DOC: update the DataFrame.cov docstring (pandas-dev#20245)

* DOC: Revise docstring of DataFrame cov method

Update the docstring with some examples from
elsewhere in the pandas documentation.

Some of the examples use randomly generated time series
because we need to get covariance between long series.
Used a random seed to ensure that the results are the
same each time.

* DOC: Fix See Also and min_periods explanation.

Responding to comments on PR. See also section will link
properly and number of periods explanation clearer.

pandas/core/frame.py

@@ -5501,7 +5501,22 @@ def corr(self, method='pearson', min_periods=1):
 
     def cov(self, min_periods=None):
         """
-        Compute pairwise covariance of columns, excluding NA/null values
+        Compute pairwise covariance of columns, excluding NA/null values.
+
+        Compute the pairwise covariance among the series of a DataFrame.
+        The returned data frame is the `covariance matrix
+        <https://en.wikipedia.org/wiki/Covariance_matrix>`__ of the columns
+        of the DataFrame.
+
+        Both NA and null values are automatically excluded from the
+        calculation. (See the note below about bias from missing values.)
+        A threshold can be set for the minimum number of
+        observations for each value created. Comparisons with observations
+        below this threshold will be returned as ``NaN``.
+
+        This method is generally used for the analysis of time series data to
+        understand the relationship between different measures
+        across time.
 
         Parameters
         ----------
@@ -5511,12 +5526,71 @@ def cov(self, min_periods=None):
 
         Returns
         -------
-        y : DataFrame
+        DataFrame
+            The covariance matrix of the series of the DataFrame.
+
+        See Also
+        --------
+        pandas.Series.cov : compute covariance with another Series
+        pandas.core.window.EWM.cov: expoential weighted sample covariance
+        pandas.core.window.Expanding.cov : expanding sample covariance
+        pandas.core.window.Rolling.cov : rolling sample covariance
 
         Notes
         -----
-        `y` contains the covariance matrix of the DataFrame's time series.
-        The covariance is normalized by N-1 (unbiased estimator).
+        Returns the covariance matrix of the DataFrame's time series.
+        The covariance is normalized by N-1.
+
+        For DataFrames that have Series that are missing data (assuming that
+        data is `missing at random
+        <https://en.wikipedia.org/wiki/Missing_data#Missing_at_random>`__)
+        the returned covariance matrix will be an unbiased estimate
+        of the variance and covariance between the member Series.
+
+        However, for many applications this estimate may not be acceptable
+        because the estimate covariance matrix is not guaranteed to be positive
+        semi-definite. This could lead to estimate correlations having
+        absolute values which are greater than one, and/or a non-invertible
+        covariance matrix. See `Estimation of covariance matrices
+        <http://en.wikipedia.org/w/index.php?title=Estimation_of_covariance_
+        matrices>`__ for more details.
+
+        Examples
+        --------
+        >>> df = pd.DataFrame([(1, 2), (0, 3), (2, 0), (1, 1)],
+        ...                   columns=['dogs', 'cats'])
+        >>> df.cov()
+                  dogs      cats
+        dogs  0.666667 -1.000000
+        cats -1.000000  1.666667
+
+        >>> np.random.seed(42)
+        >>> df = pd.DataFrame(np.random.randn(1000, 5),
+        ...                   columns=['a', 'b', 'c', 'd', 'e'])
+        >>> df.cov()
+                  a         b         c         d         e
+        a  0.998438 -0.020161  0.059277 -0.008943  0.014144
+        b -0.020161  1.059352 -0.008543 -0.024738  0.009826
+        c  0.059277 -0.008543  1.010670 -0.001486 -0.000271
+        d -0.008943 -0.024738 -0.001486  0.921297 -0.013692
+        e  0.014144  0.009826 -0.000271 -0.013692  0.977795
+
+        **Minimum number of periods**
+
+        This method also supports an optional ``min_periods`` keyword
+        that specifies the required minimum number of non-NA observations for
+        each column pair in order to have a valid result:
+
+        >>> np.random.seed(42)
+        >>> df = pd.DataFrame(np.random.randn(20, 3),
+        ...                   columns=['a', 'b', 'c'])
+        >>> df.loc[df.index[:5], 'a'] = np.nan
+        >>> df.loc[df.index[5:10], 'b'] = np.nan
+        >>> df.cov(min_periods=12)
+                  a         b         c
+        a  0.316741       NaN -0.150812
+        b       NaN  1.248003  0.191417
+        c -0.150812  0.191417  0.895202
         """
         numeric_df = self._get_numeric_data()
         cols = numeric_df.columns