diff --git a/powerlaw.py b/powerlaw.py
index 20176aa..a8fff50 100644
--- a/powerlaw.py
+++ b/powerlaw.py
@@ -1163,7 +1163,7 @@ def sigma(self):
         return (self.alpha - 1) / sqrt(self.n)
 
     def _in_standard_parameter_range(self):
-        return self.alpha>1
+        return self.alpha>0
 
     def fit(self, data=None):
         if data is None and hasattr(self, 'parent_Fit'):
@@ -1416,7 +1416,7 @@ def _initial_parameters(self, data):
         return (alpha, Lambda)
 
     def _in_standard_parameter_range(self):
-        return self.Lambda>0 and self.alpha>1
+        return self.Lambda>0 and self.alpha>0
 
     def _cdf_base_function(self, x):
         from mpmath import gammainc
@@ -1928,13 +1928,13 @@ def cumulative_distribution_function(data,
 
     if all_unique:
         from numpy import arange
-        CDF = arange(n)/n
+        CDF = arange(1, n+1)/n
     else:
 #This clever bit is a way of using searchsorted to rapidly calculate the
 #CDF of data with repeated values comes from Adam Ginsburg's plfit code,
 #specifically https://github.com/keflavich/plfit/commit/453edc36e4eb35f35a34b6c792a6d8c7e848d3b5#plfit/plfit.py
         from numpy import searchsorted, unique
-        CDF = searchsorted(data, data,side='left')/n
+        CDF = searchsorted(data, data,side='right')/n
         unique_data, unique_indices = unique(data, return_index=True)
         data=unique_data
         CDF = CDF[unique_indices]
diff --git a/testing/test_powerlaw.py b/testing/test_powerlaw.py
index fce8385..7e29fb5 100644
--- a/testing/test_powerlaw.py
+++ b/testing/test_powerlaw.py
@@ -115,6 +115,39 @@
         'surnames': {}
         }
 
+class BaseTestCase(unittest.TestCase):
+    """
+    Test simple mathematical properties
+    """
+
+    def test_cumulative_distribution_function(self):
+        # Test all unique, and repeated values
+        X1 = [10, 30, 20, 20]
+        X2 = [10, 30, 20]
+
+        for test_data in [X1, X2]:
+            X_sorted, CDF = powerlaw.cumulative_distribution_function(test_data)
+            # Check array is sorted
+            assert_allclose(X_sorted[-1], max(test_data))
+            # CDF gives probability of less than or equal to some value.
+            # The last element is the max of the sample. All other elements
+            # are therefore smaller than or equal, and the CDF should be 1.
+            assert_allclose(CDF[-1], 1.0)
+
+    def test_pdf(self):
+        import numpy as np
+        np.random.seed(0)
+        data = np.random.lognormal(mean=0, sigma=1, size=1000)
+        # We test limits
+        xmin = 0.1
+        xmax = 0.2
+        edges, hist = powerlaw.pdf(data, xmin=xmin, xmax=xmax)
+        area = np.sum(hist * np.diff(edges))
+        # The area under a pdf should be 1
+        assert_allclose(area, 1)
+        # Test limits
+        assert_allclose(edges[0], xmin)
+        assert_allclose(edges[-1], xmax)
 
 class FirstTestCase(unittest.TestCase):