diff --git a/sklearn/cluster/bicluster.py b/sklearn/cluster/bicluster.py
index 8f6206f9599d08f235123cd4f473dc7ae9b5477d..6d5b6e76ee658e32be5b4efcd17bb8781d997b3f 100644
--- a/sklearn/cluster/bicluster.py
+++ b/sklearn/cluster/bicluster.py
@@ -236,9 +236,11 @@ class SpectralCoclustering(BaseSpectral):
         (n_cpus + 1 + n_jobs) are used. Thus for n_jobs = -2, all CPUs but one
         are used.
 
-    random_state : int seed, RandomState instance, or None (default)
-        A pseudo random number generator used by the K-Means
-        initialization.
+    random_state : int, RandomState instance or None, optional, default: None
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     Attributes
     ----------
@@ -366,9 +368,11 @@ class SpectralBiclustering(BaseSpectral):
         (n_cpus + 1 + n_jobs) are used. Thus for n_jobs = -2, all CPUs but one
         are used.
 
-    random_state : int seed, RandomState instance, or None (default)
-        A pseudo random number generator used by the K-Means
-        initialization.
+    random_state : int, RandomState instance or None, optional, default: None
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     Attributes
     ----------
diff --git a/sklearn/cluster/k_means_.py b/sklearn/cluster/k_means_.py
index f33b3f65b714ed8c00ab4f51c4a14e5627b79afd..680edc2672a71e581671d693e8a773ed7de8a2f0 100644
--- a/sklearn/cluster/k_means_.py
+++ b/sklearn/cluster/k_means_.py
@@ -230,10 +230,11 @@ def k_means(X, n_clusters, init='k-means++', precompute_distances='auto',
     verbose : boolean, optional
         Verbosity mode.
 
-    random_state : integer or numpy.RandomState, optional
-        The generator used to initialize the centers. If an integer is
-        given, it fixes the seed. Defaults to the global numpy random
-        number generator.
+    random_state : int, RandomState instance or None, optional, default: None
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     copy_x : boolean, optional
         When pre-computing distances it is more numerically accurate to center
@@ -449,10 +450,11 @@ def _kmeans_single_lloyd(X, n_clusters, max_iter=300, init='k-means++',
     precompute_distances : boolean, default: True
         Precompute distances (faster but takes more memory).
 
-    random_state : integer or numpy.RandomState, optional
-        The generator used to initialize the centers. If an integer is
-        given, it fixes the seed. Defaults to the global numpy random
-        number generator.
+    random_state : int, RandomState instance or None, optional, default: None
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     Returns
     -------
@@ -638,10 +640,11 @@ def _init_centroids(X, k, init, random_state=None, x_squared_norms=None,
     init : {'k-means++', 'random' or ndarray or callable} optional
         Method for initialization
 
-    random_state : integer or numpy.RandomState, optional
-        The generator used to initialize the centers. If an integer is
-        given, it fixes the seed. Defaults to the global numpy random
-        number generator.
+    random_state : int, RandomState instance or None, optional, default: None
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     x_squared_norms :  array, shape (n_samples,), optional
         Squared euclidean norm of each data point. Pass it if you have it at
@@ -766,10 +769,11 @@ class KMeans(BaseEstimator, ClusterMixin, TransformerMixin):
         (n_cpus + 1 + n_jobs) are used. Thus for n_jobs = -2, all CPUs but one
         are used.
 
-    random_state : integer or numpy.RandomState, optional
-        The generator used to initialize the centers. If an integer is
-        given, it fixes the seed. Defaults to the global numpy random
-        number generator.
+    random_state : int, RandomState instance or None, optional, default: None
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     verbose : int, default 0
         Verbosity mode.
@@ -1008,10 +1012,11 @@ def _mini_batch_step(X, x_squared_norms, centers, counts,
         the distances of each sample to its closest center.
         May not be None when random_reassign is True.
 
-    random_state : integer or numpy.RandomState, optional
-        The generator used to initialize the centers. If an integer is
-        given, it fixes the seed. Defaults to the global numpy random
-        number generator.
+    random_state : int, RandomState instance or None, optional, default: None
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     random_reassign : boolean, optional
         If True, centers with very low counts are randomly reassigned
@@ -1247,10 +1252,11 @@ class MiniBatchKMeans(KMeans):
         Compute label assignment and inertia for the complete dataset
         once the minibatch optimization has converged in fit.
 
-    random_state : integer or numpy.RandomState, optional
-        The generator used to initialize the centers. If an integer is
-        given, it fixes the seed. Defaults to the global numpy random
-        number generator.
+    random_state : int, RandomState instance or None, optional, default: None
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     reassignment_ratio : float, default: 0.01
         Control the fraction of the maximum number of counts for a
diff --git a/sklearn/cluster/mean_shift_.py b/sklearn/cluster/mean_shift_.py
index 2d554c05ff80cd2f107cc8f2719070d59fa627e7..522e034a60e5fa5172d9c6481b616843925abef3 100644
--- a/sklearn/cluster/mean_shift_.py
+++ b/sklearn/cluster/mean_shift_.py
@@ -47,8 +47,11 @@ def estimate_bandwidth(X, quantile=0.3, n_samples=None, random_state=0,
     n_samples : int, optional
         The number of samples to use. If not given, all samples are used.
 
-    random_state : int or RandomState
-        Pseudo-random number generator state used for random sampling.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     n_jobs : int, optional (default = 1)
         The number of parallel jobs to run for neighbors search.
diff --git a/sklearn/cluster/spectral.py b/sklearn/cluster/spectral.py
index 8b64ca9a6dd1263a4de16e8bbe7162b731397aa5..b3526622a718cde0c966a9065b16b107edd06e18 100644
--- a/sklearn/cluster/spectral.py
+++ b/sklearn/cluster/spectral.py
@@ -39,9 +39,11 @@ def discretize(vectors, copy=True, max_svd_restarts=30, n_iter_max=20,
         Maximum number of iterations to attempt in rotation and partition
         matrix search if machine precision convergence is not reached
 
-    random_state : int seed, RandomState instance, or None (default)
-        A pseudo random number generator used for the initialization of the
-        of the rotation matrix
+    random_state : int, RandomState instance or None, optional, default: None
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     Returns
     -------
@@ -194,10 +196,13 @@ def spectral_clustering(affinity, n_clusters=8, n_components=None,
         to be installed. It can be faster on very large, sparse problems,
         but may also lead to instabilities
 
-    random_state : int seed, RandomState instance, or None (default)
-        A pseudo random number generator used for the initialization
-        of the lobpcg eigen vectors decomposition when eigen_solver == 'amg'
-        and by the K-Means initialization.
+    random_state : int, RandomState instance or None, optional, default: None
+        A pseudo random number generator used for the initialization of the
+        lobpcg eigen vectors decomposition when eigen_solver == 'amg' and by
+        the K-Means initialization. If int, random_state is the seed used by
+        the random number generator; If RandomState instance, random_state is
+        the random number generator; If None, the random number generator is
+        the RandomState instance used by `np.random`.
 
     n_init : int, optional, default: 10
         Number of time the k-means algorithm will be run with different
@@ -326,10 +331,13 @@ class SpectralClustering(BaseEstimator, ClusterMixin):
         to be installed. It can be faster on very large, sparse problems,
         but may also lead to instabilities
 
-    random_state : int seed, RandomState instance, or None (default)
-        A pseudo random number generator used for the initialization
-        of the lobpcg eigen vectors decomposition when eigen_solver == 'amg'
-        and by the K-Means initialization.
+    random_state : int, RandomState instance or None, optional, default: None
+        A pseudo random number generator used for the initialization of the
+        lobpcg eigen vectors decomposition when eigen_solver == 'amg' and by
+        the K-Means initialization.  If int, random_state is the seed used by
+        the random number generator; If RandomState instance, random_state is
+        the random number generator; If None, the random number generator is
+        the RandomState instance used by `np.random`.
 
     n_init : int, optional, default: 10
         Number of time the k-means algorithm will be run with different
diff --git a/sklearn/covariance/robust_covariance.py b/sklearn/covariance/robust_covariance.py
index 29cbd52e183d3679d2c718ae11bad8c2dc97d544..fdf0225dbdadcf39439784dcaf50c288377a6462 100644
--- a/sklearn/covariance/robust_covariance.py
+++ b/sklearn/covariance/robust_covariance.py
@@ -55,9 +55,11 @@ def c_step(X, n_support, remaining_iterations=30, initial_estimates=None,
     verbose : boolean, optional
         Verbose mode.
 
-    random_state : integer or numpy.RandomState, optional
-        The random generator used. If an integer is given, it fixes the
-        seed. Defaults to the global numpy random number generator.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     cov_computation_method : callable, default empirical_covariance
         The function which will be used to compute the covariance.
@@ -214,9 +216,11 @@ def select_candidates(X, n_support, n_trials, select=1, n_iter=30,
         Maximum number of iterations for the c_step procedure.
         (2 is enough to be close to the final solution. "Never" exceeds 20).
 
-    random_state : integer or numpy.RandomState, default None
-        The random generator used. If an integer is given, it fixes the
-        seed. Defaults to the global numpy random number generator.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     cov_computation_method : callable, default empirical_covariance
         The function which will be used to compute the covariance.
@@ -311,10 +315,11 @@ def fast_mcd(X, support_fraction=None,
           value of support_fraction will be used within the algorithm:
           `[n_sample + n_features + 1] / 2`.
 
-    random_state : integer or numpy.RandomState, optional
-        The generator used to randomly subsample. If an integer is
-        given, it fixes the seed. Defaults to the global numpy random
-        number generator.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     cov_computation_method : callable, default empirical_covariance
         The function which will be used to compute the covariance.
@@ -531,9 +536,11 @@ class MinCovDet(EmpiricalCovariance):
         value of support_fraction will be used within the algorithm:
         [n_sample + n_features + 1] / 2
 
-    random_state : integer or numpy.RandomState, optional
-        The random generator used. If an integer is given, it fixes the
-        seed. Defaults to the global numpy random number generator.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     Attributes
     ----------
diff --git a/sklearn/cross_validation.py b/sklearn/cross_validation.py
index ff327a25e49249e748611677aec79da3954ab5b8..d56845637fc48b1983366f3a5b204f2d9bf96efa 100644
--- a/sklearn/cross_validation.py
+++ b/sklearn/cross_validation.py
@@ -1,4 +1,3 @@
-
 """
 The :mod:`sklearn.cross_validation` module includes utilities for cross-
 validation and performance evaluation.
@@ -297,9 +296,11 @@ class KFold(_BaseKFold):
     shuffle : boolean, optional
         Whether to shuffle the data before splitting into batches.
 
-    random_state : None, int or RandomState
-        When shuffle=True, pseudo-random number generator state used for
-        shuffling. If None, use default numpy RNG for shuffling.
+    random_state : int, RandomState instance or None, optional, default=None
+        If int, random_state is the seed used by the random number
+        generator; If RandomState instance, random_state is the random number
+        generator; If None, the random number generator is the RandomState
+        instance used by `np.random`. Used when ``shuffle`` == True.
 
     Examples
     --------
@@ -499,9 +500,11 @@ class StratifiedKFold(_BaseKFold):
         Whether to shuffle each stratification of the data before splitting
         into batches.
 
-    random_state : None, int or RandomState
-        When shuffle=True, pseudo-random number generator state used for
-        shuffling. If None, use default numpy RNG for shuffling.
+    random_state : int, RandomState instance or None, optional, default=None
+        If int, random_state is the seed used by the random number
+        generator; If RandomState instance, random_state is the random number
+        generator; If None, the random number generator is the RandomState
+        instance used by `np.random`. Used when ``shuffle`` == True.
 
     Examples
     --------
@@ -822,8 +825,11 @@ class ShuffleSplit(BaseShuffleSplit):
         int, represents the absolute number of train samples. If None,
         the value is automatically set to the complement of the test size.
 
-    random_state : int or RandomState
-        Pseudo-random number generator state used for random sampling.
+    random_state : int, RandomState instance or None, optional (default None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     Examples
     --------
@@ -1031,8 +1037,11 @@ class StratifiedShuffleSplit(BaseShuffleSplit):
         int, represents the absolute number of train samples. If None,
         the value is automatically set to the complement of the test size.
 
-    random_state : int or RandomState
-        Pseudo-random number generator state used for random sampling.
+    random_state : int, RandomState instance or None, optional (default None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     Examples
     --------
@@ -1225,8 +1234,11 @@ class LabelShuffleSplit(ShuffleSplit):
         int, represents the absolute number of train labels. If None,
         the value is automatically set to the complement of the test size.
 
-    random_state : int or RandomState
-        Pseudo-random number generator state used for random sampling.
+    random_state : int, RandomState instance or None, optional (default None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     """
     def __init__(self, labels, n_iter=5, test_size=0.2, train_size=None,
@@ -1889,9 +1901,11 @@ def permutation_test_score(estimator, X, y, cv=None,
         Labels constrain the permutation among groups of samples with
         a same label.
 
-    random_state : RandomState or an int seed (0 by default)
-        A random number generator instance to define the state of the
-        random permutations generator.
+    random_state : int, RandomState instance or None, optional (default=0)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     verbose : integer, optional
         The verbosity level.
@@ -1977,8 +1991,11 @@ def train_test_split(*arrays, **options):
         int, represents the absolute number of train samples. If None,
         the value is automatically set to the complement of the test size.
 
-    random_state : int or RandomState
-        Pseudo-random number generator state used for random sampling.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     stratify : array-like or None (default is None)
         If not None, data is split in a stratified fashion, using this as
diff --git a/sklearn/datasets/olivetti_faces.py b/sklearn/datasets/olivetti_faces.py
index 7dfd4dec1624792ccb1fdb8b38225b11bc95d8f9..ac80d49e937d299e43a70de682355e838d88ebd7 100644
--- a/sklearn/datasets/olivetti_faces.py
+++ b/sklearn/datasets/olivetti_faces.py
@@ -71,9 +71,11 @@ def fetch_olivetti_faces(data_home=None, shuffle=False, random_state=0,
         If False, raise a IOError if the data is not locally available
         instead of trying to download the data from the source site.
 
-    random_state : optional, integer or RandomState object
-        The seed or the random number generator used to shuffle the
-        data.
+    random_state : int, RandomState instance or None, optional (default=0)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     Returns
     -------
diff --git a/sklearn/datasets/samples_generator.py b/sklearn/datasets/samples_generator.py
index e7f61b3227331b396bafb13ffeddd3cad5960673..7a4543aa2068a4f99fc1864ad7d34b86f95bfc2c 100644
--- a/sklearn/datasets/samples_generator.py
+++ b/sklearn/datasets/samples_generator.py
@@ -1059,8 +1059,11 @@ def make_sparse_coded_signal(n_samples, n_components, n_features,
     n_nonzero_coefs : int
         number of active (non-zero) coefficients in each sample
 
-    random_state : int or RandomState instance, optional (default=None)
-        seed used by the pseudo random number generator
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     Returns
     -------
diff --git a/sklearn/decomposition/dict_learning.py b/sklearn/decomposition/dict_learning.py
index baf79544dd172e2f863bde10ae6ca0c027b070d2..154987a6279c4afa874ed418391a63a723e34b15 100644
--- a/sklearn/decomposition/dict_learning.py
+++ b/sklearn/decomposition/dict_learning.py
@@ -328,8 +328,11 @@ def _update_dict(dictionary, Y, code, verbose=False, return_r2=False,
         Whether to compute and return the residual sum of squares corresponding
         to the computed solution.
 
-    random_state : int or RandomState
-        Pseudo number generator state used for random sampling.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     Returns
     -------
@@ -434,8 +437,11 @@ def dict_learning(X, n_components, alpha, max_iter=100, tol=1e-8,
     verbose :
         Degree of output the procedure will print.
 
-    random_state : int or RandomState
-        Pseudo number generator state used for random sampling.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     return_n_iter : bool
         Whether or not to return the number of iterations.
@@ -616,8 +622,11 @@ def dict_learning_online(X, n_components=2, alpha=1, n_iter=100,
         Number of previous iterations completed on the dictionary used for
         initialization.
 
-    random_state : int or RandomState
-        Pseudo number generator state used for random sampling.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     return_inner_stats : boolean, optional
         Return the inner statistics A (dictionary covariance) and B
@@ -1000,8 +1009,11 @@ class DictionaryLearning(BaseEstimator, SparseCodingMixin):
     verbose :
         degree of verbosity of the printed output
 
-    random_state : int or RandomState
-        Pseudo number generator state used for random sampling.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     Attributes
     ----------
@@ -1160,8 +1172,11 @@ class MiniBatchDictionaryLearning(BaseEstimator, SparseCodingMixin):
     shuffle : bool,
         whether to shuffle the samples before forming batches
 
-    random_state : int or RandomState
-        Pseudo number generator state used for random sampling.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     Attributes
     ----------
diff --git a/sklearn/decomposition/factor_analysis.py b/sklearn/decomposition/factor_analysis.py
index 16e198164a5cd4037aa0b439fd28d9de3ea22ff7..3326ac197b3af9362b538914a94cb22d0c7ba02f 100644
--- a/sklearn/decomposition/factor_analysis.py
+++ b/sklearn/decomposition/factor_analysis.py
@@ -88,9 +88,11 @@ class FactorAnalysis(BaseEstimator, TransformerMixin):
         Number of iterations for the power method. 3 by default. Only used
         if ``svd_method`` equals 'randomized'
 
-    random_state : int or RandomState
-        Pseudo number generator state used for random sampling. Only used
-        if ``svd_method`` equals 'randomized'
+    random_state : int, RandomState instance or None, optional (default=0)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`. Only used when ``svd_method`` equals 'randomized'.
 
     Attributes
     ----------
diff --git a/sklearn/decomposition/fastica_.py b/sklearn/decomposition/fastica_.py
index fbbbbec1b713d2e6e4b8cf430f99b9602a16317e..3cca0b7d6e89ce9b374f2d090554f1e52128bdfd 100644
--- a/sklearn/decomposition/fastica_.py
+++ b/sklearn/decomposition/fastica_.py
@@ -199,8 +199,11 @@ def fastica(X, n_components=None, algorithm="parallel", whiten=True,
         Initial un-mixing array of dimension (n.comp,n.comp).
         If None (default) then an array of normal r.v.'s is used.
 
-    random_state : int or RandomState
-        Pseudo number generator state used for random sampling.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     return_X_mean : bool, optional
         If True, X_mean is returned too.
@@ -415,8 +418,11 @@ class FastICA(BaseEstimator, TransformerMixin):
     w_init : None of an (n_components, n_components) ndarray
         The mixing matrix to be used to initialize the algorithm.
 
-    random_state : int or RandomState
-        Pseudo number generator state used for random sampling.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     Attributes
     ----------
diff --git a/sklearn/decomposition/kernel_pca.py b/sklearn/decomposition/kernel_pca.py
index 1fb6b55f43aaaad5ae9482d9507ee034b2edcd43..2bd9a5bbddd88a2cb0724e9b66bf65cd79935895 100644
--- a/sklearn/decomposition/kernel_pca.py
+++ b/sklearn/decomposition/kernel_pca.py
@@ -74,9 +74,11 @@ class KernelPCA(BaseEstimator, TransformerMixin):
         When n_components is None, this parameter is ignored and components
         with zero eigenvalues are removed regardless.
 
-    random_state : int seed, RandomState instance, or None, default=None
-        A pseudo random number generator used for the initialization of the
-        residuals when eigen_solver == 'arpack'.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`. Used when ``eigen_solver`` == 'arpack'.
 
         .. versionadded:: 0.18
 
diff --git a/sklearn/decomposition/nmf.py b/sklearn/decomposition/nmf.py
index 63026e3ad43bdb224aad1138c4ae664578db4149..522bf150aa253473fa0c2ca3bfdcf3d8d2d3bb41 100644
--- a/sklearn/decomposition/nmf.py
+++ b/sklearn/decomposition/nmf.py
@@ -268,9 +268,11 @@ def _initialize_nmf(X, n_components, init=None, eps=1e-6,
     eps : float
         Truncate all values less then this in output to zero.
 
-    random_state : int seed, RandomState instance, or None (default)
-        Random number generator seed control, used in 'nndsvdar' and
-        'random' modes.
+    random_state : int, RandomState instance or None, optional, default: None
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`. Used when ``random`` == 'nndsvdar' or 'random'.
 
     Returns
     -------
@@ -445,8 +447,11 @@ def _fit_coordinate_descent(X, W, H, tol=1e-4, max_iter=200, l1_reg_W=0,
     shuffle : boolean, default: False
         If true, randomize the order of coordinates in the CD solver.
 
-    random_state : integer seed, RandomState instance, or None (default)
-        Random number generator seed control.
+    random_state : int, RandomState instance or None, optional, default: None
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     Returns
     -------
@@ -910,8 +915,11 @@ def non_negative_factorization(X, W=None, H=None, n_components=None,
         Select whether the regularization affects the components (H), the
         transformation (W), both or none of them.
 
-    random_state : integer seed, RandomState instance, or None (default)
-        Random number generator seed control.
+    random_state : int, RandomState instance or None, optional, default: None
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     verbose : integer, default: 0
         The verbosity level.
@@ -1097,8 +1105,11 @@ class NMF(BaseEstimator, TransformerMixin):
     max_iter : integer, default: 200
         Maximum number of iterations before timing out.
 
-    random_state : integer seed, RandomState instance, or None (default)
-        Random number generator seed control.
+    random_state : int, RandomState instance or None, optional, default: None
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     alpha : double, default: 0.
         Constant that multiplies the regularization terms. Set it to zero to
diff --git a/sklearn/decomposition/online_lda.py b/sklearn/decomposition/online_lda.py
index 8e0c5bfe6b4151a7eef7532942df52c8a9e5b8ce..d24743b3e78d5c017081ced8a2683c7f6d1981d5 100644
--- a/sklearn/decomposition/online_lda.py
+++ b/sklearn/decomposition/online_lda.py
@@ -219,8 +219,11 @@ class LatentDirichletAllocation(BaseEstimator, TransformerMixin):
     verbose : int, optional (default=0)
         Verbosity level.
 
-    random_state : int or RandomState instance or None, optional (default=None)
-        Pseudo-random number generator seed control.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     Attributes
     ----------
diff --git a/sklearn/decomposition/pca.py b/sklearn/decomposition/pca.py
index f7cb01a422645f65ee7d02a5e3e176d835bc523b..eb11d9b0321066c398c722894be678cd7f1a0677 100644
--- a/sklearn/decomposition/pca.py
+++ b/sklearn/decomposition/pca.py
@@ -183,9 +183,11 @@ class PCA(_BasePCA):
 
         .. versionadded:: 0.18.0
 
-    random_state : int or RandomState instance or None (default None)
-        Pseudo Random Number generator seed control. If None, use the
-        numpy.random singleton. Used by svd_solver == 'arpack' or 'randomized'.
+    random_state : int, RandomState instance or None, optional (default None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`. Used when ``svd_solver`` == 'arpack' or 'randomized'.
 
         .. versionadded:: 0.18.0
 
@@ -601,9 +603,11 @@ class RandomizedPCA(BaseEstimator, TransformerMixin):
         improve the predictive accuracy of the downstream estimators by
         making their data respect some hard-wired assumptions.
 
-    random_state : int or RandomState instance or None (default)
-        Pseudo Random Number generator seed control. If None, use the
-        numpy.random singleton.
+    random_state : int, RandomState instance or None, optional, default=None
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     Attributes
     ----------
diff --git a/sklearn/decomposition/sparse_pca.py b/sklearn/decomposition/sparse_pca.py
index e6fde97ccb9de18c38bb6f9543479a37dfb11ca9..23d1163fdc881032ef5b93e7da9706c12dca6cac 100644
--- a/sklearn/decomposition/sparse_pca.py
+++ b/sklearn/decomposition/sparse_pca.py
@@ -60,8 +60,11 @@ class SparsePCA(BaseEstimator, TransformerMixin):
     verbose :
         Degree of verbosity of the printed output.
 
-    random_state : int or RandomState
-        Pseudo number generator state used for random sampling.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     Attributes
     ----------
@@ -228,8 +231,11 @@ class MiniBatchSparsePCA(SparsePCA):
         Lasso solution (linear_model.Lasso). Lars will be faster if
         the estimated components are sparse.
 
-    random_state : int or RandomState
-        Pseudo number generator state used for random sampling.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     Attributes
     ----------
diff --git a/sklearn/decomposition/truncated_svd.py b/sklearn/decomposition/truncated_svd.py
index 5d029d1205bd0cdd3dc8cdc7b19a141a58ab951f..7ab20926f9589a7c48313afba32594fa41e6f1e9 100644
--- a/sklearn/decomposition/truncated_svd.py
+++ b/sklearn/decomposition/truncated_svd.py
@@ -59,9 +59,11 @@ class TruncatedSVD(BaseEstimator, TransformerMixin):
         The default is larger than the default in `randomized_svd` to handle
         sparse matrices that may have large slowly decaying spectrum.
 
-    random_state : int or RandomState, optional
-        (Seed for) pseudo-random number generator. If not given, the
-        numpy.random singleton is used.
+    random_state : int, RandomState instance or None, optional, default = None
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     tol : float, optional
         Tolerance for ARPACK. 0 means machine precision. Ignored by randomized
diff --git a/sklearn/dummy.py b/sklearn/dummy.py
index 84d42e7177a0ac0926419bd7dfaa38e9537ed232..90a43791c81b61930f80f339d63f551c26609907 100644
--- a/sklearn/dummy.py
+++ b/sklearn/dummy.py
@@ -47,8 +47,11 @@ class DummyClassifier(BaseEstimator, ClassifierMixin):
              Dummy Classifier now supports prior fitting strategy using
              parameter *prior*.
 
-    random_state : int seed, RandomState instance, or None (default)
-        The seed of the pseudo random number generator to use.
+    random_state : int, RandomState instance or None, optional, default=None
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     constant : int or str or array of shape = [n_outputs]
         The explicit constant as predicted by the "constant" strategy. This
diff --git a/sklearn/ensemble/base.py b/sklearn/ensemble/base.py
index 165124d62428adf917e323ea3a201cc4e2619589..5e9d6e2e1fc3c678cf2522baa9d58b41de001042 100644
--- a/sklearn/ensemble/base.py
+++ b/sklearn/ensemble/base.py
@@ -29,8 +29,11 @@ def _set_random_states(estimator, random_state=None):
         Estimator with potential randomness managed by random_state
         parameters.
 
-    random_state : numpy.RandomState or int, optional
-        Random state used to generate integer values.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     Notes
     -----
diff --git a/sklearn/ensemble/gradient_boosting.py b/sklearn/ensemble/gradient_boosting.py
index f5eb1001b2bf51cd862721a7c3b9f0fa8de69713..2c18d338dc71531c5b78d0453da91a71a8e28de0 100644
--- a/sklearn/ensemble/gradient_boosting.py
+++ b/sklearn/ensemble/gradient_boosting.py
@@ -1767,8 +1767,7 @@ class GradientBoostingRegressor(BaseGradientBoosting, RegressorMixin):
     warm_start : bool, default: False
         When set to ``True``, reuse the solution of the previous call to fit
         and add more estimators to the ensemble, otherwise, just erase the
-        p
-revious solution.
+        previous solution.
 
     random_state : int, RandomState instance or None, optional (default=None)
         If int, random_state is the seed used by the random number generator;
diff --git a/sklearn/feature_extraction/image.py b/sklearn/feature_extraction/image.py
index 694c624f11110b912989dfab68469062560da4db..708424cb3f84367fbc8c9178b8734fcc722e7f8e 100644
--- a/sklearn/feature_extraction/image.py
+++ b/sklearn/feature_extraction/image.py
@@ -319,9 +319,12 @@ def extract_patches_2d(image, patch_size, max_patches=None, random_state=None):
         between 0 and 1, it is taken to be a proportion of the total number
         of patches.
 
-    random_state : int or RandomState
+    random_state : int, RandomState instance or None, optional (default=None)
         Pseudo number generator state used for random sampling to use if
-        `max_patches` is not None.
+        `max_patches` is not None.  If int, random_state is the seed used by
+        the random number generator; If RandomState instance, random_state is
+        the random number generator; If None, the random number generator is
+        the RandomState instance used by `np.random`.
 
     Returns
     -------
@@ -450,8 +453,11 @@ class PatchExtractor(BaseEstimator):
         float in (0, 1), it is taken to mean a proportion of the total number
         of patches.
 
-    random_state : int or RandomState
-        Pseudo number generator state used for random sampling.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     """
     def __init__(self, patch_size=None, max_patches=None, random_state=None):
diff --git a/sklearn/feature_selection/mutual_info_.py b/sklearn/feature_selection/mutual_info_.py
index b72e884704c5f7894bfab9ec1fddd8b2541b2131..0637f784c5f95c50cd8302ba696dda36bceccb22 100644
--- a/sklearn/feature_selection/mutual_info_.py
+++ b/sklearn/feature_selection/mutual_info_.py
@@ -224,9 +224,13 @@ def _estimate_mi(X, y, discrete_features='auto', discrete_target=False,
         Whether to make a copy of the given data. If set to False, the initial
         data will be overwritten.
 
-    random_state : int seed, RandomState instance or None, default None
+    random_state : int, RandomState instance or None, optional, default None
         The seed of the pseudo random number generator for adding small noise
-        to continuous variables in order to remove repeated values.
+        to continuous variables in order to remove repeated values.  If int,
+        random_state is the seed used by the random number generator; If
+        RandomState instance, random_state is the random number generator; If
+        None, the random number generator is the RandomState instance used by
+        `np.random`.
 
     Returns
     -------
@@ -327,9 +331,13 @@ def mutual_info_regression(X, y, discrete_features='auto', n_neighbors=3,
         Whether to make a copy of the given data. If set to False, the initial
         data will be overwritten.
 
-    random_state : int seed, RandomState instance or None, default None
+    random_state : int, RandomState instance or None, optional, default None
         The seed of the pseudo random number generator for adding small noise
         to continuous variables in order to remove repeated values.
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     Returns
     -------
@@ -402,9 +410,13 @@ def mutual_info_classif(X, y, discrete_features='auto', n_neighbors=3,
         Whether to make a copy of the given data. If set to False, the initial
         data will be overwritten.
 
-    random_state : int seed, RandomState instance or None, default None
+    random_state : int, RandomState instance or None, optional, default None
         The seed of the pseudo random number generator for adding small noise
-        to continuous variables in order to remove repeated values.
+        to continuous variables in order to remove repeated values.  If int,
+        random_state is the seed used by the random number generator; If
+        RandomState instance, random_state is the random number generator; If
+        None, the random number generator is the RandomState instance used by
+        `np.random`.
 
     Returns
     -------
diff --git a/sklearn/gaussian_process/gaussian_process.py b/sklearn/gaussian_process/gaussian_process.py
index 0d1b6d4fffe7bf354ca643c83d9b50419546527a..7adac552a5c1ed0e4c6fd39bcbff8cb8f841bd64 100644
--- a/sklearn/gaussian_process/gaussian_process.py
+++ b/sklearn/gaussian_process/gaussian_process.py
@@ -169,11 +169,12 @@ class GaussianProcess(BaseEstimator, RegressorMixin):
         exponential distribution (log-uniform on [thetaL, thetaU]).
         Default does not use random starting point (random_start = 1).
 
-    random_state : integer or numpy.RandomState, optional
+    random_state : int, RandomState instance or None, optional (default=None)
         The generator used to shuffle the sequence of coordinates of theta in
-        the Welch optimizer. If an integer is given, it fixes the seed.
-        Defaults to the global numpy random number generator.
-
+        the Welch optimizer. If int, random_state is the seed used by the
+        random number generator; If RandomState instance, random_state is the
+        random number generator; If None, the random number generator is the
+        RandomState instance used by `np.random`.
 
     Attributes
     ----------
diff --git a/sklearn/gaussian_process/gpc.py b/sklearn/gaussian_process/gpc.py
index bbb1feda98e0711e4fa3179eff3981c2cedde4f5..6f491b376e1dcc21979a3b391f256ceb8de9b1e8 100644
--- a/sklearn/gaussian_process/gpc.py
+++ b/sklearn/gaussian_process/gpc.py
@@ -106,10 +106,11 @@ class _BinaryGaussianProcessClassifierLaplace(BaseEstimator):
         which might cause predictions to change if the data is modified
         externally.
 
-    random_state : integer or numpy.RandomState, optional
-        The generator used to initialize the centers. If an integer is
-        given, it fixes the seed. Defaults to the global numpy random
-        number generator.
+    random_state : int, RandomState instance or None, optional (default: None)
+        The generator used to initialize the centers. If int, random_state is
+        the seed used by the random number generator; If RandomState instance,
+        random_state is the random number generator; If None, the random number
+        generator is the RandomState instance used by `np.random`.
 
     Attributes
     ----------
@@ -510,10 +511,12 @@ class GaussianProcessClassifier(BaseEstimator, ClassifierMixin):
         which might cause predictions to change if the data is modified
         externally.
 
-    random_state : integer or numpy.RandomState, optional
-        The generator used to initialize the centers. If an integer is
-        given, it fixes the seed. Defaults to the global numpy random
-        number generator.
+    random_state : int, RandomState instance or None, optional (default: None)
+        The generator used to initialize the centers.
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     multi_class : string, default : "one_vs_rest"
         Specifies how multi-class classification problems are handled.
diff --git a/sklearn/gaussian_process/gpr.py b/sklearn/gaussian_process/gpr.py
index cbf65a8430bc0a9f822a0286dd1b303a760f5038..4ee8e556c706de9e46660d71f2d17026340b55e0 100644
--- a/sklearn/gaussian_process/gpr.py
+++ b/sklearn/gaussian_process/gpr.py
@@ -103,10 +103,11 @@ class GaussianProcessRegressor(BaseEstimator, RegressorMixin):
         which might cause predictions to change if the data is modified
         externally.
 
-    random_state : integer or numpy.RandomState, optional
-        The generator used to initialize the centers. If an integer is
-        given, it fixes the seed. Defaults to the global numpy random
-        number generator.
+    random_state : int, RandomState instance or None, optional (default: None)
+        The generator used to initialize the centers. If int, random_state is
+        the seed used by the random number generator; If RandomState instance,
+        random_state is the random number generator; If None, the random number
+        generator is the RandomState instance used by `np.random`.
 
     Attributes
     ----------
@@ -336,8 +337,11 @@ class GaussianProcessRegressor(BaseEstimator, RegressorMixin):
         n_samples : int, default: 1
             The number of samples drawn from the Gaussian process
 
-        random_state : RandomState or an int seed (0 by default)
-            A random number generator instance
+        random_state : int, RandomState instance or None, optional (default=0)
+            If int, random_state is the seed used by the random number
+            generator; If RandomState instance, random_state is the
+            random number generator; If None, the random number
+            generator is the RandomState instance used by `np.random`.
 
         Returns
         -------
diff --git a/sklearn/grid_search.py b/sklearn/grid_search.py
index 5bdff14f83a76573945d502948bd857f7c5b2bd8..2f432362e37e44a1edd4eb36a4ff0bbb8fc04b42 100644
--- a/sklearn/grid_search.py
+++ b/sklearn/grid_search.py
@@ -200,9 +200,13 @@ class ParameterSampler(object):
     n_iter : integer
         Number of parameter settings that are produced.
 
-    random_state : int or RandomState
+    random_state : int, RandomState instance or None, optional (default=None)
         Pseudo random number generator state used for random uniform sampling
         from lists of possible values instead of scipy.stats distributions.
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     Returns
     -------
@@ -954,9 +958,13 @@ class RandomizedSearchCV(BaseSearchCV):
     verbose : integer
         Controls the verbosity: the higher, the more messages.
 
-    random_state : int or RandomState
+    random_state : int, RandomState instance or None, optional, default=None
         Pseudo random number generator state used for random uniform sampling
         from lists of possible values instead of scipy.stats distributions.
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     error_score : 'raise' (default) or numeric
         Value to assign to the score if an error occurs in estimator fitting.
diff --git a/sklearn/kernel_approximation.py b/sklearn/kernel_approximation.py
index a47016e448c82666d6392df200f5941edb10fa08..3fef755dfdf4e0399e8a2ce9cde277dc99d2868b 100644
--- a/sklearn/kernel_approximation.py
+++ b/sklearn/kernel_approximation.py
@@ -38,9 +38,11 @@ class RBFSampler(BaseEstimator, TransformerMixin):
         Number of Monte Carlo samples per original feature.
         Equals the dimensionality of the computed feature space.
 
-    random_state : {int, RandomState}, optional
+    random_state : int, RandomState instance or None, optional (default=None)
         If int, random_state is the seed used by the random number generator;
-        if RandomState instance, random_state is the random number generator.
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     Notes
     -----
@@ -124,9 +126,11 @@ class SkewedChi2Sampler(BaseEstimator, TransformerMixin):
         number of Monte Carlo samples per original feature.
         Equals the dimensionality of the computed feature space.
 
-    random_state : {int, RandomState}, optional
+    random_state : int, RandomState instance or None, optional (default=None)
         If int, random_state is the seed used by the random number generator;
-        if RandomState instance, random_state is the random number generator.
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     References
     ----------
@@ -394,10 +398,11 @@ class Nystroem(BaseEstimator, TransformerMixin):
         Additional parameters (keyword arguments) for kernel function passed
         as callable object.
 
-    random_state : {int, RandomState}, optional
+    random_state : int, RandomState instance or None, optional (default=None)
         If int, random_state is the seed used by the random number generator;
-        if RandomState instance, random_state is the random number generator.
-
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     Attributes
     ----------
diff --git a/sklearn/linear_model/coordinate_descent.py b/sklearn/linear_model/coordinate_descent.py
index 95cafb29e78e250aa89500475e40ab99d5d73304..a2eb3be475f834ce96ca094bdca5e750ba2e4470 100644
--- a/sklearn/linear_model/coordinate_descent.py
+++ b/sklearn/linear_model/coordinate_descent.py
@@ -582,9 +582,12 @@ class ElasticNet(LinearModel, RegressorMixin):
         (setting to 'random') often leads to significantly faster convergence
         especially when tol is higher than 1e-4.
 
-    random_state : int, RandomState instance, or None (default)
-        The seed of the pseudo random number generator that selects
-        a random feature to update. Useful only when selection is set to
+    random_state : int, RandomState instance or None, optional, default None
+        The seed of the pseudo random number generator that selects a random
+        feature to update.  If int, random_state is the seed used by the random
+        number generator; If RandomState instance, random_state is the random
+        number generator; If None, the random number generator is the
+        RandomState instance used by `np.random`. Used when ``selection`` ==
         'random'.
 
     Attributes
@@ -829,9 +832,12 @@ class Lasso(ElasticNet):
         (setting to 'random') often leads to significantly faster convergence
         especially when tol is higher than 1e-4.
 
-    random_state : int, RandomState instance, or None (default)
-        The seed of the pseudo random number generator that selects
-        a random feature to update. Useful only when selection is set to
+    random_state : int, RandomState instance or None, optional, default None
+        The seed of the pseudo random number generator that selects a random
+        feature to update.  If int, random_state is the seed used by the random
+        number generator; If RandomState instance, random_state is the random
+        number generator; If None, the random number generator is the
+        RandomState instance used by `np.random`. Used when ``selection`` ==
         'random'.
 
     Attributes
@@ -1266,9 +1272,12 @@ class LassoCV(LinearModelCV, RegressorMixin):
         (setting to 'random') often leads to significantly faster convergence
         especially when tol is higher than 1e-4.
 
-    random_state : int, RandomState instance, or None (default)
-        The seed of the pseudo random number generator that selects
-        a random feature to update. Useful only when selection is set to
+    random_state : int, RandomState instance or None, optional, default None
+        The seed of the pseudo random number generator that selects a random
+        feature to update.  If int, random_state is the seed used by the random
+        number generator; If RandomState instance, random_state is the random
+        number generator; If None, the random number generator is the
+        RandomState instance used by `np.random`. Used when ``selection`` ==
         'random'.
 
     fit_intercept : boolean, default True
@@ -1418,9 +1427,12 @@ class ElasticNetCV(LinearModelCV, RegressorMixin):
         (setting to 'random') often leads to significantly faster convergence
         especially when tol is higher than 1e-4.
 
-    random_state : int, RandomState instance, or None (default)
-        The seed of the pseudo random number generator that selects
-        a random feature to update. Useful only when selection is set to
+    random_state : int, RandomState instance or None, optional, default None
+        The seed of the pseudo random number generator that selects a random
+        feature to update.  If int, random_state is the seed used by the random
+        number generator; If RandomState instance, random_state is the random
+        number generator; If None, the random number generator is the
+        RandomState instance used by `np.random`. Used when ``selection`` ==
         'random'.
 
     fit_intercept : boolean
@@ -1588,9 +1600,12 @@ class MultiTaskElasticNet(Lasso):
         (setting to 'random') often leads to significantly faster convergence
         especially when tol is higher than 1e-4.
 
-    random_state : int, RandomState instance, or None (default)
-        The seed of the pseudo random number generator that selects
-        a random feature to update. Useful only when selection is set to
+    random_state : int, RandomState instance or None, optional, default None
+        The seed of the pseudo random number generator that selects a random
+        feature to update.  If int, random_state is the seed used by the random
+        number generator; If RandomState instance, random_state is the random
+        number generator; If None, the random number generator is the
+        RandomState instance used by `np.random`. Used when ``selection`` ==
         'random'.
 
     Attributes
@@ -1772,9 +1787,12 @@ class MultiTaskLasso(MultiTaskElasticNet):
         (setting to 'random') often leads to significantly faster convergence
         especially when tol is higher than 1e-4
 
-    random_state : int, RandomState instance, or None (default)
-        The seed of the pseudo random number generator that selects
-        a random feature to update. Useful only when selection is set to
+    random_state : int, RandomState instance or None, optional, default None
+        The seed of the pseudo random number generator that selects a random
+        feature to update.  If int, random_state is the seed used by the random
+        number generator; If RandomState instance, random_state is the random
+        number generator; If None, the random number generator is the
+        RandomState instance used by `np.random`. Used when ``selection`` ==
         'random'.
 
     Attributes
@@ -1925,9 +1943,12 @@ class MultiTaskElasticNetCV(LinearModelCV, RegressorMixin):
         (setting to 'random') often leads to significantly faster convergence
         especially when tol is higher than 1e-4.
 
-    random_state : int, RandomState instance, or None (default)
-        The seed of the pseudo random number generator that selects
-        a random feature to update. Useful only when selection is set to
+    random_state : int, RandomState instance or None, optional, default None
+        The seed of the pseudo random number generator that selects a random
+        feature to update.  If int, random_state is the seed used by the random
+        number generator; If RandomState instance, random_state is the random
+        number generator; If None, the random number generator is the
+        RandomState instance used by `np.random`. Used when ``selection`` ==
         'random'.
 
     Attributes
@@ -2089,10 +2110,13 @@ class MultiTaskLassoCV(LinearModelCV, RegressorMixin):
         (setting to 'random') often leads to significantly faster convergence
         especially when tol is higher than 1e-4.
 
-    random_state : int, RandomState instance, or None (default)
-        The seed of the pseudo random number generator that selects
-        a random feature to update. Useful only when selection is set to
-        'random'.
+    random_state : int, RandomState instance or None, optional, default None
+        The seed of the pseudo random number generator that selects a random
+        feature to update.  If int, random_state is the seed used by the random
+        number generator; If RandomState instance, random_state is the random
+        number generator; If None, the random number generator is the
+        RandomState instance used by `np.random`. Used when ``selection`` ==
+        'random'/
 
     Attributes
     ----------
diff --git a/sklearn/linear_model/logistic.py b/sklearn/linear_model/logistic.py
index 196d7f697d0e8a1efde71b3860555e94a0600112..fa7bc7f32e4becfa8aecee5c0d572a002b2e4791 100644
--- a/sklearn/linear_model/logistic.py
+++ b/sklearn/linear_model/logistic.py
@@ -543,9 +543,13 @@ def logistic_regression_path(X, y, pos_class=None, Cs=10, fit_intercept=True,
         the entire probability distribution. Works only for the 'lbfgs' and
         'newton-cg' solvers.
 
-    random_state : int seed, RandomState instance, or None (default)
-        The seed of the pseudo random number generator to use when
-        shuffling the data. Used only in solvers 'sag' and 'liblinear'.
+    random_state : int, RandomState instance or None, optional, default None
+        The seed of the pseudo random number generator to use when shuffling
+        the data.  If int, random_state is the seed used by the random number
+        generator; If RandomState instance, random_state is the random number
+        generator; If None, the random number generator is the RandomState
+        instance used by `np.random`. Used when ``solver`` == 'sag' or
+        'liblinear'.
 
     check_input : bool, default True
         If False, the input arrays X and y will not be checked.
@@ -860,9 +864,13 @@ def _log_reg_scoring_path(X, y, train, test, pos_class=None, Cs=10,
         the entire probability distribution. Does not work for
         liblinear solver.
 
-    random_state : int seed, RandomState instance, or None (default)
-        The seed of the pseudo random number generator to use when
-        shuffling the data. Used only in solvers 'sag' and 'liblinear'.
+    random_state : int, RandomState instance or None, optional, default None
+        The seed of the pseudo random number generator to use when shuffling
+        the data.  If int, random_state is the seed used by the random number
+        generator; If RandomState instance, random_state is the random number
+        generator; If None, the random number generator is the RandomState
+        instance used by `np.random`. Used when ``solver`` == 'sag' and
+        'liblinear'.
 
     max_squared_sum : float, default None
         Maximum squared sum of X over samples. Used only in SAG solver.
@@ -1024,9 +1032,13 @@ class LogisticRegression(BaseEstimator, LinearClassifierMixin,
         Useful only for the newton-cg, sag and lbfgs solvers.
         Maximum number of iterations taken for the solvers to converge.
 
-    random_state : int seed, RandomState instance, default: None
-        The seed of the pseudo random number generator to use when
-        shuffling the data. Used only in solvers 'sag' and 'liblinear'.
+    random_state : int, RandomState instance or None, optional, default: None
+        The seed of the pseudo random number generator to use when shuffling
+        the data.  If int, random_state is the seed used by the random number
+        generator; If RandomState instance, random_state is the random number
+        generator; If None, the random number generator is the RandomState
+        instance used by `np.random`. Used when ``solver`` == 'sag' or
+        'liblinear'.
 
     solver : {'newton-cg', 'lbfgs', 'liblinear', 'sag', 'saga'},
         default: 'liblinear'
@@ -1470,9 +1482,11 @@ class LogisticRegressionCV(LogisticRegression, BaseEstimator,
         To lessen the effect of regularization on synthetic feature weight
         (and therefore on the intercept) intercept_scaling has to be increased.
 
-    random_state : int seed, RandomState instance, or None (default)
-        The seed of the pseudo random number generator to use when
-        shuffling the data.
+    random_state : int, RandomState instance or None, optional, default None
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     Attributes
     ----------
diff --git a/sklearn/linear_model/passive_aggressive.py b/sklearn/linear_model/passive_aggressive.py
index 376ca92e934225ee9e3df309abb7557cbccc0df2..941f398bd6e1380376a524d0abca988b04309ca0 100644
--- a/sklearn/linear_model/passive_aggressive.py
+++ b/sklearn/linear_model/passive_aggressive.py
@@ -28,9 +28,12 @@ class PassiveAggressiveClassifier(BaseSGDClassifier):
     shuffle : bool, default=True
         Whether or not the training data should be shuffled after each epoch.
 
-    random_state : int seed, RandomState instance, or None (default)
-        The seed of the pseudo random number generator to use when
-        shuffling the data.
+    random_state : int, RandomState instance or None, optional, default=None
+        The seed of the pseudo random number generator to use when shuffling
+        the data.  If int, random_state is the seed used by the random number
+        generator; If RandomState instance, random_state is the random number
+        generator; If None, the random number generator is the RandomState
+        instance used by `np.random`.
 
     verbose : integer, optional
         The verbosity level
@@ -204,9 +207,12 @@ class PassiveAggressiveRegressor(BaseSGDRegressor):
     shuffle : bool, default=True
         Whether or not the training data should be shuffled after each epoch.
 
-    random_state : int seed, RandomState instance, or None (default)
-        The seed of the pseudo random number generator to use when
-        shuffling the data.
+    random_state : int, RandomState instance or None, optional, default=None
+        The seed of the pseudo random number generator to use when shuffling
+        the data.  If int, random_state is the seed used by the random number
+        generator; If RandomState instance, random_state is the random number
+        generator; If None, the random number generator is the RandomState
+        instance used by `np.random`.
 
     verbose : integer, optional
         The verbosity level
diff --git a/sklearn/linear_model/perceptron.py b/sklearn/linear_model/perceptron.py
index d5971817650248b72656f017ad317b409db9db9a..0b11049fc3b39f60d680cd4ebc4fe809a304a8b6 100644
--- a/sklearn/linear_model/perceptron.py
+++ b/sklearn/linear_model/perceptron.py
@@ -30,9 +30,12 @@ class Perceptron(BaseSGDClassifier):
     shuffle : bool, optional, default True
         Whether or not the training data should be shuffled after each epoch.
 
-    random_state : int seed, RandomState instance, or None (default)
-        The seed of the pseudo random number generator to use when
-        shuffling the data.
+    random_state : int, RandomState instance or None, optional, default None
+        The seed of the pseudo random number generator to use when shuffling
+        the data.  If int, random_state is the seed used by the random number
+        generator; If RandomState instance, random_state is the random number
+        generator; If None, the random number generator is the RandomState
+        instance used by `np.random`.
 
     verbose : integer, optional
         The verbosity level
diff --git a/sklearn/linear_model/randomized_l1.py b/sklearn/linear_model/randomized_l1.py
index 8f8f5c12efe87756f71a13196fba5e53a8cee883..6ebf95d2533ff85dba5ac0478289c09051a6efae 100644
--- a/sklearn/linear_model/randomized_l1.py
+++ b/sklearn/linear_model/randomized_l1.py
@@ -575,8 +575,11 @@ def lasso_stability_path(X, y, scaling=0.5, random_state=None,
         The alpha parameter in the stability selection article used to
         randomly scale the features. Should be between 0 and 1.
 
-    random_state : integer or numpy.random.RandomState, optional
-        The generator used to randomize the design.
+    random_state : int, RandomState instance or None, optional, default=None
+        The generator used to randomize the design.  If int, random_state is
+        the seed used by the random number generator; If RandomState instance,
+        random_state is the random number generator; If None, the random number
+        generator is the RandomState instance used by `np.random`.
 
     n_resampling : int, optional, default=200
         Number of randomized models.
diff --git a/sklearn/linear_model/ransac.py b/sklearn/linear_model/ransac.py
index e4e391cb101c306e2baafd6f797d02d0646393c3..ec23d9a16d4b166a83f0a5b5df6b87c9c527ec90 100644
--- a/sklearn/linear_model/ransac.py
+++ b/sklearn/linear_model/ransac.py
@@ -158,10 +158,11 @@ class RANSACRegressor(BaseEstimator, MetaEstimatorMixin, RegressorMixin):
         If the loss on a sample is greater than the ``residual_threshold``, then
         this sample is classified as an outlier.
 
-    random_state : integer or numpy.RandomState, optional
-        The generator used to initialize the centers. If an integer is
-        given, it fixes the seed. Defaults to the global numpy random
-        number generator.
+    random_state : int, RandomState instance or None, optional, default None
+        The generator used to initialize the centers.  If int, random_state is
+        the seed used by the random number generator; If RandomState instance,
+        random_state is the random number generator; If None, the random number
+        generator is the RandomState instance used by `np.random`.
 
     Attributes
     ----------
diff --git a/sklearn/linear_model/ridge.py b/sklearn/linear_model/ridge.py
index 9715e2aaef107884260278251f703819b97b1316..398016b886bdc13ac4a2cf68f229e2ee61fc342f 100644
--- a/sklearn/linear_model/ridge.py
+++ b/sklearn/linear_model/ridge.py
@@ -275,9 +275,12 @@ def ridge_regression(X, y, alpha, sample_weight=None, solver='auto',
         Verbosity level. Setting verbose > 0 will display additional
         information depending on the solver used.
 
-    random_state : int seed, RandomState instance, or None (default)
-        The seed of the pseudo random number generator to use when
-        shuffling the data. Used only in 'sag' solver.
+    random_state : int, RandomState instance or None, optional, default None
+        The seed of the pseudo random number generator to use when shuffling
+        the data.  If int, random_state is the seed used by the random number
+        generator; If RandomState instance, random_state is the random number
+        generator; If None, the random number generator is the RandomState
+        instance used by `np.random`. Used when ``solver`` == 'sag'.
 
     return_n_iter : boolean, default False
         If True, the method also returns `n_iter`, the actual number of
@@ -580,9 +583,12 @@ class Ridge(_BaseRidge, RegressorMixin):
     tol : float
         Precision of the solution.
 
-    random_state : int seed, RandomState instance, or None (default)
-        The seed of the pseudo random number generator to use when
-        shuffling the data. Used only in 'sag' solver.
+    random_state : int, RandomState instance or None, optional, default None
+        The seed of the pseudo random number generator to use when shuffling
+        the data.  If int, random_state is the seed used by the random number
+        generator; If RandomState instance, random_state is the random number
+        generator; If None, the random number generator is the RandomState
+        instance used by `np.random`. Used when ``solver`` == 'sag'.
 
         .. versionadded:: 0.17
            *random_state* to support Stochastic Average Gradient.
@@ -728,9 +734,12 @@ class RidgeClassifier(LinearClassifierMixin, _BaseRidge):
     tol : float
         Precision of the solution.
 
-    random_state : int seed, RandomState instance, or None (default)
-        The seed of the pseudo random number generator to use when
-        shuffling the data. Used in 'sag' solver.
+    random_state : int, RandomState instance or None, optional, default None
+        The seed of the pseudo random number generator to use when shuffling
+        the data.  If int, random_state is the seed used by the random number
+        generator; If RandomState instance, random_state is the random number
+        generator; If None, the random number generator is the RandomState
+        instance used by `np.random`. Used when ``solver`` == 'sag'.
 
     Attributes
     ----------
diff --git a/sklearn/linear_model/sag.py b/sklearn/linear_model/sag.py
index 61bda1b66f4172ee141b26af89ce56f949c8de8a..9bf807a18238cfc73ccbc2a8d2608b9360a02861 100644
--- a/sklearn/linear_model/sag.py
+++ b/sklearn/linear_model/sag.py
@@ -144,9 +144,12 @@ def sag_solver(X, y, sample_weight=None, loss='log', alpha=1., beta=0.,
     verbose : integer, optional
         The verbosity level.
 
-    random_state : int seed, RandomState instance, or None (default)
-        The seed of the pseudo random number generator to use when
-        shuffling the data.
+    random_state : int, RandomState instance or None, optional, default None
+        The seed of the pseudo random number generator to use when shuffling
+        the data.  If int, random_state is the seed used by the random number
+        generator; If RandomState instance, random_state is the random number
+        generator; If None, the random number generator is the RandomState
+        instance used by `np.random`.
 
     check_input : bool, default True
         If False, the input arrays X and y will not be checked.
diff --git a/sklearn/linear_model/stochastic_gradient.py b/sklearn/linear_model/stochastic_gradient.py
index c234b8eb94f0f17e85d8efce3b60fc542ceb8118..b3c61408470cc35c515415dfa51e97e758b3c153 100644
--- a/sklearn/linear_model/stochastic_gradient.py
+++ b/sklearn/linear_model/stochastic_gradient.py
@@ -608,9 +608,12 @@ class SGDClassifier(BaseSGDClassifier):
         Whether or not the training data should be shuffled after each epoch.
         Defaults to True.
 
-    random_state : int seed, RandomState instance, or None (default)
-        The seed of the pseudo random number generator to use when
-        shuffling the data.
+    random_state : int, RandomState instance or None, optional (default=None)
+        The seed of the pseudo random number generator to use when shuffling
+        the data.  If int, random_state is the seed used by the random number
+        generator; If RandomState instance, random_state is the random number
+        generator; If None, the random number generator is the RandomState
+        instance used by `np.random`.
 
     verbose : integer, optional
         The verbosity level
@@ -1134,9 +1137,12 @@ class SGDRegressor(BaseSGDRegressor):
         Whether or not the training data should be shuffled after each epoch.
         Defaults to True.
 
-    random_state : int seed, RandomState instance, or None (default)
-        The seed of the pseudo random number generator to use when
-        shuffling the data.
+    random_state : int, RandomState instance or None, optional (default=None)
+        The seed of the pseudo random number generator to use when shuffling
+        the data.  If int, random_state is the seed used by the random number
+        generator; If RandomState instance, random_state is the random number
+        generator; If None, the random number generator is the RandomState
+        instance used by `np.random`.
 
     verbose : integer, optional
         The verbosity level.
diff --git a/sklearn/linear_model/theil_sen.py b/sklearn/linear_model/theil_sen.py
index 23b3c106c9bd7135d2599269fdd6af7d45c478d0..b51f7d6dd3c32cf457068decabb7d0a0af80979b 100644
--- a/sklearn/linear_model/theil_sen.py
+++ b/sklearn/linear_model/theil_sen.py
@@ -243,9 +243,12 @@ class TheilSenRegressor(LinearModel, RegressorMixin):
     tol : float, optional, default 1.e-3
         Tolerance when calculating spatial median.
 
-    random_state : RandomState or an int seed, optional, default None
-        A random number generator instance to define the state of the
-        random permutations generator.
+    random_state : int, RandomState instance or None, optional, default None
+        A random number generator instance to define the state of the random
+        permutations generator.  If int, random_state is the seed used by the
+        random number generator; If RandomState instance, random_state is the
+        random number generator; If None, the random number generator is the
+        RandomState instance used by `np.random`.
 
     n_jobs : integer, optional, default 1
         Number of CPUs to use during the cross validation. If ``-1``, use
diff --git a/sklearn/manifold/locally_linear.py b/sklearn/manifold/locally_linear.py
index 367710edc667eac9e8d01f3ef75aee88a79f5d56..82c4b61254361b157d24a245721bed3c231006c0 100644
--- a/sklearn/manifold/locally_linear.py
+++ b/sklearn/manifold/locally_linear.py
@@ -140,9 +140,11 @@ def null_space(M, k, k_skip=1, eigen_solver='arpack', tol=1E-6, max_iter=100,
     max_iter : maximum number of iterations for 'arpack' method
         not used if eigen_solver=='dense'
 
-    random_state : numpy.RandomState or int, optional
-        The generator or seed used to determine the starting vector for arpack
-        iterations.  Defaults to numpy.random.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`. Used when ``solver`` == 'arpack'.
 
     """
     if eigen_solver == 'auto':
@@ -245,9 +247,11 @@ def locally_linear_embedding(
         Tolerance for modified LLE method.
         Only used if method == 'modified'
 
-    random_state : numpy.RandomState or int, optional
-        The generator or seed used to determine the starting vector for arpack
-        iterations.  Defaults to numpy.random.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`. Used when ``solver`` == 'arpack'.
 
     n_jobs : int, optional (default = 1)
         The number of parallel jobs to run for neighbors search.
@@ -568,9 +572,11 @@ class LocallyLinearEmbedding(BaseEstimator, TransformerMixin):
         algorithm to use for nearest neighbors search,
         passed to neighbors.NearestNeighbors instance
 
-    random_state : numpy.RandomState or int, optional
-        The generator or seed used to determine the starting vector for arpack
-        iterations.  Defaults to numpy.random.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`. Used when ``eigen_solver`` == 'arpack'.
 
     n_jobs : int, optional (default = 1)
         The number of parallel jobs to run.
diff --git a/sklearn/manifold/mds.py b/sklearn/manifold/mds.py
index b2fe62040bb9362d5a4018ab7079cc5739ee5a9c..5f7327ef4dc84d408f985b713383ec674dc17d78 100644
--- a/sklearn/manifold/mds.py
+++ b/sklearn/manifold/mds.py
@@ -19,8 +19,7 @@ from ..isotonic import IsotonicRegression
 
 def _smacof_single(dissimilarities, metric=True, n_components=2, init=None,
                    max_iter=300, verbose=0, eps=1e-3, random_state=None):
-    """
-    Computes multidimensional scaling using SMACOF algorithm
+    """Computes multidimensional scaling using SMACOF algorithm
 
     Parameters
     ----------
@@ -50,10 +49,11 @@ def _smacof_single(dissimilarities, metric=True, n_components=2, init=None,
         Relative tolerance with respect to stress at which to declare
         convergence.
 
-    random_state : integer or numpy.RandomState, optional
-        The generator used to initialize the centers. If an integer is
-        given, it fixes the seed. Defaults to the global numpy random
-        number generator.
+    random_state : int, RandomState instance or None, optional, default: None
+        The generator used to initialize the centers.  If int, random_state is
+        the seed used by the random number generator; If RandomState instance,
+        random_state is the random number generator; If None, the random number
+        generator is the RandomState instance used by `np.random`.
 
     Returns
     -------
@@ -134,8 +134,7 @@ def _smacof_single(dissimilarities, metric=True, n_components=2, init=None,
 def smacof(dissimilarities, metric=True, n_components=2, init=None, n_init=8,
            n_jobs=1, max_iter=300, verbose=0, eps=1e-3, random_state=None,
            return_n_iter=False):
-    """
-    Computes multidimensional scaling using the SMACOF algorithm.
+    """Computes multidimensional scaling using the SMACOF algorithm.
 
     The SMACOF (Scaling by MAjorizing a COmplicated Function) algorithm is a
     multidimensional scaling algorithm which minimizes an objective function
@@ -198,10 +197,11 @@ def smacof(dissimilarities, metric=True, n_components=2, init=None, n_init=8,
         Relative tolerance with respect to stress at which to declare
         convergence.
 
-    random_state : integer or numpy.RandomState, optional, default: None
-        The generator used to initialize the centers. If an integer is given,
-        it fixes the seed. Defaults to the global numpy random number
-        generator.
+    random_state : int, RandomState instance or None, optional, default: None
+        The generator used to initialize the centers.  If int, random_state is
+        the seed used by the random number generator; If RandomState instance,
+        random_state is the random number generator; If None, the random number
+        generator is the RandomState instance used by `np.random`.
 
     return_n_iter : bool, optional, default: False
         Whether or not to return the number of iterations.
@@ -314,10 +314,11 @@ class MDS(BaseEstimator):
         (``n_cpus + 1 + n_jobs``) are used. Thus for ``n_jobs = -2``, all CPUs
         but one are used.
 
-    random_state : integer or numpy.RandomState, optional, default: None
-        The generator used to initialize the centers. If an integer is given,
-        it fixes the seed. Defaults to the global numpy random number
-        generator.
+    random_state : int, RandomState instance or None, optional, default: None
+        The generator used to initialize the centers.  If int, random_state is
+        the seed used by the random number generator; If RandomState instance,
+        random_state is the random number generator; If None, the random number
+        generator is the RandomState instance used by `np.random`.
 
     dissimilarity : 'euclidean' | 'precomputed', optional, default: 'euclidean'
         Dissimilarity measure to use:
diff --git a/sklearn/manifold/spectral_embedding_.py b/sklearn/manifold/spectral_embedding_.py
index 6250565645cd2dc813d9b8b7cab918f3b2ca4d81..31c90aa8b30aa6c6d4f298013eb1f5f4b9de7100 100644
--- a/sklearn/manifold/spectral_embedding_.py
+++ b/sklearn/manifold/spectral_embedding_.py
@@ -166,10 +166,13 @@ def spectral_embedding(adjacency, n_components=8, eigen_solver=None,
         to be installed. It can be faster on very large, sparse problems,
         but may also lead to instabilities.
 
-    random_state : int seed, RandomState instance, or None (default)
+    random_state : int, RandomState instance or None, optional, default: None
         A pseudo random number generator used for the initialization of the
-        lobpcg eigenvectors decomposition when eigen_solver == 'amg'.
-        By default, arpack is used.
+        lobpcg eigenvectors decomposition.  If int, random_state is the seed
+        used by the random number generator; If RandomState instance,
+        random_state is the random number generator; If None, the random number
+        generator is the RandomState instance used by `np.random`. Used when
+        ``solver`` == 'amg'.
 
     eigen_tol : float, optional, default=0.0
         Stopping criterion for eigendecomposition of the Laplacian matrix
@@ -345,9 +348,13 @@ class SpectralEmbedding(BaseEstimator):
         to be installed. It can be faster on very large, sparse problems,
         but may also lead to instabilities.
 
-    random_state : int seed, RandomState instance, or None, default : None
+    random_state : int, RandomState instance or None, optional, default: None
         A pseudo random number generator used for the initialization of the
-        lobpcg eigenvectors decomposition when eigen_solver == 'amg'.
+        lobpcg eigenvectors.  If int, random_state is the seed used by the
+        random number generator; If RandomState instance, random_state is the
+        random number generator; If None, the random number generator is the
+        RandomState instance used by `np.random`. Used when ``solver`` ==
+        'amg'.
 
     affinity : string or callable, default : "nearest_neighbors"
         How to construct the affinity matrix.
diff --git a/sklearn/manifold/t_sne.py b/sklearn/manifold/t_sne.py
index a124753cb04988d1a83221f15df772a2a2c048c0..83d42c444fa5c7f4d084a4de1fc4f97cf5b8e459 100644
--- a/sklearn/manifold/t_sne.py
+++ b/sklearn/manifold/t_sne.py
@@ -582,10 +582,12 @@ class TSNE(BaseEstimator):
     verbose : int, optional (default: 0)
         Verbosity level.
 
-    random_state : int or RandomState instance or None (default)
-        Pseudo Random Number generator seed control. If None, use the
-        numpy.random singleton. Note that different initializations
-        might result in different local minima of the cost function.
+    random_state : int, RandomState instance or None, optional (default: None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.  Note that different initializations might result in
+        different local minima of the cost function.
 
     method : string (default: 'barnes_hut')
         By default the gradient calculation algorithm uses Barnes-Hut
diff --git a/sklearn/metrics/cluster/unsupervised.py b/sklearn/metrics/cluster/unsupervised.py
index 606ffcddf849452f241dd09612aebe8ebeaddf0d..3be683ae08d9810b6a1ac2251087a4b76dc52a58 100644
--- a/sklearn/metrics/cluster/unsupervised.py
+++ b/sklearn/metrics/cluster/unsupervised.py
@@ -62,10 +62,12 @@ def silhouette_score(X, labels, metric='euclidean', sample_size=None,
         on a random subset of the data.
         If ``sample_size is None``, no sampling is used.
 
-    random_state : integer or numpy.RandomState, optional
-        The generator used to randomly select a subset of samples if
-        ``sample_size is not None``. If an integer is given, it fixes the seed.
-        Defaults to the global numpy random number generator.
+    random_state : int, RandomState instance or None, optional (default=None)
+        The generator used to randomly select a subset of samples.  If int,
+        random_state is the seed used by the random number generator; If
+        RandomState instance, random_state is the random number generator; If
+        None, the random number generator is the RandomState instance used by
+        `np.random`. Used when ``sample_size is not None``.
 
     **kwds : optional keyword parameters
         Any further parameters are passed directly to the distance function.
diff --git a/sklearn/mixture/bayesian_mixture.py b/sklearn/mixture/bayesian_mixture.py
index 497b339a4f80784bf86361d5c4396c5321f8cf78..24c0ae62e4efbfb040da8d52dcafe3b2fbfa546a 100644
--- a/sklearn/mixture/bayesian_mixture.py
+++ b/sklearn/mixture/bayesian_mixture.py
@@ -163,8 +163,11 @@ class BayesianGaussianMixture(BaseMixture):
                 (n_features)             if 'diag',
                 float                    if 'spherical'
 
-    random_state : RandomState or an int seed, defaults to None.
-        A random number generator instance.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     warm_start : bool, default to False.
         If 'warm_start' is True, the solution of the last fitting is used as
diff --git a/sklearn/mixture/gaussian_mixture.py b/sklearn/mixture/gaussian_mixture.py
index edbfc08c4e07d67142d4f79c8849252287dc9325..eced5407249402603308f53ffe2634e02c79d524 100644
--- a/sklearn/mixture/gaussian_mixture.py
+++ b/sklearn/mixture/gaussian_mixture.py
@@ -500,8 +500,11 @@ class GaussianMixture(BaseMixture):
             (n_components, n_features)             if 'diag',
             (n_components, n_features, n_features) if 'full'
 
-    random_state : RandomState or an int seed, defaults to None.
-        A random number generator instance.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     warm_start : bool, default to False.
         If 'warm_start' is True, the solution of the last fitting is used as
diff --git a/sklearn/mixture/gmm.py b/sklearn/mixture/gmm.py
index 024981bda8fade2624d87272751d4bc38f4f202f..5b2dece572c34aa13c618f102d366d72d001bf09 100644
--- a/sklearn/mixture/gmm.py
+++ b/sklearn/mixture/gmm.py
@@ -152,8 +152,11 @@ class _GMMBase(BaseEstimator):
         use.  Must be one of 'spherical', 'tied', 'diag', 'full'.
         Defaults to 'diag'.
 
-    random_state : RandomState or an int seed (None by default)
-        A random number generator instance
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     min_covar : float, optional
         Floor on the diagonal of the covariance matrix to prevent
diff --git a/sklearn/model_selection/_search.py b/sklearn/model_selection/_search.py
index 3b8a0ed882cf56a46057136d325b193c64713b5a..98d9e32017e46309fdcb021c96e2f8e7846309d6 100644
--- a/sklearn/model_selection/_search.py
+++ b/sklearn/model_selection/_search.py
@@ -192,9 +192,13 @@ class ParameterSampler(object):
     n_iter : integer
         Number of parameter settings that are produced.
 
-    random_state : int or RandomState
+    random_state : int, RandomState instance or None, optional (default=None)
         Pseudo random number generator state used for random uniform sampling
         from lists of possible values instead of scipy.stats distributions.
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     Returns
     -------
@@ -1054,9 +1058,13 @@ class RandomizedSearchCV(BaseSearchCV):
     verbose : integer
         Controls the verbosity: the higher, the more messages.
 
-    random_state : int or RandomState
+    random_state : int, RandomState instance or None, optional, default=None
         Pseudo random number generator state used for random uniform sampling
         from lists of possible values instead of scipy.stats distributions.
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     error_score : 'raise' (default) or numeric
         Value to assign to the score if an error occurs in estimator fitting.
diff --git a/sklearn/model_selection/_split.py b/sklearn/model_selection/_split.py
index 992c4f6d81e6a4ef4566339770d70f8e4f8e622d..0eb51be93f5bb266b4780ea14e0b5f41b8a2420d 100644
--- a/sklearn/model_selection/_split.py
+++ b/sklearn/model_selection/_split.py
@@ -364,9 +364,11 @@ class KFold(_BaseKFold):
     shuffle : boolean, optional
         Whether to shuffle the data before splitting into batches.
 
-    random_state : None, int or RandomState
-        When shuffle=True, pseudo-random number generator state used for
-        shuffling. If None, use default numpy RNG for shuffling.
+    random_state : int, RandomState instance or None, optional, default=None
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`. Used when ``shuffle`` == True.
 
     Examples
     --------
@@ -531,9 +533,11 @@ class StratifiedKFold(_BaseKFold):
         Whether to shuffle each stratification of the data before splitting
         into batches.
 
-    random_state : None, int or RandomState
-        When shuffle=True, pseudo-random number generator state used for
-        shuffling. If None, use default numpy RNG for shuffling.
+    random_state : int, RandomState instance or None, optional, default=None
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`. Used when ``shuffle`` == True.
 
     Examples
     --------
@@ -934,9 +938,11 @@ class _RepeatedSplits(with_metaclass(ABCMeta)):
     n_repeats : int, default=10
         Number of times cross-validator needs to be repeated.
 
-    random_state : None, int or RandomState, default=None
-        Random state to be used to generate random state for each
-        repetition.
+    random_state : int, RandomState instance or None, optional, default=None
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     **cvargs : additional params
         Constructor parameters for cv. Must not contain random_state
@@ -1007,9 +1013,11 @@ class RepeatedKFold(_RepeatedSplits):
     n_repeats : int, default=10
         Number of times cross-validator needs to be repeated.
 
-    random_state : None, int or RandomState, default=None
-        Random state to be used to generate random state for each
-        repetition.
+    random_state : int, RandomState instance or None, optional, default=None
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     Examples
     --------
@@ -1180,8 +1188,11 @@ class ShuffleSplit(BaseShuffleSplit):
         int, represents the absolute number of train samples. If None,
         the value is automatically set to the complement of the test size.
 
-    random_state : int or RandomState
-        Pseudo-random number generator state used for random sampling.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     Examples
     --------
@@ -1262,8 +1273,12 @@ class GroupShuffleSplit(ShuffleSplit):
         int, represents the absolute number of train groups. If None,
         the value is automatically set to the complement of the test size.
 
-    random_state : int or RandomState
-        Pseudo-random number generator state used for random sampling.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
+
     '''
 
     def __init__(self, n_splits=5, test_size=0.2, train_size=None,
@@ -1389,8 +1404,12 @@ class StratifiedShuffleSplit(BaseShuffleSplit):
         int, represents the absolute number of train samples. If None,
         the value is automatically set to the complement of the test size.
 
-    random_state : int or RandomState
-        Pseudo-random number generator state used for random sampling.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
+
 
     Examples
     --------
@@ -1795,8 +1814,11 @@ def train_test_split(*arrays, **options):
         int, represents the absolute number of train samples. If None,
         the value is automatically set to the complement of the test size.
 
-    random_state : int or RandomState
-        Pseudo-random number generator state used for random sampling.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     stratify : array-like or None (default is None)
         If not None, data is split in a stratified fashion, using this as
diff --git a/sklearn/model_selection/_validation.py b/sklearn/model_selection/_validation.py
index e65720b709555982e887f4992fdd9f7cb87b6117..e105f0d0b122f00f77597d9f5cf78c8ccdde9063 100644
--- a/sklearn/model_selection/_validation.py
+++ b/sklearn/model_selection/_validation.py
@@ -580,9 +580,11 @@ def permutation_test_score(estimator, X, y, groups=None, cv=None,
         The number of CPUs to use to do the computation. -1 means
         'all CPUs'.
 
-    random_state : RandomState or an int seed (0 by default)
-        A random number generator instance to define the state of the
-        random permutations generator.
+    random_state : int, RandomState instance or None, optional (default=0)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     verbose : integer, optional
         The verbosity level.
@@ -743,9 +745,11 @@ def learning_curve(estimator, X, y, groups=None,
         Whether to shuffle training data before taking prefixes of it
         based on``train_sizes``.
 
-    random_state : None, int or RandomState
-        When shuffle=True, pseudo-random number generator state used for
-        shuffling. If None, use default numpy RNG for shuffling.
+    random_state : int, RandomState instance or None, optional (default=None)
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`. Used when ``shuffle`` == 'True'.
 
     -------
     train_sizes_abs : array, shape = (n_unique_ticks,), dtype int
diff --git a/sklearn/multiclass.py b/sklearn/multiclass.py
index 3de5ee319c71820bc8107569527d095759e0c2df..8f9788e6a425ce504cb71ca92700cb187499eacd 100644
--- a/sklearn/multiclass.py
+++ b/sklearn/multiclass.py
@@ -640,9 +640,11 @@ class OutputCodeClassifier(BaseEstimator, ClassifierMixin, MetaEstimatorMixin):
         one-vs-the-rest. A number greater than 1 will require more classifiers
         than one-vs-the-rest.
 
-    random_state : numpy.RandomState, optional
-        The generator used to initialize the codebook. Defaults to
-        numpy.random.
+    random_state : int, RandomState instance or None, optional, default: None
+        The generator used to initialize the codebook.  If int, random_state is
+        the seed used by the random number generator; If RandomState instance,
+        random_state is the random number generator; If None, the random number
+        generator is the RandomState instance used by `np.random`.
 
     n_jobs : int, optional, default: 1
         The number of jobs to use for the computation. If -1 all CPUs are used.
diff --git a/sklearn/neighbors/kde.py b/sklearn/neighbors/kde.py
index dfb349a8dc424f227810d191df186d502698f36f..3cfdbc63042b77aad2fe06e330bd972f4637a196 100644
--- a/sklearn/neighbors/kde.py
+++ b/sklearn/neighbors/kde.py
@@ -184,8 +184,11 @@ class KernelDensity(BaseEstimator):
         n_samples : int, optional
             Number of samples to generate. Defaults to 1.
 
-        random_state : RandomState or an int seed (0 by default)
-            A random number generator instance.
+        random_state : int, RandomState instance or None. default to None
+            If int, random_state is the seed used by the random number
+            generator; If RandomState instance, random_state is the random
+            number generator; If None, the random number generator is the
+            RandomState instance used by `np.random`.
 
         Returns
         -------
diff --git a/sklearn/neural_network/multilayer_perceptron.py b/sklearn/neural_network/multilayer_perceptron.py
index 720a3fef21d84086c4f276353ec7c973aad226ff..1d329f8074c20dc12118cd39789d1b564ebeb155 100644
--- a/sklearn/neural_network/multilayer_perceptron.py
+++ b/sklearn/neural_network/multilayer_perceptron.py
@@ -755,8 +755,11 @@ class MLPClassifier(BaseMultilayerPerceptron, ClassifierMixin):
         Maximum number of iterations. The solver iterates until convergence
         (determined by 'tol') or this number of iterations.
 
-    random_state : int or RandomState, optional, default None
-        State or seed for random number generator.
+    random_state : int, RandomState instance or None, optional, default None
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     shuffle : bool, optional, default True
         Whether to shuffle samples in each iteration. Only used when
@@ -1126,8 +1129,11 @@ class MLPRegressor(BaseMultilayerPerceptron, RegressorMixin):
         Maximum number of iterations. The solver iterates until convergence
         (determined by 'tol') or this number of iterations.
 
-    random_state : int or RandomState, optional, default None
-        State or seed for random number generator.
+    random_state : int, RandomState instance or None, optional, default None
+        If int, random_state is the seed used by the random number generator;
+        If RandomState instance, random_state is the random number generator;
+        If None, the random number generator is the RandomState instance used
+        by `np.random`.
 
     shuffle : bool, optional, default True
         Whether to shuffle samples in each iteration. Only used when
diff --git a/sklearn/random_projection.py b/sklearn/random_projection.py
index 0d47b2886ff0ec5090a45af1dfbc390f471280d5..1ec4d0d21e678657a4d1c2c94d1c8756c2474fa3 100644
--- a/sklearn/random_projection.py
+++ b/sklearn/random_projection.py
@@ -154,7 +154,7 @@ def _check_input_size(n_components, n_features):
 
 
 def gaussian_random_matrix(n_components, n_features, random_state=None):
-    """ Generate a dense Gaussian random matrix.
+    """Generate a dense Gaussian random matrix.
 
     The components of the random matrix are drawn from
 
@@ -170,9 +170,12 @@ def gaussian_random_matrix(n_components, n_features, random_state=None):
     n_features : int,
         Dimensionality of the original source space.
 
-    random_state : int, RandomState instance or None (default=None)
-        Control the pseudo random number generator used to generate the
-        matrix at fit time.
+    random_state : int, RandomState instance or None, optional (default=None)
+        Control the pseudo random number generator used to generate the matrix
+        at fit time.  If int, random_state is the seed used by the random
+        number generator; If RandomState instance, random_state is the random
+        number generator; If None, the random number generator is the
+        RandomState instance used by `np.random`.
 
     Returns
     -------
@@ -226,9 +229,12 @@ def sparse_random_matrix(n_components, n_features, density='auto',
         Use density = 1 / 3.0 if you want to reproduce the results from
         Achlioptas, 2001.
 
-    random_state : integer, RandomState instance or None (default=None)
-        Control the pseudo random number generator used to generate the
-        matrix at fit time.
+    random_state : int, RandomState instance or None, optional (default=None)
+        Control the pseudo random number generator used to generate the matrix
+        at fit time.  If int, random_state is the seed used by the random
+        number generator; If RandomState instance, random_state is the random
+        number generator; If None, the random number generator is the
+        RandomState instance used by `np.random`.
 
     Returns
     -------
@@ -446,9 +452,12 @@ class GaussianRandomProjection(BaseRandomProjection):
         Smaller values lead to better embedding and higher number of
         dimensions (n_components) in the target projection space.
 
-    random_state : integer, RandomState instance or None (default=None)
-        Control the pseudo random number generator used to generate the
-        matrix at fit time.
+    random_state : int, RandomState instance or None, optional (default=None)
+        Control the pseudo random number generator used to generate the matrix
+        at fit time.  If int, random_state is the seed used by the random
+        number generator; If RandomState instance, random_state is the random
+        number generator; If None, the random number generator is the
+        RandomState instance used by `np.random`.
 
     Attributes
     ----------
@@ -552,9 +561,12 @@ class SparseRandomProjection(BaseRandomProjection):
         If False, the projected data uses a sparse representation if
         the input is sparse.
 
-    random_state : integer, RandomState instance or None (default=None)
-        Control the pseudo random number generator used to generate the
-        matrix at fit time.
+    random_state : int, RandomState instance or None, optional (default=None)
+        Control the pseudo random number generator used to generate the matrix
+        at fit time.  If int, random_state is the seed used by the random
+        number generator; If RandomState instance, random_state is the random
+        number generator; If None, the random number generator is the
+        RandomState instance used by `np.random`.
 
     Attributes
     ----------
diff --git a/sklearn/svm/base.py b/sklearn/svm/base.py
index cff4c35a58b46a588776920a343ca930b44738f6..208a69f3720c8b7ced2db8ffcafa3ba534858f18 100644
--- a/sklearn/svm/base.py
+++ b/sklearn/svm/base.py
@@ -804,9 +804,13 @@ def _fit_liblinear(X, y, C, fit_intercept, intercept_scaling, class_weight,
     tol : float
         Stopping condition.
 
-    random_state : int seed, RandomState instance, or None (default)
-        The seed of the pseudo random number generator to use when
-        shuffling the data.
+    random_state : int, RandomState instance or None, optional (default=None)
+        The seed of the pseudo random number generator to use when shuffling
+        the data.  If int, random_state is the seed used by the random number
+        generator; If RandomState instance, random_state is the random number
+        generator; If None, the random number generator is the RandomState
+        instance used by `np.random`.
+
 
     multi_class : str, {'ovr', 'crammer_singer'}
         `ovr` trains n_classes one-vs-rest classifiers, while `crammer_singer`
diff --git a/sklearn/svm/classes.py b/sklearn/svm/classes.py
index 7e920011d002db701da77efba35392ec932e2d54..2de3029cb2f26d1298d2f6bd76739ded543df10c 100644
--- a/sklearn/svm/classes.py
+++ b/sklearn/svm/classes.py
@@ -86,9 +86,12 @@ class LinearSVC(BaseEstimator, LinearClassifierMixin,
         per-process runtime setting in liblinear that, if enabled, may not work
         properly in a multithreaded context.
 
-    random_state : int seed, RandomState instance, or None (default=None)
-        The seed of the pseudo random number generator to use when
-        shuffling the data.
+    random_state : int, RandomState instance or None, optional (default=None)
+        The seed of the pseudo random number generator to use when shuffling
+        the data.  If int, random_state is the seed used by the random number
+        generator; If RandomState instance, random_state is the random number
+        generator; If None, the random number generator is the RandomState
+        instance used by `np.random`.
 
     max_iter : int, (default=1000)
         The maximum number of iterations to be run.
@@ -277,9 +280,12 @@ class LinearSVR(LinearModel, RegressorMixin):
         per-process runtime setting in liblinear that, if enabled, may not work
         properly in a multithreaded context.
 
-    random_state : int seed, RandomState instance, or None (default=None)
-        The seed of the pseudo random number generator to use when
-        shuffling the data.
+    random_state : int, RandomState instance or None, optional (default=None)
+        The seed of the pseudo random number generator to use when shuffling
+        the data.  If int, random_state is the seed used by the random number
+        generator; If RandomState instance, random_state is the random number
+        generator; If None, the random number generator is the RandomState
+        instance used by `np.random`.
 
     max_iter : int, (default=1000)
         The maximum number of iterations to be run.
@@ -468,9 +474,12 @@ class SVC(BaseSVC):
         .. versionchanged:: 0.17
            Deprecated *decision_function_shape='ovo' and None*.
 
-    random_state : int seed, RandomState instance, or None (default)
-        The seed of the pseudo random number generator to use when
-        shuffling the data for probability estimation.
+    random_state : int, RandomState instance or None, optional (default=None)
+        The seed of the pseudo random number generator to use when shuffling
+        the data.  If int, random_state is the seed used by the random number
+        generator; If RandomState instance, random_state is the random number
+        generator; If None, the random number generator is the RandomState
+        instance used by `np.random`.
 
     Attributes
     ----------
@@ -621,9 +630,12 @@ class NuSVC(BaseSVC):
         .. versionchanged:: 0.17
            Deprecated *decision_function_shape='ovo' and None*.
 
-    random_state : int seed, RandomState instance, or None (default)
-        The seed of the pseudo random number generator to use when
-        shuffling the data for probability estimation.
+    random_state : int, RandomState instance or None, optional (default=None)
+        The seed of the pseudo random number generator to use when shuffling
+        the data.  If int, random_state is the seed used by the random number
+        generator; If RandomState instance, random_state is the random number
+        generator; If None, the random number generator is the RandomState
+        instance used by `np.random`.
 
     Attributes
     ----------
@@ -972,9 +984,12 @@ class OneClassSVM(BaseLibSVM):
     max_iter : int, optional (default=-1)
         Hard limit on iterations within solver, or -1 for no limit.
 
-    random_state : int seed, RandomState instance, or None (default)
-        The seed of the pseudo random number generator to use when
-        shuffling the data for probability estimation.
+    random_state : int, RandomState instance or None, optional (default=None)
+        The seed of the pseudo random number generator to use when shuffling
+        the data.  If int, random_state is the seed used by the random number
+        generator; If RandomState instance, random_state is the random number
+        generator; If None, the random number generator is the RandomState
+        instance used by `np.random`.
 
     Attributes
     ----------
diff --git a/sklearn/utils/__init__.py b/sklearn/utils/__init__.py
index 0bc4d6de33c3f17d33e693ce970a36a0f961b237..b3e41e1c130fbc15af7d3a5009586f56866d666b 100644
--- a/sklearn/utils/__init__.py
+++ b/sklearn/utils/__init__.py
@@ -175,8 +175,12 @@ def resample(*arrays, **options):
         If replace is False it should not be larger than the length of
         arrays.
 
-    random_state : int or RandomState instance
-        Control the shuffling for reproducible behavior.
+    random_state : int, RandomState instance or None, optional (default=None)
+        The seed of the pseudo random number generator to use when shuffling
+        the data.  If int, random_state is the seed used by the random number
+        generator; If RandomState instance, random_state is the random number
+        generator; If None, the random number generator is the RandomState
+        instance used by `np.random`.
 
     Returns
     -------
@@ -271,8 +275,12 @@ def shuffle(*arrays, **options):
         Indexable data-structures can be arrays, lists, dataframes or scipy
         sparse matrices with consistent first dimension.
 
-    random_state : int or RandomState instance
-        Control the shuffling for reproducible behavior.
+    random_state : int, RandomState instance or None, optional (default=None)
+        The seed of the pseudo random number generator to use when shuffling
+        the data.  If int, random_state is the seed used by the random number
+        generator; If RandomState instance, random_state is the random number
+        generator; If None, the random number generator is the RandomState
+        instance used by `np.random`.
 
     n_samples : int, None by default
         Number of samples to generate. If left to None this is
diff --git a/sklearn/utils/extmath.py b/sklearn/utils/extmath.py
index df1f56dbcb8913e8dc9e14c404bbdf87aa6bdad7..d797950ba8efa5c1e511d942a8afb844c20a7c97 100644
--- a/sklearn/utils/extmath.py
+++ b/sklearn/utils/extmath.py
@@ -215,8 +215,12 @@ def randomized_range_finder(A, size, n_iter,
 
         .. versionadded:: 0.18
 
-    random_state : RandomState or an int seed (0 by default)
-        A random number generator instance
+    random_state : int, RandomState instance or None, optional (default=None)
+        The seed of the pseudo random number generator to use when shuffling
+        the data.  If int, random_state is the seed used by the random number
+        generator; If RandomState instance, random_state is the random number
+        generator; If None, the random number generator is the RandomState
+        instance used by `np.random`.
 
     Returns
     -------
@@ -320,8 +324,12 @@ def randomized_svd(M, n_components, n_oversamples=10, n_iter='auto',
         set to `True`, the sign ambiguity is resolved by making the largest
         loadings for each component in the left singular vectors positive.
 
-    random_state : RandomState or an int seed (0 by default)
-        A random number generator instance to make behavior
+    random_state : int, RandomState instance or None, optional (default=None)
+        The seed of the pseudo random number generator to use when shuffling
+        the data.  If int, random_state is the seed used by the random number
+        generator; If RandomState instance, random_state is the random number
+        generator; If None, the random number generator is the RandomState
+        instance used by `np.random`.
 
     Notes
     -----