diff --git a/src/main/java/org/apache/commons/math4/distribution/UniformIntegerDistribution.java b/src/main/java/org/apache/commons/math4/distribution/UniformIntegerDistribution.java
index 1e89c4309..2909cc87d 100644
--- a/src/main/java/org/apache/commons/math4/distribution/UniformIntegerDistribution.java
+++ b/src/main/java/org/apache/commons/math4/distribution/UniformIntegerDistribution.java
@@ -22,10 +22,8 @@ import org.apache.commons.math4.exception.util.LocalizedFormats;
 import org.apache.commons.rng.UniformRandomProvider;
 
 /**
- * Implementation of the uniform integer distribution.
- *
- * @see <a href="http://en.wikipedia.org/wiki/Uniform_distribution_(discrete)"
- * >Uniform distribution (discrete), at Wikipedia</a>
+ * Implementation of the <a href="http://en.wikipedia.org/wiki/Uniform_distribution_(discrete)">
+ * uniform integer distribution</a>.
  *
  * @since 3.0
  */
@@ -36,6 +34,10 @@ public class UniformIntegerDistribution extends AbstractIntegerDistribution {
     private final int lower;
     /** Upper bound (inclusive) of this distribution. */
     private final int upper;
+    /** "upper" + "lower" (to avoid overflow). */
+    private final double upperPlusLower;
+    /** "upper" - "lower" (to avoid overflow). */
+    private final double upperMinusLower;
 
     /**
      * Creates a new uniform integer distribution using the given lower and
@@ -55,6 +57,8 @@ public class UniformIntegerDistribution extends AbstractIntegerDistribution {
         }
         this.lower = lower;
         this.upper = upper;
+        upperPlusLower = (double) upper + (double) lower;
+        upperMinusLower = (double) upper - (double) lower;
     }
 
     /** {@inheritDoc} */
@@ -63,7 +67,7 @@ public class UniformIntegerDistribution extends AbstractIntegerDistribution {
         if (x < lower || x > upper) {
             return 0;
         }
-        return 1.0 / (upper - lower + 1);
+        return 1.0 / (upperMinusLower + 1);
     }
 
     /** {@inheritDoc} */
@@ -75,7 +79,7 @@ public class UniformIntegerDistribution extends AbstractIntegerDistribution {
         if (x > upper) {
             return 1;
         }
-        return (x - lower + 1.0) / (upper - lower + 1.0);
+        return (x - lower + 1.0) / (upperMinusLower + 1.0);
     }
 
     /**
@@ -86,7 +90,7 @@ public class UniformIntegerDistribution extends AbstractIntegerDistribution {
      */
     @Override
     public double getNumericalMean() {
-        return 0.5 * (lower + upper);
+        return 0.5 * upperPlusLower;
     }
 
     /**
@@ -97,7 +101,7 @@ public class UniformIntegerDistribution extends AbstractIntegerDistribution {
      */
     @Override
     public double getNumericalVariance() {
-        double n = upper - lower + 1;
+        double n = upperMinusLower + 1;
         return (n * n - 1) / 12.0;
     }
 
diff --git a/src/test/java/org/apache/commons/math4/distribution/UniformIntegerDistributionTest.java b/src/test/java/org/apache/commons/math4/distribution/UniformIntegerDistributionTest.java
index 3ae15dda0..d1fc3161e 100644
--- a/src/test/java/org/apache/commons/math4/distribution/UniformIntegerDistributionTest.java
+++ b/src/test/java/org/apache/commons/math4/distribution/UniformIntegerDistributionTest.java
@@ -22,6 +22,7 @@ import org.junit.Test;
 import org.apache.commons.math4.distribution.IntegerDistribution;
 import org.apache.commons.math4.distribution.UniformIntegerDistribution;
 import org.apache.commons.math4.exception.NumberIsTooLargeException;
+import org.apache.commons.math4.util.Precision;
 
 /**
  * Test cases for UniformIntegerDistribution. See class javadoc for
@@ -112,4 +113,30 @@ public class UniformIntegerDistributionTest extends IntegerDistributionAbstractT
         // Degenerate case is allowed.
         new UniformIntegerDistribution(0, 0);
     }
+
+    // MATH-1396
+    @Test
+    public void testLargeRangeSubtractionOverflow() {
+        final int hi = Integer.MAX_VALUE / 2 + 10;
+        UniformIntegerDistribution dist = new UniformIntegerDistribution(-hi, hi - 1);
+
+        final double tol = Math.ulp(1d);
+        Assert.assertEquals(0.5 / hi, dist.probability(123456), tol);
+        Assert.assertEquals(0.5, dist.cumulativeProbability(-1), tol);
+
+        Assert.assertTrue(Precision.equals((Math.pow(2d * hi, 2) - 1) / 12, dist.getNumericalVariance(), 1));
+    }
+
+    // MATH-1396
+    @Test
+    public void testLargeRangeAdditionOverflow() {
+        final int hi = Integer.MAX_VALUE / 2 + 10;
+        UniformIntegerDistribution dist = new UniformIntegerDistribution(hi - 1, hi + 1);
+
+        final double tol = Math.ulp(1d);
+        Assert.assertEquals(1d / 3d, dist.probability(hi), tol);
+        Assert.assertEquals(2d / 3d, dist.cumulativeProbability(hi), tol);
+
+        Assert.assertTrue(Precision.equals(hi, dist.getNumericalMean(), 1));
+    }
 }