Added explanation of density functions.

I felt the description was a bit sparse, so I expanded it a bit.

03-Gaussians.ipynb

@@ -239,11 +239,14 @@
        "                },\n",
        "                displayAlign: 'center', // Change this to 'center' to center equations.\n",
        "                \"HTML-CSS\": {\n",
-       "                    scale:95,\n",
+       "                    scale: 95,\n",
        "                        availableFonts: [],\n",
        "                        preferredFont:null,\n",
        "                        webFont: \"TeX\",\n",
        "                    styles: {'.MathJax_Display': {\"margin\": 4}}\n",
+       "                },\n",
+       "                 \"fast-preview\": {\n",
+       "                    \"disabled\": false\n",
        "                }\n",
        "        });\n",
        "</script>\n"
@@ -608,7 +611,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x1c6ee41ad68>"
+       "<matplotlib.figure.Figure at 0x1deb10b0f60>"
       ]
      },
      "metadata": {},
@@ -639,9 +642,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAwgAAAEvCAYAAADhM+awAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x1c6ef489cf8>"
+       "<matplotlib.figure.Figure at 0x1deb1db3f28>"
       ]
      },
      "metadata": {},
@@ -651,7 +654,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "mean = 1.821\n",
+      "mean = 1.817\n",
       "std  = 0.140\n"
      ]
     }
@@ -765,7 +768,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x1c6ef6c2908>"
+       "<matplotlib.figure.Figure at 0x1deb2302dd8>"
       ]
      },
      "metadata": {},
@@ -844,7 +847,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x1c6ee41af98>"
+       "<matplotlib.figure.Figure at 0x1deb1e1e668>"
       ]
      },
      "metadata": {},
@@ -883,7 +886,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x1c6f074fd68>"
+       "<matplotlib.figure.Figure at 0x1deb337d358>"
       ]
      },
      "metadata": {},
@@ -921,7 +924,7 @@
      "data": {
       "image/png": 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       "text/plain": [
-       "<matplotlib.figure.Figure at 0x1c6ed3a8f60>"
+       "<matplotlib.figure.Figure at 0x1deb340edd8>"
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      },
      "metadata": {},
@@ -936,9 +939,9 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "The y-axis depicts the *probability density* — the relative amount of cars that are going the speed at the corresponding x-axis.\n",
+    "The y-axis depicts the *probability density* — the relative amount of cars that are going the speed at the corresponding x-axis. I will explain this further in the next section.\n",
     "\n",
-    "You may object that human heights or automobile speeds cannot be less than zero, let alone $-\\infty$ or $\\infty$. This is true, but this is a common limitation of mathematical modeling. “The map is not the territory” is a common expression, and it is true for Bayesian filtering and statistics. The Gaussian distribution above models the distribution of the measured automobile speeds, but being a model it is necessarily imperfect. The difference between model and reality will come up again and again in these filters. Gaussians are used in many branches of mathematics, not because they perfectly model reality, but because they are easier to use than any other relatively accurate choice. However, even in this book Gaussians will fail to model reality, forcing us to use computationally expensive alternatives. \n",
+    "The Gaussian model is imperfect. For example, you may object that human heights or automobile speeds cannot be less than zero, let alone $-\\infty$ or $\\infty$. “The map is not the territory” is a common expression, and it is true for Bayesian filtering and statistics. The Gaussian distribution above models the distribution of the measured automobile speeds, but being a model it is necessarily imperfect. The difference between model and reality will come up again and again in these filters. Gaussians are used in many branches of mathematics, not because they perfectly model reality, but because they are easier to use than any other relatively accurate choice. However, even in this book Gaussians will fail to model reality, forcing us to use computationally expensive alternatives. \n",
     "\n",
     "You will hear these distributions called *Gaussian distributions* or *normal distributions*.  *Gaussian* and *normal* both mean the same thing in this context, and are used interchangeably. I will use both throughout this book as different sources will use either term, and I want you to be used to seeing both. Finally, as in this paragraph, it is typical to shorten the name and talk about a *Gaussian* or *normal* — these are both typical shortcut names for the *Gaussian distribution*. "
    ]
@@ -987,7 +990,7 @@
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x1c6f07f1978>"
+       "<matplotlib.figure.Figure at 0x1deb341f358>"
       ]
      },
      "metadata": {},
@@ -1008,7 +1011,17 @@
     "\n",
     "What is this curve? It is something we call the *probability density function.* The area under the curve at any region gives you the probability of those values. So, for example, if you compute the area under the curve between 20 and 22 the resulting area will be the probability of the temperature reading being between those two temperatures. \n",
     "\n",
-    "We can think of this in Bayesian terms or frequentist terms. As a Bayesian, if the thermometer reads exactly 22°C, then our belief is described by the curve - our belief that the actual (system) temperature is near 22 is very high, and our belief that the actual temperature is near 18 is very low. As a frequentist we would say that if we took 1 billion temperature measurements of a system at exactly 22°C, then a histogram of the measurements would look like this curve. \n",
+    "Here is another way to understand it. What is the *density* of a rock, or a sponge? It is a measure of how much mass is compacted into a given space. Rocks are dense, sponges less so. So, if you wanted to know how much a rock weighed but didn't have a scale, you could take its volume and multiply by its density. This would give you its mass. In practice density varies in most objects, so you would integrate the local density across the rock's volume.\n",
+    "\n",
+    "$$M = \\iiint_R p(x,y,z)\\, dV$$\n",
+    "\n",
+    "We do the same with *probability density*. If you want to know the temperature being between 20°C and 21°C kph you would integrate the curve above from 20 to 21. As you know the integral of a curve gives you the area under the curve. Since this is a curve of the probability density, the integral of the density is the probability. \n",
+    "\n",
+    "What is the probability of a the temperature being exactly 22°C? Intuitively, 0. These are real numbers, and the odds of 22°C vs, say, 22.00000000000017°C is infitesimal. Mathematically, what would we get if we integrate from 22 to 22? Zero. \n",
+    "\n",
+    "In practice our sensors do not have infinite precision, so a reading of 22°C implies a range, such as 22$\\pm$0.1°C, and we can compute the probability of that range by integrating from 21.9 to 22.1.\n",
+    "\n",
+    "We can think of this in Bayesian terms or frequentist terms. As a Bayesian, if the thermometer reads exactly 22°C, then our belief is described by the curve - our belief that the actual (system) temperature is near 22°C is very high, and our belief that the actual temperature is near 18 is very low. As a frequentist we would say that if we took 1 billion temperature measurements of a system at exactly 22°C, then a histogram of the measurements would look like this curve. \n",
     "\n",
     "How do you compute the probability, or area under the curve? You integrate the equation for the Gaussian \n",
     "\n",
@@ -1100,7 +1113,7 @@
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x1c6f0917128>"
+       "<matplotlib.figure.Figure at 0x1deb3469cc0>"
       ]
      },
      "metadata": {},
@@ -1153,7 +1166,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x1c6f09c7c18>"
+       "<matplotlib.figure.Figure at 0x1deb35f1d30>"
       ]
      },
      "metadata": {},
@@ -1182,7 +1195,7 @@
     {
      "data": {
       "application/vnd.jupyter.widget-view+json": {
-       "model_id": "3e38ed0d56f045db94523a2266729107",
+       "model_id": "9349728a16de442db16d694bd0d176ab",
        "version_major": 2,
        "version_minor": 0
       },
@@ -1439,9 +1452,9 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[ 0.824  3.24   3.113  4.934 -3.799 -1.775 -1.02\n",
-      "  7.542 -0.76   1.875  3.862  1.77   4.846 -0.818\n",
-      "  5.726]\n"
+      "[ 1.42  -2.72   0.378  3.368  1.608  2.191  2.704\n",
+      "  2.972 -0.632  4.722  1.322  4.977  5.128 -1.258\n",
+      "  1.485]\n"
      ]
     }
    ],
@@ -1527,7 +1540,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x1c6f0a3a2b0>"
+       "<matplotlib.figure.Figure at 0x1deb3512a20>"
       ]
      },
      "metadata": {},
@@ -1580,9 +1593,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x1c6f086b8d0>"
+       "<matplotlib.figure.Figure at 0x1deb34a0898>"
       ]
      },
      "metadata": {},
@@ -1631,9 +1644,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x1c6ef6ce978>"
+       "<matplotlib.figure.Figure at 0x1deb1d8e2e8>"
       ]
      },
      "metadata": {},

05-Multivariate-Gaussians.ipynb

@@ -241,11 +241,14 @@
        "                },\n",
        "                displayAlign: 'center', // Change this to 'center' to center equations.\n",
        "                \"HTML-CSS\": {\n",
-       "                    scale:95,\n",
+       "                    scale: 95,\n",
        "                        availableFonts: [],\n",
        "                        preferredFont:null,\n",
        "                        webFont: \"TeX\",\n",
        "                    styles: {'.MathJax_Display': {\"margin\": 4}}\n",
+       "                },\n",
+       "                 \"fast-preview\": {\n",
+       "                    \"disabled\": false\n",
        "                }\n",
        "        });\n",
        "</script>\n"
@@ -334,7 +337,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c6f92d278>"
+       "<matplotlib.figure.Figure at 0x157252b3390>"
       ]
      },
      "metadata": {},
@@ -562,7 +565,7 @@
      "data": {
       "image/png": "iVBORw0KGgoAAAANSUhEUgAAAlYAAAEaCAYAAADT1xmVAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAAPYQAAD2EBqD+naQAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDIuMS4wLCBo\ndHRwOi8vbWF0cGxvdGxpYi5vcmcvpW3flQAAIABJREFUeJzs3XlYlOX+x/H3zDAKsojkksrimgsu\nmJprpUaaueaSS4ulZm6lmac6Zamp2WbqqUzNrFwyzX1LM7dScd+OmrggAorihuwIzPz+8ODPBQV0\nGBj4vK7L63SGZ2buxxvwM9/7+9yPwWq1WhERERGRB2bM7QGIiIiI5BcKViIiIiI2omAlIiIiYiMK\nViIiIiI2omAlIiIiYiNOuT2A21ksFiwWyy2PGQwGDAZDLo1IRERECiKr1crtmycYjUaMxrvXpfJk\nsIqPj8/tYYiIiIjcwdXV9Z7BSkuBIiIiIjaiYCUiIiJiIwpWIiIiIjaS53qsMmpSz2w9UzJ2+PBh\nUlNTcXJywt/fP7eHI9mguXNMmjfHpblzXDk5dxn1fWd2MZ1DBKvMOvAlYxaLhbS0NP39OSDNnWPS\nvDkuzZ3jsvfcZRas7PLdExsbyzvvvEPLli0pUaIEBoOBUaNG2eOtRUREROzGLsHq0qVLTJ8+neTk\nZDp27GiPtxQRERGxO7ssBfr5+XHlyhUMBgMXL15kxowZ9nhbEREREbuyS7DSrukiIiJSEOS55vWM\nHD58+I7b3EjmUlJSbvzvgQMHcnk0kh2aO8ekeXNcmjvHlZNzZzQa8fX1zdZzHCJYpaamkpaWltvD\ncGjp33jieDR3jknz5rg0d47L1nNnMpmy/ZxsB6tNmzbRvHnzLB27b98+AgICsj2o2zk5OeXYJZRX\nrlxh6tSp9OzZEz8/vxx5j9xy8zeY2WzOxZFIdmnuHJPmzXFp7hxXTs7d/WSPbAerKlWq8P3332fp\n2OyWz+7G398/x4LV2LFjmT9/PvPnz6dt27YMGzaMZs2a5Yu+sAMHDpCSkoLZbKZ27dq5PRzJBs2d\nY9K8OS7NnePKybmzWCzExsZm6znZDlalS5emb9++2X1antW8eXPatWvHihUrWLlyJStXriQgIIC3\n3nqL7t27U6hQodweooiIiDiIAr+9bJMmTVi+fDnBwcEMHDgQFxcX9u/fT69evShXrhzjxo3j0qVL\nuT1MERERcQB2C1a///47CxcuZMWKFQAcOXKEhQsXsnDhQhISEuw1jAxZrVYeeeQRvv32WyIiIhg/\nfjxlypQhMjKSESNG4OPjQ//+/Tl69GiujlNERETyNoPVarXa443KlSvH6dOnM/zaqVOnKFeuHJDx\neqa7u7vNe6ziklP5cm0wf/5znpQ0K2aTgcBqpRjeqgpuhZ24du0av/32G1999RV79+698bxnn32W\nYcOG0aJFizzfh6WeAceluXNMmjfHpblzXPbuscosk9itYhUaGorVas3wT3qospe45FQ6TdnKrKBQ\nIq4kcj4miYgricwKCqXTlK3EJadiNpt54YUX2L17N5s3b6ZDhw4YDAZWr15NYGAgAQEB/PjjjyQn\nJ9t17CIiIpJ3Fcgeqy/XBnMiKg7LbbU6ixWOnY+j4Sd/0nD8Bpp+toHRK47waIPGLF26lGPHjjF4\n8GBcXV05ePAgvXv3xs/Pj48//pgLFy7kzsmIiIhInlEgg9Wf/5y/I1TdLC45LcMqVqVKlfj6668J\nDw/ns88+w9vbm/PnzzNy5Eh8fHx47bXXOHLkiN3OQ0RERPKWAhesrFYrKWlZbyuzWOFEVBwT1gaT\n3o5WrFgx3nnnHUJCQvjll1+oV68eycnJzJgxA39/f5555hn++OMP7NS+JiIiInlEgQtWBoMBsyl7\nTecWK8zafvrG8uCo5Ydv9GH16NGDnTt38vfff/Pcc89hMBhYu3YtrVq1ombNmsycOZOkpKQcOhsR\nERHJSwpcsAIIrFYKYzYv6EuzWDNcHoTrYa1p06YsXryYEydOMGTIENzc3Dh8+DB9+vTBz8+PUaNG\nERUVZfuTERERkTyjQAar4a2qUKmkW7bDVbqMlgfTVahQgUmTJhEeHs4XX3yBj48PUVFRjB49Gl9f\nX/r06cOhQ4dscBYiIiKS1xTIYOVW2InFA5vQq1E5vIu58LCHM66FsncH67stD6bz9PRk+PDhhISE\n8Ouvv9KgQQOSk5OZOXMmNWvWpGXLlqxZs0Z9WCIiIvlIgQxWcD1cjWzvz5Z3WxD07xbs+CCQR0pl\nr4p1t+XBm8OSk5MT3bp1Y/v27Wzbto0uXbpgNBpZt24drVu3xt/fn++//57ExETbn6SIiIjYVYEN\nVjczGAwZVrFM2UhZGe2BdXsVq1GjRvz222+cPHmSt956C3d3d/755x/69euHr68vH374IefOncuJ\nUxQRERE7ULC6ye1VrJca+mW7D+tue2DdXMUqV64cX331FREREUycOJFy5cpx8eJFxo4di5+fH6+8\n8goHDhyw7cmJiIhIjlOwuguDwWCTJvd7VbE8PDwYOnQox48f57fffqNx48Zcu3aNn3/+mYCAAAID\nA1m1ahUWi8WGZyYiIiI5RcHqHh50eTDd3apY6ZycnOjSpQtbt25l+/btdOvWDZPJxPr162nbti3V\nq1dn6tSpJCQk2PDsRERExNYUrDJhi+XBm91rqwaABg0a8OuvvxISEsLw4cPx8PAgODiYAQMG4OPj\nw4gRI4iMjHyAMxIREZGcomCVDbZYHoTMt2oA8PX15YsvviAiIoJJkyZRvnx5Ll++zLhx4/Dz86NX\nr17s37//Ac9IREREbEnBKptssQcWZG2rBgB3d3eGDBnC8ePHWbRoEU2bNiUlJYVZs2ZRp04dWrRo\nwYoVK9SHJSIikgcoWN0HW+yBdbOsbNVgMpno1KkTf//9Nzt37qRHjx6YTCY2btxI+/btqVq1KlOm\nTCE+Pt5GZykiIiLZpWD1gO62B9b9VLGyslUDQP369fnll184deoU77zzDp6enhw/fpxBgwbh4+PD\nv//9b86cOWOjMxQREZGsMljz2D1VLBYLsbGxtzzm7u6O0eg4GdBqtRJ/LY1OU7ZyIioOywP8DbsV\nNuFa2IzZZCCwWimGt6qCW2GnW46Ji4vjp59+YtKkSZw8eRK4fqVhy5Yt6d69O7Vq1aJ27doPckpi\nZwcOHCAlJQWz2ay5cyCaN8eluXNcOTl395NJHCetOBBb7OSeLitVLDc3NwYPHkxwcDBLly7liSee\nIDU1ldWrV/Pyyy/Tp08fli5dSlpamg3PUkRERG6nYJWDcmKrhnv1YplMJjp06MDmzZvZvXs3rVu3\nxmQysWfPHp577jmqVKnC119/TVxcnI3OUERERG6mYGUnttqqAbJWxapbty7jx49n2bJl9O7dm2LF\ninHy5EnefPNNvL29eeeddwgPD3+wgYiIiMgtFKzsyFZN7jfLrIpVqlQp3nzzTcLDw/n222+pXLky\nV69e5YsvvqB8+fL06NGDXbt22eL0RERECjwFKzuz9VYN6TKqYiWkWG5UsVxdXRk4cCBHjx5l+fLl\nNGvWjLS0NH799Vcee+wxHn/8cRYvXqw+LBERkQegYJWLbLlVw83Sq1i9l0by+qqLvLY88kYVy2g0\n0q5dOzZu3MjevXt5+eWXMZvNbNmyhc6dO1O5cmUmT558x1UQIiIikjkFqzwgp6pYialWLidZiIpP\ny7AXq06dOvz888+EhobywQcf4OXlxalTpxg6dCje3t4MHz6c06dPP/gJioiIFBAKVnlMTlexMurF\nKlOmDGPGjCE8PJypU6dSpUoVYmJimDBhAhUrVqRbt25s377dRmcoIiKSfzllfojkhvQq1sj2/jbd\ncDQuOY245Ot9VD9vC2XpvgiKFDaTZrH+bxPSpuzc9yp/b1jHxIkTWb9+PQsWLGDBggU0atSIt956\ni+eeew4nJ33riIiI3E4VKweQU1UsKxCdmMrZ6MRbGt+7TA3iycBWrFu3jv379/PKK69gNpsJCgri\n+eefp1KlSkycOJGYmBibnJ+IiEh+oVvaOChbVrEycvutdF6s5cHPP0znu+++49KlS8D1eenTpw9v\nvvkm5cuXt+0ACjjdXsMxad4cl+bOcemWNmITOVXFSnf79g0Dl4Tw7oiRhIWFMW3aNKpVq0ZsbCyT\nJk2iUqVKdOnShW3btt1xw2gREZGCRBWrfMSeVaynqpQgwBTGd19PZt26dTeOeeyxxxg2bBidO3dW\nH9YD0Kdnx6R5c1yaO8elipXkmNurWCVdTXg5G3GxUb65pYq1PYzRe52Ia/EeNd6YTsl6z2ByMrNz\n5066d+9OhQoV+PLLL4mOjrbNm4uIiDgABat8KP2Kwu/bl2Zam+LM7FjGJvti3ezmxvfYImVweWow\nZfrPpNSTL+Dk6kl4eDj/+te/KFW6LIPeeJOQkBDbvbmIiEgepWCVzxkMBoqYjXf0YpXxdMbTxWzT\nsGV0LYZzwx6U6T+Th1q/ibm4L9eSEpjyzddUrlyZzp07s2XLFvVhiYhIvqUmmALi9n2xDAYDccmp\nTFgbzLp/zpOaZiU2KYX4aw9+r0CDUyHcarXEtebTJIXuI2bXMpJO7WHx4sUsXryYMpVrMHbEu7zY\noxtOTk4YDDZMdyIiIrlIwaoASg8yObUJ6c3v41L+UVzKP8q1i2HE7l5G3KENnD1+iN69XqL/m8Mo\n1bAjZRq04Zm6lRjeqgpuhZ1uBD8RERFHo2AlwK2N7zlRxSpU3JeHnnkDzydeJnbfamL3ruLa1QuE\nr/2eiA2zOVIzkPl/dqJYGb+bdoEvdSNsiYiIOAL9iyW3yOkqlqlIUTyb9KBogy7EHLine truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c7087a7f0>"
+       "<matplotlib.figure.Figure at 0x15725fac4a8>"
       ]
      },
      "metadata": {},
@@ -600,9 +603,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c703089e8>"
+       "<matplotlib.figure.Figure at 0x15725ca9dd8>"
       ]
      },
      "metadata": {},
@@ -612,8 +615,8 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[[ 10.7  3.03]\n",
-      " [ 3.03  2.25]]\n"
+      "[[ 10.5  2.86]\n",
+      " [ 2.86  2.18]]\n"
      ]
     }
    ],
@@ -641,9 +644,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c6f92d0b8>"
+       "<matplotlib.figure.Figure at 0x157252b3128>"
       ]
      },
      "metadata": {},
@@ -653,8 +656,8 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[[ 1.0  -0.00434]\n",
-      " [ -0.00434  0.995]]\n"
+      "[[ 1.0  -0.00104]\n",
+      " [ -0.00104  1.01]]\n"
      ]
     }
    ],
@@ -704,7 +707,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c6e8a8b70>"
+       "<matplotlib.figure.Figure at 0x157252b3278>"
       ]
      },
      "metadata": {},
@@ -862,9 +865,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c72312e80>"
+       "<matplotlib.figure.Figure at 0x15727cc4ef0>"
       ]
      },
      "metadata": {},
@@ -902,7 +905,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c729036d8>"
+       "<matplotlib.figure.Figure at 0x15727fb8160>"
       ]
      },
      "metadata": {},
@@ -940,7 +943,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c72985160>"
+       "<matplotlib.figure.Figure at 0x157283b1978>"
       ]
      },
      "metadata": {},
@@ -974,7 +977,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c729c8ef0>"
+       "<matplotlib.figure.Figure at 0x15728578518>"
       ]
      },
      "metadata": {},
@@ -1007,7 +1010,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c72977278>"
+       "<matplotlib.figure.Figure at 0x15725c8ab38>"
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      },
      "metadata": {},
@@ -1050,7 +1053,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c70ee89b0>"
+       "<matplotlib.figure.Figure at 0x15727fe9b38>"
       ]
      },
      "metadata": {},
@@ -1093,7 +1096,7 @@
      "data": {
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       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c704cd940>"
+       "<matplotlib.figure.Figure at 0x15725fb3e48>"
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@@ -1149,7 +1152,7 @@
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c7285be48>"
+       "<matplotlib.figure.Figure at 0x157284df358>"
       ]
      },
      "metadata": {},
@@ -1180,7 +1183,7 @@
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c7295df98>"
+       "<matplotlib.figure.Figure at 0x157284437f0>"
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      },
      "metadata": {},
@@ -1214,7 +1217,7 @@
     {
      "data": {
       "application/vnd.jupyter.widget-view+json": {
-       "model_id": "0b366d69a8f844d8a0f379569a461908",
+       "model_id": "bfb35469dd51488192ac84683ff9b6f6",
        "version_major": 2,
        "version_minor": 0
       },
@@ -1336,7 +1339,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c72bd0d30>"
+       "<matplotlib.figure.Figure at 0x15728585a58>"
       ]
      },
      "metadata": {},
@@ -1367,7 +1370,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c72b3f828>"
+       "<matplotlib.figure.Figure at 0x15725fb3710>"
       ]
      },
      "metadata": {},
@@ -1410,7 +1413,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c7240ff60>"
+       "<matplotlib.figure.Figure at 0x15727dc2da0>"
       ]
      },
      "metadata": {},
@@ -1469,7 +1472,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c723ecb38>"
+       "<matplotlib.figure.Figure at 0x15727d6d4e0>"
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      "metadata": {},
@@ -1499,7 +1502,7 @@
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c723a0da0>"
+       "<matplotlib.figure.Figure at 0x15727d6d5c0>"
       ]
      },
      "metadata": {},
@@ -1532,7 +1535,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c729925c0>"
+       "<matplotlib.figure.Figure at 0x1572833b550>"
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      },
      "metadata": {},
@@ -1568,7 +1571,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c70889438>"
+       "<matplotlib.figure.Figure at 0x1572833b6a0>"
       ]
      },
      "metadata": {},
@@ -1597,7 +1600,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c7237eb00>"
+       "<matplotlib.figure.Figure at 0x157284df940>"
       ]
      },
      "metadata": {},
@@ -1629,7 +1632,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c72997550>"
+       "<matplotlib.figure.Figure at 0x15727d086a0>"
       ]
      },
      "metadata": {},
@@ -1660,7 +1663,7 @@
      "data": {
       "image/png": 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       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c704f5c50>"
+       "<matplotlib.figure.Figure at 0x15727cc4c50>"
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      },
      "metadata": {},
@@ -1691,7 +1694,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c7295c518>"
+       "<matplotlib.figure.Figure at 0x157263015c0>"
       ]
      },
      "metadata": {},
@@ -1724,7 +1727,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c72859a90>"
+       "<matplotlib.figure.Figure at 0x15727fdd4a8>"
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      },
      "metadata": {},
@@ -1763,7 +1766,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c723b0c50>"
+       "<matplotlib.figure.Figure at 0x15727d08710>"
       ]
      },
      "metadata": {},
@@ -1790,7 +1793,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c7295c4e0>"
+       "<matplotlib.figure.Figure at 0x15728523128>"
       ]
      },
      "metadata": {},
@@ -1824,7 +1827,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c72454ba8>"
+       "<matplotlib.figure.Figure at 0x157284ed4a8>"
       ]
      },
      "metadata": {},
@@ -1851,7 +1854,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c72b69400>"
+       "<matplotlib.figure.Figure at 0x15728578f60>"
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      },
      "metadata": {},
@@ -1878,7 +1881,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c72b6a7b8>"
+       "<matplotlib.figure.Figure at 0x15727fe79e8>"
       ]
      },
      "metadata": {},
@@ -1907,7 +1910,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x25c7244fd68>"
+       "<matplotlib.figure.Figure at 0x15727fe75c0>"
       ]
      },
      "metadata": {},