code: from __future__ import annotations
  import numpy as np
  from numpy.linalg import inv, det, slogdet
  
  
  class UnivariateGaussian:
      """
      Class for univariate Gaussian Distribution Estimator
      """
  
      def __init__(self, biased_var: bool = False) -> UnivariateGaussian:
          """
          Estimator for univariate Gaussian mean and variance parameters
  
          Parameters
          ----------
          biased_var : bool, default=False
              Should fitted estimator of variance be a biased or unbiased estimator
  
          Attributes
          ----------
          fitted_ : bool
              Initialized as false indicating current estimator instance has not been fitted.
              To be set as True in `UnivariateGaussian.fit` function.
  
          mu_: float
              Estimated expectation initialized as None. To be set in `UnivariateGaussian.fit`
              function.
  
          var_: float
              Estimated variance initialized as None. To be set in `UnivariateGaussian.fit`
              function.
          """
          self.biased_ = biased_var
          self.fitted_, self.mu_, self.var_ = False, None, None
  
      def fit(self, X: np.ndarray) -> UnivariateGaussian:
          """
          Estimate Gaussian expectation and variance from given samples
  
          Parameters
          ----------
          X: ndarray of shape (n_samples, )
              Training data
  
          Returns
          -------
          self : returns an instance of self.
  
          Notes
          -----
          Sets `self.mu_`, `self.var_` attributes according to calculated estimation (where
          estimator is either biased or unbiased). Then sets `self.fitted_` attribute to `True`
          """
  
          self.mu_ = sum(X) / X.size
          var_sum = 0
          for s in X:
              var_sum += (s - self.mu_) ** 2
  
          self.var_ = (var_sum / (X.size if self.biased_ else X.size - 1)) ** 0.5
  
          self.fitted_ = True
          return self
  
      def pdf(self, X: np.ndarray) -> np.ndarray:
          """
          Calculate PDF of observations under Gaussian model with fitted estimators
  
          Parameters
          ----------
          X: ndarray of shape (n_samples, )
              Samples to calculate PDF for
  
          Returns
          -------
          pdfs: ndarray of shape (n_samples, )
              Calculated values of given samples for PDF function of N(mu_, var_)
  
          Raises
          ------
          ValueError: In case function was called prior fitting the model
          """
          if not self.fitted_:
              raise ValueError(
                  "Estimator must first be fitted before calling `pdf` function")
          pdfs = np.ndarray(X.size)
          for i in range(X.size):
              pdfs[i] = np.exp(
                  -((X[i] - self.mu_) ** 2) / 2 * self.var_) / np.sqrt(
                  2 * np.pi * self.var_)
          return pdfs
  
      @staticmethod
      def log_likelihood(mu: float, sigma: float, X: np.ndarray) -> float:
          """
          Calculate the log-likelihood of the data under a specified Gaussian model
  
          Parameters
          ----------
          mu : float
              Expectation of Gaussian
          sigma : float
              Variance of Gaussian
          X : ndarray of shape (n_samples, )
              Samples to calculate log-likelihood with
  
          Returns
          -------
          log_likelihood: float
              log-likelihood calculated
          """
          temp_sum = 0
          for i in range(X.size):
              temp_sum += (X[i] - mu) ** 2
          return -(X.size / 2) * (
                  np.log(2 * np.pi) + np.log(sigma)) - temp_sum / (
                         2 * sigma)
  
  
  class MultivariateGaussian:
      """
      Class for multivariate Gaussian Distribution Estimator
      """
  
      def __init__(self):
          """
          Initialize an instance of multivariate Gaussian estimator
  
          Attributes
          ----------
          fitted_ : bool
              Initialized as false indicating current estimator instance has not been fitted.
              To be set as True in `MultivariateGaussian.fit` function.
  
          mu_: ndarray of shape (n_features,)
              Estimated expectation initialized as None. To be set in `MultivariateGaussian.fit`
              function.
  
          cov_: ndarray of shape (n_features, n_features)
              Estimated covariance initialized as None. To be set in `MultivariateGaussian.fit`
              function.
          """
          self.mu_, self.cov_ = None, None
          self.fitted_ = False
  
      def fit(self, X: np.ndarray) -> MultivariateGaussian:
          """
          Estimate Gaussian expectation and covariance from given samples
  
          Parameters
          ----------
          X: ndarray of shape (n_samples, n_features)
              Training data
  
          Returns
          -------
          self : returns an instance of self
  
          Notes
          -----
          Sets `self.mu_`, `self.cov_` attributes according to calculated estimation.
          Then sets `self.fitted_` attribute to `True`
          """
          rows, cols = X.shape
          self.mu_ = np.sum(X, axis=0) / rows
          X_gal = np.array([X[i] - self.mu_ for i in range(rows)])
          self.cov_ = np.dot(X_gal.transpose(), X_gal) / (rows - 1)
          self.fitted_ = True
          return self
  
      def pdf(self, X: np.ndarray):
          """
          Calculate PDF of observations under Gaussian model with fitted estimators
  
          Parameters
          ----------
          X: ndarray of shape (n_samples, n_features)
              Samples to calculate PDF for
  
          Returns
          -------
          pdfs: ndarray of shape (n_samples, )
              Calculated values of given samples for PDF function of N(mu_, cov_)
  
          Raises
          ------
          ValueError: In case function was called prior fitting the model
          """
          if not self.fitted_:
              raise ValueError(
                  "Estimator must first be fitted before calling `pdf` function")
          mahalanobis = np.einsum("bi,ij,bj->b", X-self.mu_, inv(self.cov_), X-self.mu_)
  
          return np.exp(-.5 * mahalanobis) / \
                 np.sqrt((2*np.pi) ** len(X) * det(self.cov_))
  
  
      @staticmethod
      def log_likelihood(mu: np.ndarray, cov: np.ndarray,
                         X: np.ndarray) -> float:
          """
          Calculate the log-likelihood of the data under a specified Gaussian model
  
          Parameters
          ----------
          mu : ndarray of shape (n_features,)
              Expectation of Gaussian
          cov : ndarray of shape (n_features, n_features)
              covariance matrix of Gaussian
          X : ndarray of shape (n_samples, n_features)
              Samples to calculate log-likelihood with
  
          Returns
          -------
          log_likelihood: float
              log-likelihood calculated over all input data and under given parameters of Gaussian
          """
          rows, cols = X.shape
          X_gal = np.array([X[i] - mu for i in range(rows)])
  
          temp_sun = 0
          for i in range(rows):
              temp_sun += np.linalg.multi_dot([X_gal[i].transpose(),
                                   np.linalg.inv(cov),
                                   X_gal[i]])
          return -(X.size / 2) * (cols * np.log(2 * np.pi) + np.log(
              np.linalg.det(cov))) - 0.5 * temp_sun
  
  
  
apis: numpy.sum numpy.linalg.inv numpy.linalg.det numpy.exp numpy.ndarray numpy.log numpy.sqrt