diff --git a/pescador/mux.py b/pescador/mux.py
index ec6fd0a..70deb23 100644
--- a/pescador/mux.py
+++ b/pescador/mux.py
@@ -289,6 +289,7 @@ def __init__(
         weights=None,
         mode="with_replacement",
         prune_empty_streams=True,
+        dist="binomial",
         random_state=None,
     ):
         """Given an array (pool) of streamer types, do the following:
@@ -335,7 +336,12 @@ def __init__(
                 Run every selected stream once to exhaustion.
 
         prune_empty_streams : bool
-            Disable streamers that produce no data. See `BaseMux`
+            Disable streamers that produce no data.
+
+        dist : ["constant", "binomial", "poisson"]
+            Distribution governing the (maximum) number of samples taken
+            from an active streamer.
+            In each case, the expected number of samples will be `rate`.
 
         random_state : None, int, or np.random.RandomState
             See `BaseMux`
@@ -344,6 +350,7 @@ def __init__(
         self.n_active = n_active
         self.rate = rate
         self.prune_empty_streams = prune_empty_streams
+        self.dist = dist
 
         super().__init__(streamers, random_state=random_state)
 
@@ -354,6 +361,10 @@ def __init__(
             raise PescadorError(
                 f"{self.mode} is not a valid mode for StochasticMux"
             )
+        if self.dist not in ["constant", "binomial", "poisson"]:
+            raise PescadorError(
+                f"{self.dist} is not a valid distribution"
+            )
 
         self.weights = weights
         if self.weights is None:
@@ -436,7 +447,7 @@ def _on_stream_exhausted(self, idx):
             else:
                 self.distribution_[self.stream_idxs_[idx]] = 1.0
 
-    def _activate_stream(self, idx):
+    def _activate_stream(self, idx, old_idx):
         """Randomly select and create a stream.
 
         StochasticMux adds mode handling to _activate_stream, making it so that
@@ -447,16 +458,34 @@ def _activate_stream(self, idx):
         Parameters
         ----------
         idx : int, [0:n_streams - 1]
-            The stream index to replace
+            The stream index to activate
+
+        old_idx : int
+            The index of the stream being replaced in the active set.
+            This is needed for computing binomial probabilities.
         """
+        weight = self.weights[idx]
+
         # Get the number of samples for this streamer.
         n_samples_to_stream = None
         if self.rate is not None:
-            n_samples_to_stream = 1 + self.rng.poisson(lam=self.rate)
+            if self.dist == "constant":
+                n_samples_to_stream = self.rate
+            elif self.dist == "poisson":
+                n_samples_to_stream = 1 + self.rng.poisson(lam=self.rate - 1)
+            elif self.dist == "binomial":
+                # Bin((rate-1) / (1-p), 1-p)  where p = prob of selecting the new
+                # streamer from the active set
+                p = weight / (np.sum(self.stream_weights_) - self.stream_weights_[old_idx] + weight)
+                if p > 0.9999:
+                    # If we effectively have only one streamer, use the poisson limit
+                    # theorem
+                    n_samples_to_stream = 1 + self.rng.poisson(lam=self.rate - 1)
+                else:
+                    n_samples_to_stream = 1 + self.rng.binomial((self.rate-1) / (1-p), 1-p)
 
         # instantiate a new streamer
         streamer = self.streamers[idx].iterate(max_iter=n_samples_to_stream)
-        weight = self.weights[idx]
 
         # If we're sampling without replacement, zero this one out
         # This effectively disables this stream as soon as it is chosen,
@@ -484,7 +513,7 @@ def _new_stream(self, idx):
 
         # Activate the Streamer, and get the weights
         self.streams_[idx], self.stream_weights_[idx] = self._activate_stream(
-            self.stream_idxs_[idx]
+            self.stream_idxs_[idx], idx
         )
 
         # Reset the sample count to zero
diff --git a/tests/test_mux.py b/tests/test_mux.py
index 2ab0d39..f94b0ad 100644
--- a/tests/test_mux.py
+++ b/tests/test_mux.py
@@ -544,7 +544,10 @@ def test_mux_inf_loop(self, mux_class):
 
         assert len(list(mux(max_iter=100))) == 0
 
-    def test_mux_stacked_uniform_convergence(self, mux_class):
+    @pytest.mark.parametrize(
+        'dist', ['constant', 'binomial', 'poisson',
+                 pytest.param('gaussian', marks=pytest.mark.xfail(raises=pescador.PescadorError))])
+    def test_mux_stacked_uniform_convergence(self, mux_class, dist):
         """This test is designed to check that bootstrapped streams of data
         (Streamer subsampling, rate limiting) cascaded through multiple
         multiplexors converges in expectation to a flat, uniform sample of the
@@ -553,19 +556,19 @@ def test_mux_stacked_uniform_convergence(self, mux_class):
         ab = pescador.Streamer(_choice, 'ab')
         cd = pescador.Streamer(_choice, 'cd')
         ef = pescador.Streamer(_choice, 'ef')
-        mux1 = mux_class([ab, cd, ef], 2, rate=2, random_state=1357)
+        mux1 = mux_class([ab, cd, ef], 2, rate=4, random_state=1357, dist=dist)
 
         gh = pescador.Streamer(_choice, 'gh')
         ij = pescador.Streamer(_choice, 'ij')
         kl = pescador.Streamer(_choice, 'kl')
 
-        mux2 = mux_class([gh, ij, kl], 2, rate=2, random_state=2468)
+        mux2 = mux_class([gh, ij, kl], 2, rate=4, random_state=2468, dist=dist)
 
         stacked_mux = mux_class([mux1, mux2], 2, rate=None,
                                 random_state=12345)
 
-        max_iter = 1000
         chars = 'abcdefghijkl'
+        max_iter = len(chars) * 500
         samples = list(stacked_mux.iterate(max_iter=max_iter))
         counter = collections.Counter(samples)
         assert set(chars) == set(counter.keys())
@@ -574,7 +577,7 @@ def test_mux_stacked_uniform_convergence(self, mux_class):
 
         # Check that the pvalue for the chi^2 test is at least 0.95
         test = scipy.stats.chisquare(counts)
-        assert test.pvalue >= 0.95
+        assert test.pvalue >= 0.5, counts
 
 
 class TestShuffledMux: