diff --git a/core/src/main/scala/dimwit/stats/IndependentDistributions.scala b/core/src/main/scala/dimwit/stats/IndependentDistributions.scala
index 2132c49..6929598 100644
--- a/core/src/main/scala/dimwit/stats/IndependentDistributions.scala
+++ b/core/src/main/scala/dimwit/stats/IndependentDistributions.scala
@@ -216,12 +216,22 @@ class Exponential[T <: Tuple: Labels, V: IsFloating](val rate: Tensor[T, V]) ext
     )
     stndExp / rate
 
-class Poisson[T <: Tuple: Labels, V: IsInteger](val rate: Tensor[T, V]) extends IndependentDistribution[T, V]:
+class Poisson[T <: Tuple: Labels, VSample: IsInteger, VRate: IsFloating](val rate: Tensor[T, VRate]) extends IndependentDistribution[T, VSample]:
 
-  override def elementWiseLogProb(x: Tensor[T, V]): Tensor[T, LogProb] =
+  override def elementWiseLogProb(x: Tensor[T, VSample]): Tensor[T, LogProb] =
     liftPyTensor(jstats.poisson.logpmf(x.jaxValue, mu = rate.jaxValue))
 
-  override def sample(k: Random.Key): Tensor[T, V] =
-    liftPyTensor(
-      Jax.jrandom.poisson(k.jaxKey, lam = rate.jaxValue, shape = rate.shape.dimensions.toPythonProxy)
+  override def sample(k: Random.Key): Tensor[T, VSample] =
+    liftPyTensor(rate.shape, VType[VSample])(
+      Jax.jrandom.poisson(k.jaxKey, lam = rate.jaxValue, shape = rate.shape.dimensions.toPythonProxy, dtype = VType[VSample].dtype.jaxType)
     )
+
+object Poisson:
+
+  /** Create a Poisson distribution from a rate tensor, sampling as Int32 counts */
+  def apply[T <: Tuple: Labels, VRate: IsFloating](rate: Tensor[T, VRate]): Poisson[T, Int32, VRate] =
+    new Poisson[T, Int32, VRate](rate)
+
+  /** Create a Poisson distribution from a rate tensor, sampling counts of the given integer type */
+  def apply[T <: Tuple: Labels, VSample: IsInteger, VRate: IsFloating](rate: Tensor[T, VRate], vtype: VType[VSample]): Poisson[T, VSample, VRate] =
+    new Poisson[T, VSample, VRate](rate)
diff --git a/core/src/test/scala/dimwit/stats/DistributionSuite.scala b/core/src/test/scala/dimwit/stats/DistributionSuite.scala
index ab0de81..cb280bd 100644
--- a/core/src/test/scala/dimwit/stats/DistributionSuite.scala
+++ b/core/src/test/scala/dimwit/stats/DistributionSuite.scala
@@ -401,7 +401,7 @@ class DistributionSuite extends DimwitTest:
 
   describe("Poisson"):
     it("logProbs matches JAX"):
-      val rate = Tensor(Shape(Axis[A] -> 3)).fromArray(Array(1, 3, 10))
+      val rate = Tensor(Shape(Axis[A] -> 3)).fromArray(Array(1.0f, 3.0f, 10.0f))
       val x = Tensor(Shape(Axis[A] -> 3)).fromArray(Array(1, 2, 8))
 
       val dist = Poisson(rate)
@@ -413,7 +413,7 @@ class DistributionSuite extends DimwitTest:
 
     it("sample means approximates rate"):
       val poisson = Poisson(
-        Tensor(Shape(Axis[A] -> 2)).fromArray(Array(1, 5))
+        Tensor(Shape(Axis[A] -> 2)).fromArray(Array(1.0f, 5.0f))
       )
       val key = Random.Key(42)
       val samples = key.splitvmap(Axis[Samples] -> 10000)(k => poisson.sample(k))