diff --git a/jaxampler/_src/rvs/poisson.py b/jaxampler/_src/rvs/poisson.py
index 6ebd0c0..699a528 100644
--- a/jaxampler/_src/rvs/poisson.py
+++ b/jaxampler/_src/rvs/poisson.py
@@ -27,34 +27,50 @@
 
 
 class Poisson(DiscreteRV):
-    def __init__(self, lmbda: Numeric | Any, name: Optional[str] = None) -> None:
-        shape, self._lmbda = jx_cast(lmbda)
+    def __init__(self, mu: Numeric | Any, loc: Numeric | Any = 0.0, name: Optional[str] = None) -> None:
+        shape, self._mu, self._loc = jx_cast(mu, loc)
         self.check_params()
         super().__init__(name=name, shape=shape)
 
     def check_params(self) -> None:
-        assert jnp.all(self._lmbda > 0.0), "Lambda must be positive"
+        assert jnp.all(self._mu > 0.0), "Lambda must be positive"
 
     @partial(jit, static_argnums=(0,))
     def logpmf_x(self, x: Numeric) -> Numeric:
-        return jax_poisson.logpmf(x, self._lmbda)
+        return jax_poisson.logpmf(
+            k=x,
+            mu=self._mu,
+            loc=self._loc,
+        )
 
     @partial(jit, static_argnums=(0,))
     def pmf_x(self, x: Numeric) -> Numeric:
-        return jax_poisson.pmf(x, self._lmbda)
+        return jax_poisson.pmf(
+            k=x,
+            mu=self._mu,
+            loc=self._loc,
+        )
+
+    @partial(jit, static_argnums=(0,))
+    def logcdf_x(self, x: Numeric) -> Numeric:
+        return jnp.log(self.cdf_x(x))
 
     @partial(jit, static_argnums=(0,))
     def cdf_x(self, x: Numeric) -> Numeric:
-        return jax_poisson.cdf(x, self._lmbda)
+        return jax_poisson.cdf(
+            k=x,
+            mu=self._mu,
+            loc=self._loc,
+        )
 
     def rvs(self, shape: tuple[int, ...], key: Optional[Array] = None) -> Array:
         if key is None:
             key = self.get_key()
         new_shape = shape + self._shape
-        return jax.random.poisson(key, self._lmbda, shape=new_shape)
+        return jax.random.poisson(key, self._mu, shape=new_shape)
 
     def __repr__(self) -> str:
-        string = f"Poisson(lmbda={self._lmbda}"
+        string = f"Poisson(lmbda={self._mu}"
         if self._name is not None:
             string += f", name={self._name}"
         string += ")"