DOC: Correct doc errors

Fix typos and wrap length

doc/release/upcoming_changes/21976.new_feature.rst

@@ -1,31 +1,32 @@
 The bit generator underlying the singleton RandomState can be changed
 ---------------------------------------------------------------------
 The singleton ``RandomState`` instance exposed in the ``numpy.random`` module
-id initialized using system-provided entropy with the ``MT19937` bit generator.
-The function ``set_bit_generator`` allows the default bit generator to be
-replaced with a user-provided bit generator. This function has been introduced to
-provide a method allowing seemless integration of a high-quality, modern bit generator
-in new code with existing code that makes use of the singleton-provided random
-variate generating functions. The companion function ``get_bit_generator`` returns the current bit generator
-being used by the singleton ``RandomState``. This is provided to simplify restoring
+is initialized at startup with the ``MT19937` bit generator. The function new ``set_bit_generator`` allows the default bit generator to be
+replaced with a user-provided bit generator. This function has been introduced
+to provide a method allowing seamless integration of a high-quality, modern bit
+generator in new code with existing code that makes use of the
+singleton-provided random variate generating functions. The companion function
+``get_bit_generator`` returns the current bit generator being used by the
+singleton ``RandomState``. This is provided to simplify restoring
 the original source of randomness if required.
 
 The preferred method to generate reproducible random numbers is to use a modern
 bit generator in an instance of ``Generator``. The function ``default_rng``
 simplifies instantization.
 
-    >>> rg = np.random.default_rng(3728973198)
-    >>> rg.random()
+   >>> rg = np.random.default_rng(3728973198)
+   >>> rg.random()
 
-The same bit generator can then shared with the singleton instance so that
-calling functions in the ``random`` module will use the same bit generator.
+The same bit generator can then be shared with the singleton instance so that
+calling functions in the ``random`` module will use the same bit
+generator.
 
-    >>> orig_bit_gen = np.random.get_bit_generator()
-    >>> np.random.set_bit_generator(rg.bit_generator)
-    >>> np.random.normal()
+   >>> orig_bit_gen = np.random.get_bit_generator()
+   >>> np.random.set_bit_generator(rg.bit_generator)
+   >>> np.random.normal()
 
 The swap is permanent (until reversed) and so any call to functions
-in the ``random`` module will use the new bit generator. The orginal
+in the ``random`` module will use the new bit generator. The original
 can be restored if required for code to run correctly.
 
-    >>> np.random.set_bit_generator(orig_bit_gen)
+   >>> np.random.set_bit_generator(orig_bit_gen)

numpy/random/mtrand.pyx

@@ -4749,7 +4749,7 @@ def get_bit_generator():
     Notes
     -----
     The singleton RandomState provides the random variate generators in the
-    NumPy random namespace. This function, and its counterpart set method,
+    ``numpy.random`` namespace. This function, and its counterpart set method,
     provides a path to hot-swap the default MT19937 bit generator with a
     user provided alternative. These function are intended to provide
     a continuous path where a single underlying bit generator can be
@@ -4775,7 +4775,7 @@ def set_bit_generator(bitgen):
     Notes
     -----
     The singleton RandomState provides the random variate generators in the
-    NumPy random namespace. This function, and its counterpart get method,
+    ``numpy.random``namespace. This function, and its counterpart get method,
     provides a path to hot-swap the default MT19937 bit generator with a
     user provided alternative. These function are intended to provide
     a continuous path where a single underlying bit generator can be