03: Add TODO note, falsely solve 2a

Exercise sheet 3/exercise_sheet_03.ipynb

@@ -272,7 +272,8 @@
     "plt.ylabel(\"$p_X(k)$\")\n",
     "plt.xlabel(\"$k$\")\n",
     "plt.legend()\n",
-    "plt.show()"
+    "plt.show()\n",
+    "# TODO: The histogram overshoots the theoretical value every time."
    ]
   },
   {
@@ -337,7 +338,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 7,
    "id": "bbf5c843",
    "metadata": {
     "deletable": false,
@@ -353,10 +354,32 @@
      "task": false
     }
    },
-   "outputs": [],
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "\n",
+      "After trial and error, I found that 43402 trials were sufficient to arrive at an 1-sigma error of around 0.001,\n",
+      "with mean 0.486440 and standard deviation sigma 0.000998. Wolfram Alpha tells me it should be approximately I =\n",
+      "0.487595, so I am happy.\n",
+      "      \n"
+     ]
+    }
+   ],
    "source": [
-    "# YOUR CODE HERE\n",
+    "def sample_X():\n",
-    "raise NotImplementedError()"
+    "    x = rng.random()\n",
+    "    return np.sin(np.pi*x*(1 - x))\n",
+    "\n",
+    "n = 43402\n",
+    "sample_mean, sample_stdev = estimate_expectation(sample_X, n)\n",
+    "print(\"\"\"\n",
+    "After trial and error, I found that {} trials were sufficient to arrive at an 1-sigma error of around 0.001,\n",
+    "with mean {:.6f} and standard deviation sigma {:.6f}. Wolfram Alpha tells me it should be approximately I =\n",
+    "{}, so I am happy.\n",
+    "      \"\"\".format(n, sample_mean, sample_stdev, 0.487595))\n",
+    "# TODO: Maybe I did not answer the question. Reading is difficult."
    ]
   },
   {