fixed.

  Notes

  Setting the parameter mean to None is equivalent to having mean
    be the zero-vector. The parameter cov can be a scalar, in which case the covariance matrix is the identity times that value, a vector of diagonal entries for the covariance matrix, or a two-dimensional array_like.

  The covariance matrix cov must be a (symmetric) positive semi-definite matrix. The determinant and inverse of cov are computed as the pseudo-determinant and pseudo-inverse, respectively, so that cov does not need to have full rank.

  The probability density function for `multivariate_normal` is

  \[f(x) = \frac{1}{\sqrt{(2 \pi)^k \det \Sigma}} \exp\left( -\frac{1}{2} (x - \mu)^T \Sigma^{-1} (x - \mu) \right),\]

  where \(\mu\) is the mean, \(\Sigma\) the covariance matrix, and \(k\) is the dimension of the space where \(x\) takes values.

  New in version 0.14.0.

  Examples

      >>> import matplotlib.pyplot as plt
      >>> from scipy.stats import multivariate_normal

      >>> x = np.linspace(0, 5, 10, endpoint=False)
      >>> y = multivariate_normal.pdf(x, mean=2.5, cov=0.5); y
      array([ 0.00108914,  0.01033349,  0.05946514,  0.20755375,  0.43939129,
              0.56418958,  0.43939129,  0.20755375,  0.05946514,  0.01033349])
      >>> fig1 = plt.figure()
      >>> ax = fig1.add_subplot(111)
      >>> ax.plot(x, y)

  The input quantiles can be any shape of array, as long as the last axis labels the components. This allows us for instance to display the frozen pdf for a non-isotropic random variable in 2D as follows:

      >>> x, y = np.mgrid[-1:1:.01, -1:1:.01]
      >>> pos = np.dstack((x, y))
      >>> rv = multivariate_normal([0.5, -0.2], [[2.0, 0.3], [0.3, 0.5]])
      >>> fig2 = plt.figure()
      >>> ax2 = fig2.add_subplot(111)
      >>> ax2.contourf(x, y, rv.pdf(pos))

  (Source code)

  Methods

  `pdf(x, mean=None, cov=1, allow_singular=False)` | Probability density function.
  `logpdf(x, mean=None, cov=1, allow_singular=False)` | Log of the probability density function.
  `rvs(mean=None, cov=1, size=1, random_state=None)` | Draw random samples from a multivariate normal distribution.
  `entropy()` | Compute the differential entropy of the multivariate normal.