01: Add Metropolis algorithm attempt

Exercise sheet 1/exercise_sheet_01.ipynb

@@ -542,7 +542,8 @@
     "          \"\\td = {}\".format(d),\n",
     "          \"\\tV_d = {:.3f}\".format(volume_d_ball(d)),\n",
     "          \"\\tdirect-sampled V_d = {:5.3f}\".format(number_of_hits_in_d_dimensions(N, d)/N*2**d)\n",
-    "         )"
+    "         )\n",
+    "# TODO: Plot the results as is asked."
    ]
   },
   {
@@ -571,7 +572,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 14,
    "metadata": {
     "deletable": false,
     "nbgrader": {
@@ -586,10 +587,42 @@
      "task": false
     }
    },
-   "outputs": [],
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "N = 2000 \tdelta = 0.000 \tpi_estimate = 4.000\n",
+      "N = 2000 \tdelta = 0.333 \tpi_estimate = 3.149\n",
+      "N = 2000 \tdelta = 0.667 \tpi_estimate = 3.139\n",
+      "N = 2000 \tdelta = 1.000 \tpi_estimate = 3.135\n",
+      "N = 2000 \tdelta = 1.333 \tpi_estimate = 3.146\n",
+      "N = 2000 \tdelta = 1.667 \tpi_estimate = 3.150\n",
+      "N = 2000 \tdelta = 2.000 \tpi_estimate = 3.150\n",
+      "N = 2000 \tdelta = 2.333 \tpi_estimate = 3.141\n",
+      "N = 2000 \tdelta = 2.667 \tpi_estimate = 3.146\n",
+      "N = 2000 \tdelta = 3.000 \tpi_estimate = 3.146\n"
+     ]
+    }
+   ],
    "source": [
-    "# YOUR CODE HERE\n",
-    "raise NotImplementedError()"
+    "N = 2000 # number of trials\n",
+    "m = 200  # number of repetitions\n",
+    "\n",
+    "p0 = np.array([0., 0.]) # starting position in origin\n",
+    "\n",
+    "for delta in np.linspace(0, 3, 10):\n",
+    "    pi_estimate = 0\n",
+    "    for i in range(m):\n",
+    "        hits = markov_pebble(p0, delta, N)\n",
+    "        pi_estimate += hits/N*4\n",
+    "    pi_estimate /= m\n",
+    "    print(\"N = {}\".format(N),\n",
+    "          \"\\tdelta = {:.3f}\".format(delta),\n",
+    "          \"\\tpi_estimate = {:.3f}\".format(pi_estimate)\n",
+    "         )\n",
+    "\n",
+    "# TODO: Investigate the rejection rate and the mean square deviation. Maybe only the latter, as the next question concerns the former."
    ]
   },
   {