diff --git a/tests/test_math.py b/tests/test_math.py
new file mode 100644
index 0000000..9715c17
--- /dev/null
+++ b/tests/test_math.py
@@ -0,0 +1,58 @@
+"""Test default grad and hessian"""
+
+import numpy as np
+from spey.math import value_and_grad, hessian
+import spey
+
+
+def test_uncorrelated_background():
+    """tester for uncorrelated background model"""
+
+    pdf_wrapper = spey.get_backend("default_pdf.uncorrelated_background")
+
+    data = [36, 33]
+    signal_yields = [12.0, 15.0]
+    background_yields = [50.0, 48.0]
+    background_unc = [12.0, 16.0]
+
+    stat_model = pdf_wrapper(
+        signal_yields=signal_yields,
+        background_yields=background_yields,
+        data=data,
+        absolute_uncertainties=background_unc,
+        analysis="multi_bin",
+        xsection=0.123,
+    )
+    hess = hessian(stat_model)([1.0, 1.0])
+    nll, grad = value_and_grad(stat_model)([1.0, 1.0])
+    nll_apri, grad_apri = value_and_grad(
+        stat_model, expected=spey.ExpectationType.apriori
+    )([1.0, 1.0])
+    hess_apri = hessian(stat_model, expected=spey.ExpectationType.apriori)([1.0, 1.0])
+
+    hess_data = hessian(stat_model, expected=spey.ExpectationType.apriori, data=[22, 34])(
+        [1.0, 1.0]
+    )
+    nll_dat, grad_dat = value_and_grad(
+        stat_model, expected=spey.ExpectationType.apriori, data=[22, 34]
+    )([1.0, 1.0])
+
+    assert np.allclose(
+        hess, np.array([[2.13638959, 2.21570381], [2.21570381, 4.30030563]])
+    ), "Hessian is wrong"
+    assert np.isclose(nll, 37.47391613937222), "NLL is wrong"
+    assert np.allclose(grad, np.array([14.89633938, 17.47861786])), "Gradient is wrong"
+    assert np.isclose(nll_apri, 20.052816087791097), "NLL apriori is wrong"
+    assert np.allclose(
+        grad_apri, np.array([9.77796784, 12.1703729])
+    ), "apriori Gradient is wrong"
+    assert np.allclose(
+        hess_apri, np.array([[3.04532025, 3.16068638], [3.16068638, 5.28374358]])
+    ), "Hessian apriori is wrong"
+    assert np.allclose(
+        hess_data, np.array([[1.80428957, 1.88600725], [1.88600725, 3.97317276]])
+    ), "Hessian data is wrong"
+    assert np.isclose(nll_dat, 49.63922692607177), "NLL data is wrong"
+    assert np.allclose(
+        grad_dat, np.array([16.97673623, 19.54635648])
+    ), "data Gradient is wrong"