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     "source": [
      "[Table of Contents](http://nbviewer.ipython.org/github/rlabbe/Kalman-and-Bayesian-Filters-in-Python/blob/master/table_of_contents.ipynb)"
     ]
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     "metadata": {},
     "source": [
      "Gaussian Probabilities"
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     "input": [
      "#format the book\n",
      "%matplotlib inline\n",
      "from __future__ import division, print_function\n",
      "import matplotlib.pyplot as plt\n",
      "import sys\n",
      "sys.path.insert(0, '../code') # allow us to format the book\n",
      "import book_format\n",
      "book_format.load_style()"
     ],
     "language": "python",
     "metadata": {},
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     "metadata": {},
     "source": [
      "Introduction"
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     "metadata": {},
     "source": [
      "The last chapter ended by discussing some of the drawbacks of the Discrete Bayesian filter. For many tracking and filtering problems our desire is to have a filter that is *unimodal* and *continuous*. That is, we want to model our system using floating point math (continuous) and to have only one belief represented (unimodal). For example, we want to say an aircraft is at (12.34381, -95.54321,2389.5) where that is latitude, longitude, and altitude. We do not want our filter to tell us \"it might be at (1,65,78) or at (34,656,98)\" That doesn't match our physical intuition of how the world works, and as we discussed, it is prohibitively expensive to compute.\n",
      "\n",
      ">So we desire a unimodal, continuous way to represent probabilities that models how the real world works, and that is very computationally efficient to calculate. As you might guess from the chapter name, Gaussian distributions provide all of these features.\n",
      "\n",
      "Before we go into the math, lets just look at a graph of the Gaussian distribution to get a sense of what we are talking about."
     ]
    },
    {
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     "input": [
      "from stats import plot_gaussian\n",
      "\n",
      "plot_gaussian(mean=100, variance=15**2, xlabel='IQ')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
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/yVebEo2OIyIGUJEWEYthNpsZP38zOXmF3NqhIb1a1zU6kshl+Xm78dK9HQGY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\naYT/D5zOJt1Jfr++AAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7fd656243668>"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Probably this is immediately recognizable to you as a 'bell curve'. This curve is ubiquitous because under real world conditions most observations are distributed in such a manner. In fact, this is the bell curve for IQ (Intelligence Quotient). You've probably seen this before, and understand it. It tells us that the average IQ is 100, and that the number of people that have IQs higher or lower than that drops off as they get further away from 100. It's hard to see the exact number, but we can see that very few people have an IQ over 150 or under 50, but a lot have an IQ of 90 or 110. \n",
      "\n",
      "This curve is not unique to IQ distributions - a vast amount of natural phenomena exhibits this sort of distribution, including the sensors that we use in filtering problems. As we will see, it also has all the attributes that we are looking for - it represents a unimodal belief or value as a probability, it is continuous, and it is computationally efficient. We will soon discover that it also other desirable qualities that we do not yet recognize we need.\n",
      "<p>\n",
      "<p>"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Nomenclature"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "A bit of nomenclature before we continue - this chart depicts the *probability density* of of a *random variable* having any value between ($-\\infty..\\infty)$. What does that mean? Imagine we take an infinite number of infinitely precise measurements of the speed of automobiles on a section of highway. We could then plot the results by showing the relative number of cars going at any given speed. If the average was 120 kph, it might look like this:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot_gaussian(mean=120, variance=17**2, xlabel='speed(kph)')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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Xfpm0Wq2mI4lIGVGRFpFyl52bXzSf7sRhnajp7mw4kUjFem5oMDXdndm87xTf\n7Yo3HUdEyoiKtIiUu3e+jeb4+TRa1a/FqP6tTMcRqXDeni5MHHbpLzEvf7CN7Nx8w4lEpCyoSItI\nuTqbnMn8z/YAMH10dxzs9bYjtmn0gEAC6tci/lwa73wbbTqOiJQBfaKJSLl6dcVPZGTnMbBjI/oE\n+ZuOI2KMg70dM0Z3ByDi092cSc4wnEhEbpSKtIiUm1+PnefjjQdxtLfjpcvz6YrYsj5B/gwKbkRm\nTj6vLP/JdBwRuUEq0iJSLqxWK9OWRmG1wqODg2hap4bpSCKVwrSR3XBysGPlpkPsPnLOdBwRuQFX\nLdInT54kJCQEd3d3goOD2bdv3zVteP78+dSpUwdvb28mT55cbNmsWbNo06YN9vb2LFmypNiykJAQ\nXF1d8fT0xNPTkzFjxlzHwxGRyuLzbUf56eBZfLxc+Ms9N5uOI1JpNPbzYuzgIABNhydSxV21SI8b\nN4527dqRlJREaGgooaGhV93o9u3bCQ8PZ926dURHRxMZGcmKFSuKljdr1oy33nqL4OBgLBZLsXUt\nFgsLFiwgLS2NtLS03xVtEan8snLzmf3RDgBeHN4ZLzcnw4lEKpdn7r4Z3xqu7Dp0jk+3HjEdR0RK\nqcQinZqaypo1awgLC8PZ2ZkJEyYQHx9PdHTJZxuvXLmSoUOHEhgYSL169Rg7diyRkZFFy0eMGEH/\n/v1xdv7juWT127lI1favr34lITGd1g29eSCkpek4IpWOp5sTL97fCYDZkTvIzM4znEhESqPEIn34\n8GFcXFxwd3end+/eHDt2jGbNmrF///4SN3rw4EECAgKIiIhg4sSJtG7dmgMHDlxzqEmTJuHr68vA\ngQOvui8RqVxOJaaz4ItfAHj5oR7Y2+lUDJE/cn+flgQ19uF0UgZvf/Wr6TgiUgoOJS3MyMjAw8OD\ntLQ0YmNjSU5OxtPTk4yMkqfs+W29mJgY4uPjGTJkCOnp6dcU6I033iAoKIiCggJmzpzJXXfdRUxM\nDA4Ov4/q4+NzTduU4hwdHQE9f7aoIsb+uXe2kpWTz329ArijV1C57Ueun177lc/fnxrMLRM/4O2v\nfmX8vd1o4OtVLvvR2Ns2jX/5KbFIu7u7k56eTv369blw4QIAaWlpeHh4lLjR39aLiIgAYPXq1Vdd\n5zfBwcFF/54zZw4LFixg//79BAX9/gN55syZRf/u06cPffv2vaZ9iEj52B6bQOS6fTg72jNnbD/T\ncUQqvV5/f+AqAAAgAElEQVRBDRjWpxUrN+5nyrvrWRp2l+lIIjZvw4YNbNy4seh2v35X/jwrsUg3\nb96crKwsEhIS8Pf3Jzc3lyNHjhAQEFBigJYtWxY7JCMmJoZWrUp3WWCLxXLFY6afeOKJYrcTExNL\ntQ9b89tvpHq+bE95jn1hoZUJ//gWgHG3tcXTsUD/xyoZvfYrp+fv68CXUYf4eH0MD/ZtTueWfmW+\nD429bdP4X5+goKBiX+DGxsZe8b4lHrzo5eXFoEGDmDt3LtnZ2cybN49GjRoV23hISAhhYWHF1hs+\nfDirVq0iJiaGhIQEFi1aVGy2j/z8fLKzsyksLCQ3N5fs7GysVispKSl888035OTkkJOTQ3h4OH5+\nfrRu3fq6nwQRqVirthxm95Hz+NV04+m7OpiOI1Jl1Pf15LHb2wEwfdlWCgt1wr1IVXHVs4AWLlzI\n3r178fb25uOPP2b58uXFlsfHx3PuXPEJ5bt06cL06dPp168fbdu2JTQ0lOHDhxctHzt2LG5ubmzb\nto1x48bh5ubGpk2byMvLY8qUKdSuXZu6deuybds2vvjiC+zt7cvo4YpIecjIzuOV5ZemuwsL7Yy7\ni6PhRCJVy5N3tqdOLTd+OXqBlZsPmY4jItfIYq2ic82tXbuWwMBA0zGqJP2Jx3aV19i/tmInEZ/u\npkNTX74Ivxs7O8vVV5IKp9d+5bZy0yH+8q/1+NV0Y+Mbw/FwLbv51zX2tk3jf2NiY2MZMGDAHy7T\nvFQickNOnk9j4eWpu2aM7q4SLVJK9/Vszs3NfDl7MZO3Pv/FdBwRuQYq0iJyQ2Z9tIPsvALu6d6s\nXE6SErEVdnYWwkd3B+A/3+zl+LlUw4lE5GpUpEWk1LbvP80X24/i4mTP5BFdTMcRqfKCW/hxX8/m\n5OQVMPPDHabjiMhVqEiLSKkUFlqZvmwbAE/c0R5/n2ubK15ESjYptDOuzg58/dMxomJPm44jIiVQ\nkRaRUvl440H2xl2grrc7T9zR3nQckWqjno8HT15+TU1bupWCwkLDiUTkSlSkReS6pWXm8srynwCY\n8kAXXJ1LvLaTiFynx29vRz0fd2KOJxG5/qDpOCJyBSrSInLd5n+2mwupWXRq4cc9PZqZjiNS7bg6\nOzB1RFcAXl3xE6mZuYYTicgfUZEWkesSdzaVd76NBiB8dHcsFk13J1Ie7urWlM4t/UhMzSbi092m\n44jIH1CRFpHrMvPDbeTmFzK8dws6NPM1HUek2rJYLLz8UHcsFnj322iOnkkxHUlE/oeKtIhcs837\nEvh2Zzxuzg6EhXY2HUek2mvXxJf7+7Qkr6CQmR9sNx1HRP6HirSIXJP8gkJmXJ7u7um7O1Cnlrvh\nRCK2Iez+zri7OPL9z/Fs3HvSdBwR+S8q0iJyTT5ct5/YE0k08PVg3JC2puOI2IybarrxzN0dAJjx\n/jbyCzQdnkhloSItIleVkpHD6yt3ATB1RFdcnDTdnUhFGjs4iIa+nhw4mcz7a2NNxxGRy1SkReSq\n5q3+maS0bLq1qsPtXZqYjiNic1ycHHhp5KXp8F7/ZBfJ6dmGE4kIqEiLyFUcPnWR977fh8Wi6e5E\nTBrSqTHdA+tyMT2Heat+Nh1HRFCRFpGrePmDbeQXWBnRN4CgxrVNxxGxWRaLhfDR3bGzWFi8JoZD\nCcmmI4nYPBVpEbmidb+cYO2eE3i4OPLC/Z1MxxGxeW0a+TCiXwAFhVbC399mOo6IzVORFpE/lJdf\nyIzLH9TP3tcR3xpuhhOJCMALwzrh6erIul9PsnbPcdNxRGyairSI/KGlP8Rw+NRFGvt58cigNqbj\niMhltWu4MuHejgCEv7+NvHxNhydiioq0iPxOUlo2b14+mWn6yG44OdgbTiQi/+2RQW1oUseLI6dT\nWLxmn+k4IjZLRVpEfudvn+ziYkYOvYP8ubVjQ9NxROR/ODnYM21kNwDmrbo0PaWIVDwVaREp5sDJ\nJJatjcXOYmHGqG6a7k6kkrr15ob0CfInJTOX11fuNB1HxCapSItIEavVyrSlURQUWhk9IJBWDbxN\nRxKRK7BYLMwY3Q17Owvvr91P7PEk05FEbI6KtIgU+X5XPJv3naKmuzMThwWbjiMiVxFQ35uHbgmk\n0Gpl+vtRWK1W05FEbIqKtIgAkJNXQPgHl6a7mzgsGG9PF8OJRORa/PW+YGq6O7Nl3ym+2xVvOo6I\nTVGRFhEA3vl2L/Hn0gioX4vRAwJNxxGRa+Tt6cJzQy9Nhzfzw+3k5BUYTiRiO1SkRYSzyZlEfLoH\ngBmju+Ngr7cGkapk9IDWtKhXk7izqbz7bbTpOCI2Q5+WIsIry3eQkZ3HoOBG9AnyNx1HRK6To4Md\nM0Zfmg4v4tPdnE/JNJxIxDaoSIvYuN1HzrFi0yGcHOx46cGupuOISCmFtGvAgA4NSM/O49WPNR2e\nSEVQkRaxYYWFVl5aEgXAn4e0pUmdGoYTiciNmDayGw72FiI3HCA67oLpOCLVnoq0iA1bvfUwu4+c\n46aarjxzdwfTcUTkBjWvV5M/DWyD1QrTlmo6PJHypiItYqMysvOYE7kDgEmhXfBwdTKcSETKwrP3\ndsTb04XtB87w5Y5jpuOIVGsq0iI26q3P93AmOZMOTX0Z1quF6TgiUkZquDvz/OULKs36cDtZufmG\nE4lUXyrSIjYo/lwq//56LwDhD3XHzs5iOJGIlKUH+7UisIE3Jy+kF73WRaTsqUiL2KBZly/acF/P\n5nRq4Wc6joiUMQf7/58O7x+f7yHhQprhRCLVk4q0iI1ZtyeOr3+Kw9XZgckPdDEdR0TKSa82/gzu\n1IjMnHxeem+96Tgi1ZKKtIgNyS8o5Pl/rQXg6bs6UNfb3XAiESlPLz3YDScHOz5cu48d+0+ZjiNS\n7ahIi9iQRd/sITruPA18PRh3W1vTcUSknDX28+LPQy691p//1w+aDk+kjKlIi9iI5PRswpduAi5d\ntMHVycFwIhGpCM/c3QG/Wu5s33+K1VuPmI4jUq2oSIvYiNdX7CIxNYu+7RsypFNj03FEpIJ4uDoR\n/nAfAGZ/tIPM7DzDiUSqDxVpERsQHXeBZWtjsbez8Ob4W7FYNN2diC156NZ23NzcjzPJGfzzy19N\nxxGpNlSkRao5q9XKlMVbKbRaeeLuYNo09jUdSUQqmJ2dhdcfvwWAt7/8hYQL6YYTiVQPKtIi1dwn\nmw+z89BZanu5MnVUL9NxRMSQXkENuLNrU7LzCpgducN0HJFqQUVapBpLy8xl1kfbAZgyogs13F0M\nJxIRk6aO6IKLoz2fRR3hpwNnTMcRqfJUpEWqsTdX/cz5lCw6tfBjWK8WpuOIiGH1fT157PZ2ALy0\nNIqCwkLDiUSqNhVpkWrqwMkk3v0uGosFZj/cAzs7nWAoIvDUne2pU8udvXEX+ODH/abjiFRpKtIi\n1ZDVamXqkq0UFFoZPSCQoMa1TUcSkUrCzcWRGaO7AfDqxztJTM0ynEik6lKRFqmGvth+lK0xp6nl\n4cwLwzuZjiMilcwdXZrQO8ifixk5vLL8J9NxRKosFWmRaiYjO4+XP7h0gmFYaGdqeegEQxEpzmKx\nMGtMDxzt7fho/QF2HTprOpJIlaQiLVLNzP9sD6eTMmjXpDYjQgJMxxGRSqp5vZpFJx5OXrxFJx6K\nlIKKtEg1cuT0RRZ+demqZbMf7om9nV7iInJlf7m7A/V83ImOS2TZWp14KHK99CkrUk1YrVamL40i\nr6CQB/q2pGPzm0xHEpFKzs3FkfDR3QF47eOfuJCiEw9FroeKtEg18d2ueNb9ehIvNycmhXYxHUdE\nqoghnRoT0q4+KZm5zFmuKx6KXA8VaZFqICM7j5eWbgXgheGdqF3D1XAiEakqLBYLM8f0wMnBjuUb\nDuqKhyLXQUVapBqYt+pnTiVeOsHwoVsCTccRkSqmaZ0ajL+jPXDpxMP8Ap14KHItVKRFqrjY40n8\n+5u9WCww95FeOsFQRErl6bs6UL+2BzHHk1j6Q4zpOCJVgj5xRaqwwkIrk97bTEGhlYcGtKZ9U1/T\nkUSkinJ1duDl3048XLGT8ymZhhOJVH5XLdInT54kJCQEd3d3goOD2bdv3zVteP78+dSpUwdvb28m\nT55cbNmsWbNo06YN9vb2LFmypEz2J2KLPt54kJ8OnsW3hisv3q8rGIrIjRkY3Ij+HRqQlpXHzA+3\nm44jUuldtUiPGzeOdu3akZSURGhoKKGhoVfd6Pbt2wkPD2fdunVER0cTGRnJihUripY3a9aMt956\ni+DgYCwWyw3vT8QWJaVlM+ujSx9000Z2o4a7s+FEIlLVWSwWZj7UA2dHez7ZfJgt+06ZjiRSqZVY\npFNTU1mzZg1hYWE4OzszYcIE4uPjiY6OLnGjK1euZOjQoQQGBlKvXj3Gjh1LZGRk0fIRI0bQv39/\nnJ2Lf/CXdn8itmhO5A6S03Po2aYe9/ZoZjqOiFQTjf28eObuDgCEvbeZnLwCw4lEKq8Si/Thw4dx\ncXHB3d2d3r17c+zYMZo1a8b+/SVf/ejgwYMEBAQQERHBxIkTad26NQcOHLhqmNLuT8TW/HTgDB+t\nP4CTgx1zHu75u7/siIjciPF3tKd5vZocPZ3Cgs/3mI4jUmk5lLQwIyMDDw8P0tLSiI2NJTk5GU9P\nTzIyMkrc6G/rxcTEEB8fz5AhQ0hPT79qmOvdn4+Pz1W3Kb/n6OgI6PmrqvLyC5i69FMA/jq8G13b\nXvu30Rp726bxt12lGfu3n72NW5//kLe++IUxQ4Jp2UD/b6oqvfbLT4lF2t3dnfT0dOrXr8+FCxcA\nSEtLw8PDo8SN/rZeREQEAKtXr77qOqXZ38yZM4v+3adPH/r27XvVfYhUdQs+20l03Hma1K3Jiw90\nNx1HRKqp3m0bMmZgO5Z8/ytP/+M7vp07Qn/9EpuwYcMGNm7cWHS7X79+V7xviUW6efPmZGVlkZCQ\ngL+/P7m5uRw5coSAgIASA7Rs2bLY4RgxMTG0atXqqsGvd39PPPFEsduJiYlX3Yf8/2+ker6qnoTE\ndF5eugmAl0d1IzM9lcyr/7GniMbetmn8bVdpx37ife35IuogG345zsLPtjG8d8vyiCflTK/96xMU\nFERQUFDR7djY2Cvet8RjpL28vBg0aBBz584lOzubefPm0ahRo2IbDwkJISwsrNh6w4cPZ9WqVcTE\nxJCQkMCiRYuKzb6Rn59PdnY2hYWF5Obmkp2djdVqvab9idiyaUu3kpmTz22dm9C/QwPTcUSkmvP2\ndGHayK4AhL+/jaS0bMOJRCqXq05/t3DhQvbu3Yu3tzcff/wxy5cvL7Y8Pj6ec+fOFftZly5dmD59\nOv369aNt27aEhoYyfPjwouVjx47Fzc2Nbdu2MW7cONzc3Ni0adM17U/EVn390zG+3RmPh4sj4aO7\nmY4jIjZiWK8W9Ghdl+T0nKIpN0XkEovVarWaDlEaa9euJTAw0HSMKkl/4ql6UjJy6PfCSs5ezGT2\nmB48PLBNqbajsbdtGn/bdaNjf+T0RW4J+4Tc/EJWTr2D7oF1yzKelDO99m9MbGwsAwYM+MNlukS4\nSBUwJ3IHZy9mEtziJh66pbXpOCJiY5rVrcnTd12eW3qR5pYW+Y2KtEglt33/ad7/cT+O9na8PrY3\ndnY6a15EKt6Td3Wgad0aHD51kX9++YvpOCKVgoq0SCWWk1fAC+9uBuDJu9oTUN/bcCIRsVXOjva8\n+kgvAN76bA9HTl80nEjEPBVpkUrsrc/2cPjURZrXq8kzd99sOo6I2Lgeretxf5+W5OQV8Px/NlFY\nWCVPsxIpMyrSIpXUgZNJ/OPypXlfH9sbZ0d7w4lERGDayK7U9nJl+4EzLPvxyvPritgCFWmRSqiw\n0MoL72wmr6CQUf1b0SWgjulIIiIA1PJwYfbDPQCY/dEOEi5cx1WhRKoZFWmRSmjp2lh2HjqLX003\npozoajqOiEgxd3Rtym2dG5ORnUfYos1U0Zl0RW6YirRIJXMqMZ1XIncAMOvhHni5ORlOJCLye7PG\n9KSGmxM//nKCVVsOm44jYoSKtEglYrVaeXHRZtKz8xjcqRG3dW5iOpKIyB/yq+XG9FGXrrI6bVkU\nF1KyDCcSqXgq0iKVyMrNh/hxzwlquDkx5+FepuOIiJTo/j4t6RPkz8X0HF5autV0HJEKpyItUkmc\nSc5g+tIoAMIf6o5fLTfDiURESmaxWHhtbG/cnB34fNtRvtsZZzqSSIVSkRapBKxWK2GLNpOSmUv/\nDg0Y1quF6UgiItekga8nYfd3BmDSe1tIycgxnEik4qhIi1QCq7ceYc3Px/F0deTVR3phsegy4CJS\ndTw8sDXBLW7i7MVMZn643XQckQqjIi1i2LmLmUXHFs4Y1Z16Ph6GE4mIXB97Ozv+9uc+ODnY8dH6\nA6z/9YTpSCIVQkVaxCCr1crk97ZwMT2HkHb1Ce3b0nQkEZFSaeFfi+eGBgPw3L836RAPsQkq0iIG\nfb7tKN/sjMPDxZHXHu2tQzpEpEp7/PZ2dGx+E2eSM5i2LMp0HJFypyItYkhiahZTl1w6pOOlkV3x\nr61DOkSkanOwt2PeY31xcbRn5aZDfL8r3nQkkXKlIi1igNVq5cV3N5OUlk2vNvUY2a+V6UgiImWi\neb2ahIVemsXjhXc3kZSWbTiRSPlRkRYxYMWmQ3yzMw5PV0feHNdXh3SISLXy6KAgugfW5XxKFpPf\n22I6jki5UZEWqWAnz6fx0uVDOmaO6aFDOkSk2rGzs/C3cX1wc3bgi+1H+XzbEdORRMqFirRIBSos\ntDJh4QbSs/MY0qmxLrwiItVWo5u8mDayGwCT39vCuYuZhhOJlD0VaZEK9M530UTFnqa2lyuvPqoL\nr4hI9Taqfyv6tvUnOT2HF9/djNVqNR1JpEypSItUkIMnk5m7/CcAXv9zb3y8XA0nEhEpXxaLhdf/\n3AcvNye+/zmej9YfMB1JpEypSItUgNz8Ap55ez05eQWMCAlgYMdGpiOJiFQIfx8P5jzcE4Bpy6I4\neibFcCKRsqMiLVIB/r56N3vjLtDA14MZo7qZjiMiUqHu7dmce3s0Iysnn6cXrCMvv9B0JJEyoSIt\nUs52HTrLW5/twWKBiMdD8HB1Mh1JRKTCzX64J/4+Huw5ep55q382HUekTKhIi5Sj1MxcnlqwjkKr\nlcdva0fXVnVNRxIRMaKGuzPzx4dgscBbn+1hx4EzpiOJ3DAVaZFyYrVamfzeZo6fT6Nt49q8cH8n\n05FERIzqFliXJ+/sQKHVyjNvryM1M9d0JJEboiItUk5Wbj7E6q1HcHN2YMFT/XBysDcdSUTEuOeG\ndqRdk9qcOJ/O1CW66qFUbSrSIuXg2JkUpiy+dPXCWWN60KxuTcOJREQqBycHe956oh+uzg58svkw\nn0XpqodSdalIi5Sx3PwCnlzwIxnZedzVrSn392lpOpKISKXSvF5Npl++6mHYos2cOJ9mOJFI6ahI\ni5Sx11fs5JejF6hf24O5j+jqhSIif2RU/1YM7tSI1Mxcxr/1o6bEkypJRVqkDG2MTuCfX/6KncXC\nP57oRw13Z9ORREQqJYvFwht/7oO/jwe7j5xj7sc/mY4kct1UpEXKSGJqFn95ex0Af72vI50D6hhO\nJCJSudXycOGfT/fH3s7Cv776lR92HzcdSeS6qEiLlIHCQivPvL2ecxez6BpQh2fu6WA6kohIldCp\nhR9h93cGYMK/1nMqMd1wIpFrpyItUgbe+nwP6389SS0PZ956sh/2dnppiYhcq8dvb0e/dvVJTs/h\nqQXryC/Q8dJSNejTXuQGbdl3ijdW7gLgrSf64e/jYTiRiEjVYmdnIWJ8CHVqubH9wBneXKVLiEvV\noCItcgPOXczkyQU/XrpK190d6Ne+gelIIiJVko+XK/94sj92FgvzP9vNxugE05FErkpFWqSUCgoL\neXLBj5xPyaJ7YF2eGxpsOpKISJXWPbAuf72vI1YrPL1gHWeSM0xHEimRirRIKb256me2xpzGt4Yr\nC57sj4O9Xk4iIjfqmXs60LNNPS6kZvFYxFpy8wtMRxK5In3yi5TC+l9PEPHpbuwsFhY82R+/Wm6m\nI4mIVAv2dnb888n+1Knlzs5DZ5n14XbTkUSuSEVa5DqdOJ/GkwvWYbXCc0M70rNNPdORRESqldo1\nXPn3XwbgaG/Hu9/t49Oth01HEvlDKtIi1yErJ59H563hYnoO/Ts04Jm7bzYdSUSkWgpu4ceM0d0B\nmPjOJvafSDKcSOT3VKRFrpHVauWFdzexLz6Rxn5e/OOJftjZWUzHEhGptsbcEsjQXs3Jysln7N/X\nkJqZazqSSDEq0iLXaNF3+1i15TBuzg68++yt1HB3Nh1JRKRas1gsvPpIbwIbenPsTCrPLlyP1Wo1\nHUukiIq0yDWIij1N+AfbAHjzsb60auBtOJGIiG1wdXbgnQm34uXmxLc744n4dLfpSCJFVKRFriIh\nMZ3H5v9AQaGVJ+5ox51dm5qOJCJiUxr7eTF/fAgWC7y+chff7YwzHUkEUJEWKVF2bj7j/v4DianZ\n9A7y58X7O5uOJCJik27t2Iiwy+/BT7+9ntjjOvlQzFORFrkCq9XKxP9sZM/R89Sv7cE/n9JFV0RE\nTHryzvbc070ZGdl5PPLm9ySlZZuOJDZOrUDkCiI+3c3qrUdwd3HkvecG4u3pYjqSiIhNs1gsvDGu\nD+2b1ub4+TTGRfxAXn6h6Vhiw1SkRf7Al9uP8vrKXVgs8I8n+9G6oY/pSCIiArg6OfDuswO5qaYr\nUbGnmb4synQksWEq0iL/49dj5/nLv9YDMHVEVwZ2bGQ2kIiIFFPX2513nx2Is6M9S36IYekPMaYj\niY1SkRb5L6eTMvjT374nO7eAB/q25LHb2pqOJCIif6Bj85t47dHeAExdspV1v5wwnEhskYq0yGVZ\nOfk88ub3nEnOpFurOrzySC8sFl25UESkshrWuwVP392BgkIrj89fy774RNORxMaoSIsABYWFPPXP\nH/n12AUa3eTJfybcipODvelYIiJyFS8M68Q93ZuRnp3HQ69/x+mkDNORxIaoSIvNs1qtTFsaxbc7\n46nh5sTi5wZphg4RkSrCzs7Cm4/1pWtAHc4kZ/DQ69+SnpVrOpbYCBVpsXlvf/kri9fE4ORgx6K/\nDqRl/VqmI4mIyHVwdrTnnWdvpUkdL2KOJzH+rR/JL9C0eFL+rlqkT548SUhICO7u7gQHB7Nv375r\n2vD8+fOpU6cO3t7eTJ48udiy9evXExAQgIeHB/feey+pqalFy0JCQnB1dcXT0xNPT0/GjBlznQ9J\n5Nqt3nKY2ZE7AIgYH0K3wLqGE4mISGl4e7qw7PnBeHu68OMvJ5iyeAtWq9V0LKnmrlqkx40bR7t2\n7UhKSiI0NJTQ0NCrbnT79u2Eh4ezbt06oqOjiYyMZMWKFQBkZmYyfPhwwsPDOX/+PBaLhUmTJhWt\na7FYWLBgAWlpaaSlpbFkyZIbeHgiV7Z5XwLPLtwAwPRR3birWzPDiURE5EY0qVOD9567NC3e+z/u\n5++rd5uOJNVciUU6NTWVNWvWEBYWhrOzMxMmTCA+Pp7o6OgSN7py5UqGDh1KYGAg9erVY+zYsURG\nRgKwbt06atasyQMPPICrqysTJ05k+fLlxdbXb5BS3mKPJzF23hryCgoZOziIcUM0zZ2ISHXQqYUf\nC57sh53Fwhuf7GKJ5piWclRikT58+DAuLi64u7vTu3dvjh07RrNmzdi/f3+JGz148CABAQFEREQw\nceJEWrduzYEDBwA4cOAArVq1YsuWLQwaNIjmzZuTlJREYuL/T1kzadIkfH19GThw4FX3JXK9jp9L\nZdRr35CWlcftXZowfWQ305FERKQMDenchLmP9AJgyuItfL7tiOFEUl05lLQwIyMDDw8P0tLSiI2N\nJTk5GU9PTzIySp5a5rf1YmJiiI+PZ8iQIaSnpxdbdubMGWJjY3F2dgYgPT0dHx8f3njjDYKCgigo\nKGDmzJncddddxMTE4ODw+6g+Prpsc2k4OjoCtvn8nU5M58HXvuNMcia92jbgg6lDcXEq8WVQrdjy\n2IvG35bZ4tg/M7wn2QV2TFu8gWfeXk+Dur7c0rGJ6VhG2OL4V5QSG4S7uzvp6enUr1+fCxcuAJCW\nloaHh0eJG/1tvYiICABWr15dtM5vy4YOHcrQoUNJTk4GKFoeHBxctJ05c+awYMEC9u/fT1BQ0O/2\nM3PmzKJ/9+nTh759+171AYvtSkzN4vbJkRw7fZGOLeqwasYwmyrRIiK25vnQbpxPyeCt1TsJfXkV\n3736IJ0CdFK5lGzDhg1s3Lix6Ha/fv2ueN8SW0Tz5s3JysoiISEBf39/cnNzOXLkCAEBASUGaNmy\nZbFDMmJiYmjVqlXRsrfffrvYMm9v7yv+lmSxWK54zPQTTzxR7PZ/Hx4iV/bbc21Lz1d6Vi4PvPI1\nMfEXaFGvJkueu5W87HQSs9NNR6tQtjj28v80/rbLlsf+hfs6kHDuIqu2HObOKZGsnnYnLfxta5pT\nWx7/0ggKCir2BW5sbOwV71viMdJeXl4MGjSIuXPnkp2dzbx582jUqFGxjYeEhBAWFlZsveHDh7Nq\n1SpiYmJISEhg0aJFRbN99O/fn5SUFD766CMyMjJ44403ipalpKTwzTffkJOTQ05ODuHh4fj5+dG6\ndevrfxZELsvOzedPb37P7iPnaeDrwUeTbtMFV0REbISdnYU3x/Wlf4cGJKfnEDrna46eSTEdS6qJ\nq05/t3DhQvbu3Yu3tzcff/zx72bYiI+P59y5c8V+1qVLF6ZPn87/tXfncVWWeR/HP4d9ExAQZBFE\nURF3U3A3tSzNHdHUsqmnbWyypsdm2i0dHR2tJhvzKZts1VxKTc00xSXNrcVcUFCQfRPZZBPwnOcP\nxjORO6mH5ft+vc6L433d97l/px/k15vrvs6AAQPo0KED48ePJyoqCgAnJydWrlzJq6++ire3NwBz\n5srFLZEAACAASURBVMwBoKKighdffBEvLy98fX3Zu3cv69atw9paH9UsNVNRaeSPb0fzfUwG3u6O\nfP78Pfh6OFu6LBERuYVsbax4b+od9ArzJSu/hHGzNpCcXXj1A0WuwmCqo2vNbd26lbZt21q6jDqp\nofyKp6LSyJR/RfP1gVO4O9vzxcvDCG3mYemyLKqh9F4uTf1vuNT7KsVlFUyau5EDcVkEeLnw5cvD\n8fe68n1f9YH6//scO3aMQYMGXXJMHxEu9VJFpZEnFlaFaFcnOz7765AGH6JFRBo6ZwdbPnn2brq0\n9CY1p4hxszeQkXvllchErkRBWuqdikojf3onmg37q0L0sueG0rllE0uXJSIitUAjJzs+++vddAz2\nIjGrkPGzN5CdX2LpsqSOUpCWeqXyfFWIXr/vFI0cbVn63BCFaBERqcbN2Z6lzw2hbaAH8RkFjJu1\ngcw8XZmW66cgLfVG5fmq6RwXQvSy54fSpaW3pcsSEZFaqLGLA8ufH0poQGNOpOcTOXM9aTkNa0lU\n+f0UpKVeOFdxnj++/esr0QrRIiJyZZ6ujqx8aRjtm3uSmFXImJnrSNJqHnIdFKSlzispq+DB1zdV\nu7Gwa4hCtIiIXJ1HIwdWvHAPXUOqbkAcM2MdJ9PzLV2W1BEK0lKnFZaUM3HuRnYcTsPT1YGVLw7j\ntlY+li5LRETqEDdne5Y9N4QeoU3JzCshcuZ6jiXnWrosqQMUpKXOyikoJWrWeg7EZeHr4cyXLw+n\nffNLf9S8iIjIlbg42vHpX4bQr70/OYWljJ21np9OZl/9QGnQFKSlTko/U8SYmes4kniG4KaurHll\nOCF+7pYuS0RE6jBHexuW/O9g7uwaSH7ROcbN3kD0wRRLlyW1mIK01DlxqXmMem0d8RkFtA30YPUr\nwwlo0sjSZYmISD3gYGfD4qfuZFy/1pSeq+QPr29ixc44S5cltZSCtNQp38ekM/K1r0g7U8RtrbxZ\n9dIwmrg5WbosERGpR2xtrHjj0X78aURnzhtN/PndHSxcdxCTyWTp0qSWUZCWOmPtnngmzd1IYUk5\nQ7o1Z/kL9+DubG/pskREpB4yGAw8P747M+7vicEAsz8/wPRP92I0KkzLfylIS61nMpl4Z90vTPlX\nNOWVRv7nrna8+9QgHO1sLF2aiIjUc/9zd3sWPjEQW2sr/v3NER5bsJXSc5WWLktqCSURqdUqzxuZ\n/skePvw2BoBXJkXw6JAOGAwGC1cmIiINxcieLfF0deDhN7/l6wOnSM05y5L/HUzTxs6WLk0sTFek\npdbKLz7H5Hnf8OG3MdjZWLHoyYE8NrSjQrSIiNxyfdr589WrIwnybsShUznc8/IaDp/KsXRZYmEK\n0lIrnUjLY9gra9hxOA2PRg4sf+EeRvRoaemyRESkAWsd0Jj1M0YR0abqg1tGz1zH1wdOWbossSAF\naal1th5MZvj0tZzKLCQs0IONM0cR3qappcsSERHBo5EDy54fSlTfVpSeq+SRf25hwdqftaJHA6Ug\nLbXGhZsKH5i/ibOlFQztHsza6SO0RrSIiNQq9rbWvPlYf168NxyDAeau+IFH/rmFwpJyS5cmt5hu\nNpRaoai0nGff/46v9iYAMC3yNp4a1QUrK82HFhGR2sdgMDBleCdC/N15atF2Nv6QSGxaHu8/fQdt\nAjwsXZ7cIroiLRZ3LDmXoS+v4au9CTg72PL+03fw5zFdFaJFRKTWG9w1iK9njqJtMw8SMgq455W1\nrN0Tb+my5BZRkBaLWrEzjmHT1xCfUUBoQGO+njmKId2DLV2WiIjINQtu6sa610YypncIpecqmfKv\naKZ/sofyyvOWLk1uMgVpsYjS8kqmLd7Jn9/dQVn5ecb1a836GaMI8XO3dGkiIiLXzdHehgV/vJ1Z\nD/TCxtrA+98cYfRr60jMKrR0aXITKUjLLXc8JZfhr6xl2fZYHGytef2Rfrz5WH8c7TVlX0RE6i6D\nwcAfBrfji5eHE+DlwsGE0wx+4Uu+2HXC0qXJTaIgLbeM0Whi8cbDDH15DcdScglu6sq6GSO59/Y2\nli5NRETkhunWyofNs8cwLCKY4rIKpi7aztRF2ygq1aoe9Y2CtNwSmXnFTJq7kVc/3cu5ivNMGhDK\nplljCAv0tHRpIiIiN5ybsz3/9+Qg5j/SF0d7G77YdZK7XlzNTyezLV2a3EAK0nLTbdh/ikHPfcHO\nI1WfUvjBn+/kHw/3xdnB1tKliYiI3DQGg4EJt4fyzd9GExboQWJWISNf/YrZn++nrLzS0uXJDaAg\nLTfNmcJSnvhXNI++tYX8onMM6BjA1jmR3NWtuaVLExERuWVC/NxZP2MUf7ynIwAL1/3CkJdWczD+\ntIUrk99LQVpuOJPJxOrdJ+n/7ErW7InH0d6Gvz3Qi0/+cjfe7k6WLk9EROSWs7e15qWJEayePpwW\nvm7EpeUz4tW1zFlxgHMVWiavrlKQlhsq/UwRf3h9M396Zxt5Refo086PrXMieXBwOwwGfcCKiIg0\nbBduRHxsaAeMJhNvrz3I3S9+yf7YTEuXJjWg9cbkhjhvNPLJ1uPMWb6fs6UVuDrZMX1SD8b3b60A\nLSIi8iuOdja8MqkHQ7o158/v7SAuLZ/RM9Zxb//WvDghAo9GDpYuUa6RrkjL7/bDiSyGvryGFz/c\nzdnSCu66LYht/xjLvbe3UYgWERG5jO5tmvLt3yP58+iu2NlY8fmOOPpNW8HyHXGYTCZLlyfXQFek\npcZyCkqZ9fl+VuyMA8Df04VX76/6F7YCtIiIyNU52tkwbextjOrVkhc+3M3uo+k8894Olu+IZcbk\nXrRvrmViazMFabluFZVGPtkaw7xVP1JYUo6djRWP39ORqSO76NMJRUREaiDEz53lzw9l9ffxvPbp\nXvbFZnL3S19yb/82/CWqm27Wr6WUeuSamUwmNv6QyOzP93MqsxCAAR0DmPFAL1o0dbNwdSIiInWb\nwWBgTO8QBnVuxpurf2LJ5qMs2x7LV3sT+NOITjwypAOOdoputYm6IdfkQFwWf1u6jx9OZAHQwteN\nl+4NZ/BtQZrGISIicgO5Odvz6n09uX9QW/62dD+bf0pi7oof+Cz6OH8d152RPVtgbaXb3GoDBWm5\norjUPOat+oGvDyQC4OnqwDNjbmPSgFBsbfRDLCIicrO09HVnyf8O5rsjabz22V6OJefy5Dvb+NdX\nB3l27G3crXuSLE5BWi7pRFoeb67+ma/2xmMygYOdNY8O6cCUYZ1o5GRn6fJEREQajL7t/dk0azSr\nvjvBG1/+RGxqHg//cwudWnjxl6hu9O8QoEBtIQrSUs3J9Hze/PIn1v4nQNtaWzFhQBumjuyCr4ez\npcsTERFpkKytrBjfvw2jeoWwdNtxFqz9mV8Scpg09xvC2/gwdWQXbu+oQH2rKUgLAIdOnWbR+kOs\n33cKo8mErbUV997ehidHdMbfy8XS5YmIiAhVHzX+4OB23Nu/DUs2H2Xh+l/YH5vFff/4hvbNPXli\neCfuCQ/WHOpbREG6ATOZTOw4nMo76w+x+2g6UHUFemL/qgAd0KSRhSsUERGRS3G0t2HK8E7cP6gt\nn0Yf492vD3Mk8Qx/fDua4KauPDG8E6N7heCgVT5uKv3XbYDKyitZtfMYr6/Yw7HkXACcHWyZNCCU\nh4e0x99TV6BFRETqgkZOdvxxWCceHNyOFTvjeGf9L5zKLGTa4u/4+/ID3D+oLU+N7Y2v/m6/KRSk\nG5Dk7EI+jT7O8p1x5BSUAuDt7sjDd7fnvoFtcXO2t3CFIiIiUhMOdjZMviOMiQNCWbsnnv/bcIiY\n5Fz+ufpnFq77hci+bblvQAhdWnpbutR6RUG6njMaTWw/lMpHW2LYejAZk6lqe6eW3kweGMro3iHY\n21pbtkgRERG5IWysrYjs04oxvUPYdzyTf286wjc/JvH5tqN8vu0onVp4MXFAKCN7tNQqXDeAgnQ9\nlZRdyKrvTrDquxMknz4LgJ2NFcN7tGBqZE/CQ/3Izc21cJUiIiJyMxgMBnq09aVHW1+KKm14d91P\nfLDxIL8k5PBLwi5e/XQvwyNaMPH2NnRr7aPVPmpIQboeOVtSzvr9CazceYJ9sZnm7c2auHD/oLbc\n278Nnq6OeHp6WrBKERERuZWCfNyY/fAAptwTxsYDiSzddpw9xzJYsTOOFTvjCPFzJ6pvK0b2bEkz\nLTRwXRSk67jS8kp2HEpl3d4EvvkxkbLy80DV3bxDujUnql9reof5ahkcERGRBs7RzoYxvUMY0zuE\nhMwClm+PZcV3cZxMz+fvyw/w9+UH6NbKh1G9WjI8ogVebo6WLrnWU5Cug0rLK9n+Swrr953i25+T\nKS6rMI/1bOtLVN9W3BMejIuj5j6JiIjIxVo0deP5e8OZNrYb2w6lsOb7eDb/lMQPJ7L44UQW0z/Z\nQ592fgyLaMGdXQNp4uZk6ZJrJQXpOiL3bBnbfklhy8/JbD2YUi08dwz2YlhEMMMjWhDo7WrBKkVE\nRKQusbWxYnDXIAZ3DaK4rILNPyaxZk882w+lsONwGjsOp2EwwG0hPtzdLYjBtwXR0tfd0mXXGgrS\ntZTJZOJYSm5VcP45hZ9OZmO8sOQG0KmFF8PCW3BPRDBBCs8iIiLyOzk72DK6dwije4eQV1TGxgOJ\nbPwhkV1H0sxXqv+2bD+t/Ny5o0sg/ToGEN7ap0F/6EvDfee1UPqZInbHpLP7aDq7jqaTkVtsHrOz\nsaJHqB+DugQyuGugrjyLiIjITdPYxYGJA0KZOCCUotJyth9KZdOPSWz9OZkT6fmcSM9n0YZDONhZ\n0zPUl34dA+jfwZ/W/o0b1AogCtIWlFNQyp7jGew+ms7umHQSMgqqjXu7OzKocyCDOjejb3t/zXkW\nERGRW87F0Y5hES0YFtGCikoj+2Iz2HEolR2H0ziadIZth1LZdigVgKaNnegR6ktEaFMiQpvSyq8x\nVlb1N1grSN8i541G4lLzzb8a+SEui8Sswmr7uDjY0qOtL73b+dE7zI+2zTzq9TefiIiI1C22Nlb0\naedPn3b+vDgBsvNL+O5IGjsOp7LzcBqZeSWs2RPPmj3xALi72BPeuipUd2/tQ7sgz3o1FaT+vJNa\nxGQykXz6LEcSz3A4MYdf4k/zc3w2Z0srqu3naG9D1xBveof50aedHx2Dm2Bro2XqREREpG7wdnci\nsk8rIvu0wmQycTwlj32xmeyPzWTf8Uwy84rZ/FMSm39KAsDG2kBoMw86BTehU4uqR5uAxnU2/yhI\n/07nKs5zKrOAmORcDp/K4XBiDjFJZygoKb9o3wAvF7q18qFbax+6tfKhbaAHNtZ18xtHRERE5NcM\nBgNtAz1oG+jBH+4Mw2QykXL6bFWwPp7JjyeziUvL40jiGY4knuGzbccBcLC1JizIk7BAD9o28yC0\nmQehgR64O9tb+B1dnYL0NSotryQ+vYATaXnEpeVxIi2fuLQ8ErMKOW80XbS/l6sjHYO9aNfckw7N\nvega4o2vh7MFKhcRERG59QwGA4HergR6uxLVtzUAxWUVHEnM4WDCaQ4lVH1NzCrkp5PZ/HQyu9rx\nTRs707ZZY0KbedDKvzEtfN1o6etGYxf7WnNDo4L0rxSWlJOUVUhS9n8eWWfNz1NzijBdnJcxGKC5\njyuhzRrTvrkXHZp70b65Jz7uTrWmySIiIiK1gbODLRGhvkSE+pq35Ref4/CpHI6l5HI8JZfYlDxi\n0/LIzCsmM6/YfCPjBe7O9gQ3dSW4qRstfN1o0dSNIG9XArxc8HR1uKX5q8EE6YpKI1l5xWTkFpOe\nW/X1wiP9TBGJWYXkFZ277PHWVgaCfd1o7e9OK//G5q8tfN1wrEeT5kVERERuJXdne/q296dve3/z\ntvNGI8nZZzmeksvxlDxOpudzKquAhIwC8ovP8XP8aX6OP33RaznYWePv6UKAlwsBXo3w97rw3IWm\nHs54uzvd0Nx21VdKTU3lvvvu48CBA4SGhvLxxx/Trl27q77wggULmD17NuXl5Tz++OPMnj3bPLZ9\n+3Yee+wx0tLSuPPOO/noo49wdXW97vOdNxrJO3uOnMJScgpLOVNYRk7Br54XlpKVV0J6bhGnC0ov\neUX51xzsrAnydiXI25VA70Y096n6GuTtSrMmjbC3tb7q+xYRERGR38fayorgpm4EN3VjSPdg83aT\nyUROYSkJGQWcyiwkIbMqXCdlF5KWU0RBSTnxGQXE/2ZJ4V9zdbLD290Jb3dHfNyd/vPcCR93J5q4\nO+LZyBFPVwc8Gjlc9V62qwbpRx99lI4dO7Jp0ybeeustxo8fz5EjR654zL59+3jttdfYtWsXbm5u\n9OnThy5duhAVFUVJSQlRUVG8/fbbjBw5kkmTJvH888+zcOHC6z5f0OR/XzUcX2BlMODT2AlfD+eq\nh6czfheeezgT5O2Kt7tjg5mOcezYMby9vS1dhliAet+wqf8Nl3rfsNWX/hsMBpq4OdHEzana9JAL\nzpaUk5pTRGrOWdJyiv7zvOqRnV9Cdn4JhSXlFJaUczI9/6rnc3e2Z9Wfu192/IpBurCwkG+//Zb3\n338fe3t7nn76aWbOnMmRI0do3779ZY9btWoVkZGRtG3bFoCHH36Yzz//nKioKLZt24a7uzv33nsv\nANOmTWPEiBEsXLiwRufzaOSAl6sDnq6OeLk64uX23+eerg74uFeFZ293J62Q8Sv15QdKrp9637Cp\n/w2Xet+wNZT+N3KyM68ccikmk4m8onNk5VWF6qz/hOsLz88UlnGmsJQzZ8vIO3uO/OLLT/uFqwTp\nkydP4uDggLOzM3379uX999+nZcuWHD9+/IpBOi4ujn79+vHWW2+RkpJCnz59WLp0KQCxsbGEhoay\ne/duZsyYwSeffEJubi5nzpwhKSnpus6X+NH/KByLiIiIyDUxGAx4NKqatnG5sH3BeaORguJyslJP\nXXafKwbp4uJiXFxcOHv2LMeOHSMvL49GjRpRXFx8xRNfOC4mJoakpCSGDBlCUVFRtbHMzEyOHTuG\nvX3VGoFFRUXXfT4f7yZXrEMuzdbWloEDB+Lu7m7pUuQWU+8bNvW/4VLvGzb1v+a8m1DzIO3s7ExR\nUREBAQHk5OQAcPbsWVxcXK540gvHvfXWWwCsXr3afMyFscjISCIjI8nLywPAxcXlus+3a9euK9Yh\nIiIiInKzXDFIh4SEUFpaSlpaGv7+/pSXlxMfH0+bNm2u+KKtW7fm+PHj5j/HxMQQGhpqHlu0aFG1\nMQ8PDzw9PbG1tb3m8w0aNOi63qiIiIiIyI10xQnGrq6u3HXXXcyZM4eysjLefPNNgoKCqs1Xvv32\n23nuueeqHRcVFcWXX35JTEwMaWlpfPDBB4wfPx6AgQMHUlBQwLJlyyguLmb+/PnmsWs5n4iIiIhI\nbXDVO/XeffddDh8+jIeHBytWrGD58uXVxpOSksjOrv6RjuHh4UyfPp0BAwbQoUMHxo8fT1RUFABO\nTk6sXLmSV1991Xz36Jw5c675fCIiIiIitYHBZLrWlZhFREREROQCrR0nIiIiIlIDCtIiIiIiIjWg\nIC0iIiIiUgNXXP5O6r7k5GQWL15McnIyjRs3ZuLEiYSHh1NZWcnixYvZu3cvzs7O3H///fTs2dPS\n5crvcODAAdasWUNiYiK9e/dmypQpAFft9ddff83q1auprKzkzjvvZOLEiZZ6C/I7XK7/a9euJTo6\nmvz8fLy8vJgwYQLdunUzH6f+132X6/0FRUVFPPXUU3Tu3Jknn3zSvF29rx+u1P8NGzawYcMGioqK\n8PX1Zc6cORgMBkD9v1EUpOu5t99+m549ezJz5kwOHTrEvHnzeOedd4iOjiY1NZVFixaRmJjInDlz\naN26NZ6enpYuWWrI2dmZkSNHcujQIcrLy83bN2zYcNlenzhxglWrVjFjxgycnJx45ZVXCA4O1j+q\n6qDL9d/a2ppp06bRrFkzYmNjmT17NvPmzcPb21v9rycu1/sLli1bho+PjzlAAep9PXK5/u/evZv1\n69fzl7/8heDgYJKTk83fA+r/jaOpHfVceno6PXr0AKBjx47Y2dmRnZ3N3r17GTJkCE5OToSFhdG6\ndWv2799v4Wrl9wgLCyM8PPyiTwK9Uq/37t1LREQEAQEBeHh4MHDgQHbv3m2J8uV3ulz/hw0bRrNm\nzQBo06YNPj4+JCQkAOp/fXG53gMkJCRw+vRpunTpwq8X6VLv64/L9f/bb79l9OjRBAcHAxAYGGge\nU/9vHAXpeq5Tp07s3bsXo9HIwYMHcXR0pFmzZqSnp+Pn58eCBQv4/vvvCQgIID093dLlyk1wpV5n\nZGTg5+fH119/zccff0xAQAAZGRkWrlhulqKiIjIyMsx/oar/9ZvJZGLJkiVMnjyZ3650q97Xf0lJ\nSRQUFPDkk08yZcoUVqxYYR5T/28cBel6bvLkyURHRzNp0iTeeOMNHn30Uezs7Dh37hwODg6kpKSQ\nm5uLg4MDZWVlli5XboIr9frCWFZWFpmZmTg6Our7oB5777336N+/P35+foD6X99FR0cTFBREQEBA\ntWkdoN43BCUlJRw8eJBZs2YxY8YMduzYYf5tpPp/42iOdD1WXl7OzJkzeeCBB+jevTuxsbHMmzeP\nuXPnYm9vT1lZGfPmzQNgyZIlODo6WrhiuRmu1OsLYw8++CAA+/fvx8HBwWK1ys2zdOlSiouLeeqp\np8zb1P/6q6SkhDVr1jBr1iyAi65Iq/f1n4ODA7fffjuurq4AREREEBMTQ3h4uPp/AylI12PJycmU\nlpYSHh4OQGhoKN7e3sTGxuLn50daWhotWrQAIDU1le7du1uyXLlJrtRrX19f0tLSzPumpqaar1ZK\n/bF+/XoOHz7M9OnTsba2Nm9X/+uv7OxssrOzeeSRR6ptT01NZe7cuep9A+Dj41Ptz7/+x5T6f+No\nakc95u3tTXl5OQcOHMBkMnHy5EnS0tIICAigZ8+ebNy4kZKSEo4ePcqJEyfMgVvqJqPRSHl5OUaj\nEaPRSEVFBefPn79ir3v27Mn+/ftJTU0lNzeXbdu20atXLwu/E6mJy/V/+/btbNmyheeff/6iK07q\nf/1wqd4HBgayfPly82Ps2LH07duXuXPnAup9fXK5n/3w8HCio6MpKioiNzeXAwcO0K5dO0D9v5EM\npt/+vkfqlR9++IFly5aRk5ODq6sro0aNYtCgQZw/f5733ntP60jXI9u3b2fRokXVtkVFRTF69Ogr\n9lpridYPl+r/2LFj2bFjB3l5edWuRI8ZM4ZRo0YB6n99cLmf/bFjx5r/vHLlSrKysvjTn/5k3qbe\n1w+X6//IkSNZvHgx+/fvx97enjvuuIOoqCjzPur/jaEgLSIiIiJSA5raISIiIiJSAwrSIiIiIiI1\noCAtIiIiIlIDCtIiIiIiIjWgIC0iIiIiUgMK0iIiIiIiNaAgLSIiIiJSAwrSIiL13IoVK3jiiScu\nOZabm8tDDz3EmTNnqm1fuHAhr7322u8678KFC3nnnXeuut+sWbPYtGnT7zqXiIglKEiLiDQABoPh\nkttXrlxJz5498fT0vGXn/K3IyEhWrVpFWVnZDa9BRORmUpAWEWkALvUhtkVFRXz33XcMGDDAAhX9\nV2hoKE5OTuzatcuidYiIXC8bSxcgIlKfrFq1iujoaAoKCnB3d6dnz57cd999AIwfP57Ro0dz6NAh\nkpKSCAoK4rHHHiMoKMh8fGxsLEuXLiU+Ph5XV1d69OjBhAkTsLW1Ne+zZs0atmzZQn5+Pn5+fowb\nN45u3bqZxw8dOsSSJUvIzs4mLCwMHx+fS9a6Z88eGjVqREhIyFXf1/bt2/noo4/461//SmhoKEeP\nHmXGjBk8+OCDrFu3jsLCQjp37szjjz+Os7Oz+Tij0ciSJUvYuXMndnZ2REZGMnjw4Itev3v37nz3\n3XfccccdV/+PLCJSS+iKtIjIDbJ//36++OILHnjgAd566y2mTp2Kh4dHtX02btzIiBEj+Mc//oGH\nhwfz58/HaDQCkJSUxMyZM+nYsSPz589n6tSpHDx4kM8++8x8/NKlS9m0aRMPPfQQb7zxBnfddRev\nv/46CQkJQNVV5vnz59OhQwfmz59Pt27diI6OvuQ0i5iYGFq2bHnV97VlyxY+/vhjXnjhBUJDQ6uN\nbd68maeffprp06eTmprKkiVLzGMmk4l9+/bRpEkT5syZQ69evViyZAm5ubkXnSMkJISTJ09SXl5+\n1XpERGoLBWkRkRskKysLBwcHbrvtNry8vGjTpg1Dhw6ttk+fPn3o0aMH/v7+PPzww2RnZ/PLL78A\nsHbtWtq3b09kZCRNmzYlNDSUyMhItm7dCkBZWRkbNmzggQceoGvXrnh7ezNo0CDat29PdHQ0ALt2\n7cLa2prJkyfj6+vLXXfdRVhY2CXrzcjIwMvL64rvafPmzSxbtoyXXnqJVq1aXTQ+duxYWrVqRUhI\nCFFRUezevZvS0lLzeGBgIMOGDcPHx4fIyEiMRiOJiYkXvY63tzeVlZWcPn36ivWIiNQmmtohInKD\nREREsGHDBp588kk6dOhA69at6d27N46OjuZ9AgMDzc/d3d1xcXEhMzMTgFOnTpGVlcXkyZPN+xiN\nRioqKsjPzycnJ4fKysqLVsOoqKgwP8/IyMDX1xcbm//+7z0wMJCMjIyL6i0tLa1W228lJCRw/Phx\nGjdufNnpIb9+PwEBARiNRk6fPm3e7uvrax53cXEBqq6a/5aTkxMAxcXFl61HRKS2UZAWEblBvL29\nWbBgATExMRw7doyVK1eyYcMG5s2bZw62l5piceFGQIPBQL9+/Rg9evRF+7i6upKTkwPAM888g5+f\nX7VxOzs782tc6sbCS3Fycqp29fi37OzsmDFjBm+//TbvvvsuzzzzzDW97q/Pb2V18S8+L1VfSUkJ\nQLX51SIitZ2mdoiI3EB2dnZ07tyZCRMm8Oyzz5Kenk5KSop5PCkpyfw8Ly+PoqIimjZtCkDzqUza\negAAAq1JREFU5s1JSUnBx8fnooeVlRX+/v7Y2Nhw+vTpi8YbN24MVF0BzszMpLKy0nye5OTkSwZ4\nX19fsrOzL/teAgICCAoKYurUqfz444/m6SO/lpycbH6ekpKClZUV3t7ewLUvfweQnZ2NtbU1TZo0\nueZjREQsTUFaROQG2bFjB1u2bCE5OZnMzEyio6NxcHCoNi3i+++/Z+/evaSmpvLvf/+bJk2a0Llz\nZwBGjBhBYmIiixcvJjExkdTUVHbs2MEHH3wAgKOjI0OHDuXzzz9n+/btZGZmEhcXx/Lly9mzZw9Q\nNQfbaDTy0UcfkZ6ezubNm4mJiblkvWFhYcTHx1/1fQUGBjJhwgQ+/PDDi6aIrFq1ihMnTnDy5ElW\nrlxJr169zNNFrvXKOEBcXBytWrUyX1kXEakLNLVDROQGcXFxYc2aNXz66aeYTCYCAwN57rnnzPN/\nAQYPHsyaNWtISUkhKCiIadOmmac/NG/enOnTp7N8+XJefvllDAYD/v7+1dZ5njhxIq6urqxevZrT\np0/j7OxMq1atiIiIAKqmRjz77LN88MEHREdHExYWxsCBAzl48OBF9fbo0YMPP/yQEydOXHQj4W+v\nJg8bNoyff/6ZBQsWMHPmTPP2O+64gzfffJPCwkK6dOnCQw89dNnXuJIDBw4wcuTIa95fRKQ2MJiu\n55KBiIjU2Pjx45kyZQr9+/e3dClm7733HiaTiccee+y6jruwjvTChQuvuvLH1Rw7dozXX3+dhQsX\nYm9v/7teS0TkVtLUDhGRBmzcuHHs27ePM2fOWKyGL7/8knHjxilEi0ido6kdIiINmLu7u3kOtqW8\n+OKLFj2/iEhNaWqHiIiIiEgNaGqHiIiIiEgNKEiLiIiIiNSAgrSIiIiISA0oSIuIiIiI1ICCtIiI\niIhIDfw/+FQD7VmoM+oAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7fd655a4cb70>"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "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",
      "\n",
      "> *Random variable* will be precisely defined later. For now just think of it as a variable that can 'freely' and 'randomly' vary. A dog's position in a hallway, air temperature, and a drone's height above the ground are all random variables. The position of the North Pole is not, nor is a sin wave (a sin wave is anything but 'free').\n",
      "\n",
      "You may object that human IQs or automobile speeds cannot be less than zero, let alone $-\\infty$. 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 very closely models the distribution of IQ test results, but being a model it is necessarily imperfect. The difference between model and reality will come up again and again in these filters. \n",
      "\n",
      "You will see these distributions called *Gaussian distributions*, *normal distributions*, and *bell curves*. Bell curve is ambiguous because there are other distributions which also look bell shaped but are not Gaussian distributions, so we will not use it further in this book. But *Gaussian* and *normal* both mean the same thing, and are used interchangeably. I will use both throughout this book as different sources will use either term, and so I want you to be used to seeing both. Finally, as in this paragraph, it is typical to shorten the name and just talk about a *Gaussian* or *normal* - these are both typical shortcut names for the *Gaussian distribution*. "
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Gaussian Distributions"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So let us explore how Gaussians work. A Gaussian is a *continuous probability distribution* that is completely described with two parameters, the mean ($\\mu$) and the variance ($\\sigma^2$). It is defined as:\n",
      "$$ \n",
      "f(x, \\mu, \\sigma) = \\frac{1}{\\sigma\\sqrt{2\\pi}} e^{-\\frac{1}{2}{(x-\\mu)^2}/\\sigma^2 }\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<p> Don't be dissuaded by the equation if you haven't seen it before; you will not need to memorize or manipulate it. The computation of this function is stored in `stats.py` with the function `gaussian(x, mean, var)`.\n",
      "\n",
      "> **Optional:** Let's remind ourselves how to look at a function stored in a file by using the *%load* magic. If you type *%load -s gaussian stats.py* into a code cell and then press CTRL-Enter, the notebook will create a new input cell and load the function into it.\n",
      "\n",
      "    %load -s gaussian stats.py\n",
      "    \n",
      "    def gaussian(x, mean, var):\n",
      "        \"\"\"returns normal distribution for x given a \n",
      "        gaussian with the specified mean and variance. \n",
      "        \"\"\"\n",
      "        return math.exp((-0.5*(x-mean)**2)/var) / \\\n",
      "                         math.sqrt(_two_pi*var)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<p><p><p><p>We will plot a Gaussian with a mean of 22 $(\\mu=22)$, with a variance of 4 $(\\sigma^2=4)$, and then discuss what this means. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from stats import gaussian, norm_cdf\n",
      "plot_gaussian(22, 4, mean_line=True, xlabel='$^{\\circ}C$')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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K81L/FkDucnKnz6UZTiQiRZnKsogUSanpWTz94UocDhjSuxENa2iZuKLkvq51aVuvAvFJ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\nYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7fd65572d630>"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So what does this curve *mean*? Assume for a moment that we have a thermometer, which reads 22$\\,^{\\circ}C$. No thermometer is perfectly accurate, and so we normally expect that thermometer will read $\\pm$ that temperature by some amount each time we read it. Furthermore, a theorem called  **Central Limit Theorem** states that if we make many measurements that the measurements will be normally distributed. So, when we look at this chart we can *sort of* think of it as representing the probability of the thermometer reading a particular value given the actual temperature of 22$^{\\circ}C$. However, that is not quite accurate mathematically. \n",
      "\n",
      "Recall that we said that the distribution is *continuous*. Think of an infinitly long straight line - what is the probability that a point you pick randomly is at, say, 2.0. Clearly 0%, as there is an infinite number of choices to choose from. The same is true for normal distributions; in the graph above the probability of being *exactly* 22$^{\\circ}C$ is 0% because there are an infinite number of values the reading can take.\n",
      "\n",
      "So what then is this curve? It is something we call the *probability density function.* Later we will delve into this in greater detail; for now just understand that 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 result will be the probability of the temperature reading being between those two temperatures."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So how do you compute the probability, or area under the curve? Well, you integrate the equation for the Gaussian \n",
      "\n",
      "$$ \\int^{x_1}_{x_0}  \\frac{1}{\\sigma\\sqrt{2\\pi}} e^{-\\frac{1}{2}{(x-\\mu)^2}/\\sigma^2 } dx$$\n",
      "\n",
      "I wrote `stats.norm_cdf` which computes the integral for you. So, for example, we can compute"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print('Probability of value in range 21.5 to 22.5 is {:.2f}%'.format(\n",
      "      norm_cdf((21.5, 22.5), 22,4)*100))\n",
      "print('Probability of value in range 23.5 to 24.5 is {:.2f}%'.format(\n",
      "      norm_cdf((23.5, 24.5), 22,4)*100))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Probability of value in range 21.5 to 22.5 is 19.74%\n",
        "Probability of value in range 23.5 to 24.5 is 12.10%\n"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So the mean ($\\mu$) is what it sounds like - the average of all possible probabilities. Because of the symmetric shape of the curve it is also the tallest part of the curve. The thermometer reads $22^{\\circ}C$, so that is what we used for the mean.  \n",
      "\n",
      "> *Important*: I will repeat what I wrote at the top of this section: \"A Gaussian...is completely described with two parameters\"\n",
      "\n",
      "The standard notation for a normal distribution for a random variable $X$ is $X \\sim\\ \\mathcal{N}(\\mu,\\sigma^2)$. This means I can express the temperature reading of our thermometer as\n",
      "\n",
      "$$temp = \\mathcal{N}(22,4)$$\n",
      "\n",
      "This is an **extremely important** result. Gaussians allow me to capture an infinite number of possible values with only two numbers! With the values $\\mu=22$ and $\\sigma^2=4$ I can compute the distribution of measurements for over any range."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "###### The Variance\n",
      "\n",
      "Since this is a probability density distribution it is required that the area under the curve always equals one. This should be intuitively clear - the area under the curve represents all possible occurences, which must sum to one. We can prove this ourselves with a bit of code. (If you are mathematically inclined, integrate the Gaussian equation from $-\\infty$ to $\\infty$)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print(norm_cdf((-1e8, 1e8), mu=0, var=4))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "1.0\n"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This leads to an important insight. If the variance is small the curve will be narrow. this is because the variance is a measure of *how much* the samples vary from the mean. To keep the area equal to 1, the curve must also be tall. On the other hand if the variance is large the curve will be wide, and thus it will also have to be short to make the area equal to 1.\n",
      "\n",
      "Let's look at that graphically:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "import matplotlib.pyplot as plt\n",
      "\n",
      "xs = np.arange(15, 30, 0.05)\n",
      "p1, = plt.plot(xs, [gaussian(x, 23, .2) for x in xs], label='var=0.2')\n",
      "p2, = plt.plot(xs, [gaussian(x, 23, 1) for x in xs], label='var=1')\n",
      "p3, = plt.plot(xs, [gaussian(x, 23, 5) for x in xs], label='var=5')\n",
      "plt.legend()\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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2X1AOMxlJiXOdztfcoVVlEQksBWURERmRZ0U53n+tF5e+ntovRCTQFJRFROQy\nTqf/T+Vzc79eg7aIE5EAU1AWEZHL2Lr66O4bICYyDHNUuF9f23NBX7uCsogEloKyiIhcJlCryQBp\ng60X2ktZRAJNQVlERC7T1B6Y/mS4dEVZPcoiElgKyiIichn3inJqvP9XlFMH923WMdYiEmgKyiIi\nchn3am6aHw8bcXNvR9eoXS9EJMAUlEVE5DJNnhVl/wflVE9Q1oqyiASWgrKIiFym0bOi7P/WC8+K\ncvtFnE6n319fRMRNQVlERC7TFMBdL8xR4cREhtHdO4Ctq8/vry8i4qagLCIil2kMYI+y63XVfiEi\ngaegLCIilwnkrhcwFNAVlEUkkBSURURkmAGHg5aObgBS4gK7oqy9lEUkkMYMyjU1NRQVFWE2myks\nLOTUqVPjGvg3v/kN1157LRaLhfz8fD788MMpFysiItOvpaMbh9NJUmwU4WGBWU9JG1zJbtCKsogE\n0JjfATdt2sSSJUtobW2luLiY4uLiMQc9f/48n/3sZ/mXf/kXOjo6ePPNN8nMzPRJwSIiMr3c+xen\nBWBrODfPirKCsogEkNeg3NHRQUlJCTt27CAyMpJt27ZRWVnJyZMnvQ76/e9/nzvvvJP77rsPo9FI\neno6qampPi1cRESmR1P7YH9yAHa8cPP0KKv1QkQCyGtQLisrIyoqCrPZzLJly6ioqCAvL4/Tp097\nHfTDDz8kMTGRm2++mYyMDB544AE6Ojp8WriIiEwP94pyIA4bcfPsemHVirKIBE6YtzvtdjsWiwWb\nzUZpaSlWq5XY2FjsdrvXQdvb2zl27Bivv/46V155JcXFxezcuZPnnntuxMcnJydP/h0IAOHh4YDm\n0lc0n76l+fQdf8ylfXDr4tyMpID9nV2Z2w9Aa2fvtNagr03f0nz6jubSt9zzOVFeg7LZbKazs5Ps\n7Gyam5sBsNlsWCwWr4OazWZWr17NtddeC7j6nL/yla+M+vhdu3Z5/rx8+XJWrFgx7jcgIiK+1WB1\nLYakJ5kDVkNGkuvnTL21M2A1iEhoO3bsGG+88QYAJpOJ5cuXT3gMr0E5Pz+frq4uamtrycrKore3\nl/LychYuXOh10Ly8POrr6z2fO51Or8eQbt26ddjnLS0t46ldLuH+jVNz5xuaT9/SfPqOP+ayqr4V\nAEt44P7ODA4HRoOB5vYu6huapm33DX1t+pbm03c0l1NXUFBAQUEB4JrP48ePT3gMr9954uLiWLNm\nDbt376bBMKTnAAAgAElEQVS7u5tnn32W3Nxcz4sCFBUVsWPHjmHP27BhA4cOHeLkyZN0d3fzne98\nh5UrV064OBER8T/33sWpATqVD8BkNHp6pN0XF4qI+NuYv6Lv3buXEydOkJSUxIEDB9i/f/+w+ysr\nK2lsbBx22/Lly9m5cyef+tSnyMrKwmw288QTT/i2chERmRbuvYvTAnQqn5s7qOvQEREJFK+tFwDZ\n2dkcPXp01PsrKipGvP2RRx7hkUcemXRhIiISGO69iwO5ogzunS9adOiIiASMjrAWERGPrp5+bF19\nhJuMJJgjA1qL+8CTpjatKItIYCgoi4iIx9BhI9EYDIaA1uLZS1kryiISIArKIiLi4T4JL9D9yXBJ\nUNbFfCISIArKIiLiESz9yTB0MqBWlEUkUBSURUTEI5hWlNM9rRfqURaRwFBQFhERD/eFc0Gxoqwe\nZREJMAVlERHxcPcDu/uDA8mz60V7l9fTXUVEpouCsoiIeLhXlN0hNZBiosKxRIXT0zdA+8XeQJcj\nIrOQgrKIiHg0ei7mC/yKMkBaoquOJrVfiEgAKCiLiIiHp/UiCFaUYagOnc4nIoGgoCwiIgA4nc6h\ni/mCYNcLGOqV1ul8IhIICsoiIgJAm72HvgEHsdHhREeGBbocYKgFRCvKIhIICsoiIgJcujVccKwm\nA6QnDO18ISLibwrKIiICBF9/Mgy1gGgvZREJBAVlEREBCLr+ZLj0dD4FZRHxPwVlEREBLj1sJIhW\nlAdrUVAWkUBQUBYREeCSw0aCqkd5cEVZPcoiEgAKyiIiAgytKAdT60WiJYowk4G2zh56+gYCXY6I\nzDIKyiIiAkCjZ0U5eFovjEYDKXGu4N6sVWUR8TMFZRERAYaOiQ6mFWUYCu7uFW8REX9RUBYREWCo\nDziYVpRhqGe60aqgLCL+paAsIiL09TtotXVjNBhIjosKdDnDuPd11gV9IuJvCsoiIkJzhyuEJsdF\nYTIG14+GtETtpSwigRFc3w1FRCQgmjw7XgRX2wXodD4RCRwFZRERuWTHi+C6kA8g3XPoiFovRMS/\nFJRFRCS4V5QHw3uTdr0QET9TUBYREc9qbXpQrii7ampQ64WI+NmYQbmmpoaioiLMZjOFhYWcOnVq\nzEGPHj2K0WgkNjbW83HmzBmfFCwiIr7nWVEOwqDsXuVuauvC6XQGuBoRmU3GDMqbNm1iyZIltLa2\nUlxcTHFx8bgGzsrKwmazeT4WLlw45WJFRGR6NFhdK8rB2HoRFRFGfEwEfQMOrJ09gS5HRGYRr0G5\no6ODkpISduzYQWRkJNu2baOyspKTJ0/6qz4REfED94pyMF7MB+pTFpHA8BqUy8rKiIqKwmw2s2zZ\nMioqKsjLy+P06dNjDtzY2EhGRgb5+fk8+eSTPitYRER8r6k9eFeUYaiuBu18ISJ+FObtTrvdjsVi\nwWazUVpaitVqJTY2Frvd7nXQq666io8++oi8vDz++Mc/sn79ejIzM/n85z8/4uOTk5Mn/QbEJTw8\nHNBc+orm07c0n74zXXPpDsqL5mcRbw6uk/kActITebv0Al39Rp++d31t+pbm03c0l77lns+J8hqU\nzWYznZ2dZGdn09zcDIDNZsNisXgdNC0tjbS0NACuvfZavvjFL/Lyyy+PGpR37drl+fPy5ctZsWLF\nRN6DiIhMQWdXL/buPqIiwoiLiQx0OSPKSDQDUN/qfaFGRMTt2LFjvPHGGwCYTCaWL18+4TG8BuX8\n/Hy6urqora0lKyuL3t5eysvLfX5h3tatW4d93tLS4tPxZwP3b5yaO9/QfPqW5tN3pmMuK+rbAUiN\nj6K1tdVn4/pSXJQBgPN1zT597/ra9C3Np+9oLqeuoKCAgoICwDWfx48fn/AYXnuU4+LiWLNmDbt3\n76a7u5tnn32W3Nxcz4sCFBUVsWPHjmHPO3LkCFVVVQCUlpbyrW99i7vvvnvCxYmIyPQb6k8Ozgv5\nQMdYi0hgjLk93N69ezlx4gRJSUkcOHCA/fv3D7u/srKSxsbGYbe9//773HTTTZjNZu666y42b948\natuFiIgEljt8BuNhI25piYNBWbteiIgfeW29AMjOzubo0aOj3l9RUXHZbY888giPPPLIlAoTERH/\n8KwoJwTnjhcAaYO7XjRq1wsR8SMdYS0iMsu5V5TTgrj1wr2/c5NaL0TEjxSURURmOXdQDuYV5URL\nJOEmI+0Xe+nq7Q90OSIySygoi4jMco2DrRfBvKJsMBg8Qb65Xe0XIuIfCsoiIrNcKKwow1CQb1D7\nhYj4iYKyiMgs5+lRDuJdL0B9yiLifwrKIiKz2IDDQXN7NwCp8cG9ouxe8W7Qzhci4icKyiIis1hz\nezcOp5Ok2CgiwkyBLscr9z7PTdpLWUT8REFZRGQWC4XDRtzcK96NVgVlEfEPBWURkVmswdOfHNxt\nFzAU5hu164WI+ImCsojILBYqF/IBpLqDsi7mExE/UVAWEZnF3CvK6YnmAFcyNs+Ksi7mExE/UVAW\nEZnFhnqUg7/1ImWwR7m54yIOhzPA1YjIbKCgLCIyi4VS60VkuIkESyT9A06snd2BLkdEZgEFZRGR\nWazB6mpjCIVdLwDS3DtfqP1CRPxAQVlEZBbzrCgnhkhQ1gV9IuJHCsoiIrOU0+kMqdYLuCQo69AR\nEfEDBWURkVnK2tlD34CDuJgIoiPCAl3OuGhFWUT8SUFZRGSWarCG1moyXHI6n3qURcQPFJRFRGap\nxhA6lc8tXSvKIuJHCsoiIrOU57CRUFpRHgz1TTrGWkT8QEFZRGSWCrUL+QDS4l21NmhFWUT8QEFZ\nRGSW8pzKFyJbw8HQNnZNCsoi4gcKyiIis1Qotl7Ex0QQGW7C1tVHV09/oMsRkRlOQVlEZJYKxdYL\ng8EwtPOF9lIWkWmmoCwiMku5t1gLpaAMkDrYp9xoVVAWkemloCwiMgs5nc6QbL0ASE90ryhr5wsR\nmV4KyiIis1DnYI9vdGQYlujwQJczIZ4VZV3QJyLTbMygXFNTQ1FREWazmcLCQk6dOjWhF1i1ahU5\nOTmTLlBERHzv0tVkg8EQ4GomRoeOiIi/jBmUN23axJIlS2htbaW4uJji4uJxD75//35sNlvIfRMW\nEZnpho6vDp1T+dzch44oKIvIdPMalDs6OigpKWHHjh1ERkaybds2KisrOXny5JgDd3Z28tRTT/HP\n//zPOJ1OnxUsIiJTF4o7Xri5a1aPsohMN69BuaysjKioKMxmM8uWLaOiooK8vDxOnz495sBPPPEE\nmzdvJi4uzmfFioiIb4TqhXwwdDqfVpRFZLqFebvTbrdjsViw2WyUlpZitVqJjY3Fbrd7HbS0tJTD\nhw/z1FNPcezYsTGLSE5OnljVcpnwcNfFOJpL39B8+pbm03d8NZe2Hte/9M3LSvXZ30tPv5269hNY\nL1bT1WtlwNlPdHg8sZFpzElYQlxUuk9eZ6HDNQctHT1Trl1fm76l+fQdzaVvuedzorwGZbPZTGdn\nJ9nZ2TQ3NwNgs9mwWCxeB/3Sl77Ev/7rv467N3nXrl2ePy9fvpwVK1aM63kiIjI5F1o7AchIMk9p\nHIdzgPMt/8uJupepazuBE8eoj42PnsNVGXdyVeYaIsO8/xzxJi3BVXNDm52BAQcmkzZwEpHLHTt2\njDfeeAMAk8nE8uXLJzyG16Ccn59PV1cXtbW1ZGVl0dvbS3l5OQsXLvQ66Hvvvcedd9457DaTyYTV\nah2xFWPr1q3DPm9paRlv/TLI/Run5s43NJ++pfn0HV/NZXWDFQBzmGPSY13o/JAPLvwQW289AAZM\npMQsICk6j+iweIyGMHr6O2jvqaPR/hHtXXW8XfEd3q18kcUp97AweS0mo9cfQ6NKio2i1dbN2cpa\nz3Zxk6GvTd/SfPqO5nLqCgoKKCgoAFzzefz48QmP4fU7VFxcHGvWrGH37t08/fTTfOMb3yA3N9fz\nogBFRUXcfPPN7N6923Ob1Wr1/PnYsWM8+OCDVFdXT7g4ERGZHlM5la93wM57dd+juuN/ATCHp7Ig\neQ1XJCwjwjTyeA6ng/rOP/Jxy69psJ/iRONPOd/2Jp/M2kRyTN6Ea0iLj6bV1k2DtWtKQVlExJsx\nf5Xfu3cvDz74IElJSSxevJj9+/cPu7+yspL58+eP+nyn06nt4UREgsxkd72wdp3nzernsfc1YTJE\nUJD2GRYkr8Fo8P7jxGgwMif2eubEXk9950nev/Bf2HrreP38/8f1GQ+Sl7hyQj8r0hJiOF1jpan9\nIqAeThGZHmMG5ezsbI4ePTrq/RUVFV6fX1RURFVV1YQLExGR6dHV20/HxV4iwowkWiLH/bzajt/z\nds0eBpx9JEbN49acL2KJSJvw62dYCliT96/8oeEnlLWW8PsL36etu5rCzI0YDOPrN9ZeyiLiD5Nr\nDhMRkZDlDpep8eM/le9823Heqd2HEwfzEpZTmPk5TMaISddgMoZTmLmR5Og83q37DuXWw/Q5LvLJ\nrE1jrk7DpafzaS9lEZk+CsoiIrNMo3VibRcVbcd5p3YvAFel3ENB2r0+a6m7IuHPiAlP4rdV/4eq\n9rdxOPu5JfuLGMdYWU4drN3VeiEiMj20p46IyCxT7z5sJHHs46vrbB/wbu0+AJakFXNN+n0+v+4k\nzbyYoiseI9wYQ03Hu/z+wvfGPNHVvaLcoNYLEZlGCsoiIrPMeFeUmy+e5a3qF3DiYHHKPSxOXTdt\nNSVHz2fZ3H/AZAjnnPUoJ5t+7vXxqfGukN+k1gsRmUYKyiIis0zjOI6vvtjXypvV32DA2cf8hCKu\nSbt32utKNS/klpwvYsDIR02/pKr9f0d9bJpWlEXEDxSURURmmQZP68XIQXnA0cub1d+gu7+dNPNi\nCud8zm/bfGbF3sC1GX8BwDu1/0lb98i7JqV5epS1oiwi00dBWURklhlrD+UP6n9Ea9c5zOEp3Jr9\n8Lh2ofClBUlruCJhKQPOXo5XfYO+gcvDcGx0OFERJuzdfdi7+/xan4jMHgrKIiKzjHtLtZFaL6o7\n3qXc+jpGQzh/lrONyLBYf5eHwWDgxsy/JiEqF3tfI++NcHGfwWAgLd69RZzaL0Rkeigoi4jMMg2j\nrCjbe5t5t/bbAFyX/hckRuf6vTY3kzGCW7MfIswYSVX725xvf/Oyx+jQERGZbgrKIiKzSE/fAK22\nbkxGA8lxUZ7bnU4H79T+J32Oi8yJvZ78pFUBrNIlNjKT6zM2AvD+hf/C3ts07P70BDMA9VYFZRGZ\nHgrKIiKzSIPVDrgu5DMZh34ElLUepvFiKZGmOD4x5wt+u3hvLPMSlpEd9wn6Hd28W/edYS0Ymcmu\noHyh1R6o8kRkhlNQFhGZRdyhMiPR7Lmts7eRDxv3A3DjnM8TFRYXkNpGYjAYKMz8HJGmWBrspyi3\nHvHclzm4a0e9VUFZRKaHgrKIyCziblPITHIFZafTyXt136Xf0UNO3CfJjvtEIMsbUVRYPDdkfg6A\nPzb8hIt9rcDQe9CKsohMFwVlEZFZxLOiPBgyq9rfpsF+igiThRsyNwayNK9y4m4iK/ZG+h3dfFD/\nIjC0Kl7fqh5lEZkeCsoiIrOIOyjPSTLTO2Dng/ofAXBt+p8HVcvFnzIYDNyQ+VeEGaOo6XiXOtsf\n1KMsItNOQVlEZBZx9/NmJMbwYcNP6RnoICVmAfMSlgW4srHFhCdRkPoZwLULRnKcCYCGNjsOh9Pb\nU0VEJkVBWURkFnGvviYltlNufR0DRgozP4/BEBo/Dq5MXk1C1Fzsfc1U2X5DUmwU/QNOmjt0lLWI\n+F5ofGcUERGfcPXzOuk0vQo4yU/6FAlROYEua9yMBhPXZTwAQGnzy8zLdP0Y084XIjIdFJRFRGYJ\nh8NJQ5udJYuasfWdJcJkoSBtQ6DLmrB081Vkxd5Av6OHZZ88C8CFFgVlEfE9BWURkVnC1Z4wwKdX\nVQBwTdp9RJjM3p8UpK5N/3MMmMjJLmdOWicXdDqfiEwDBWURkVniQqudW2+oIyGum/jIbOYnFgW6\npEmLjcwkP2kVBgPcs+ocF1o7A12SiMxACsoiIrPEhdYWVv1ZFQBL0u/HGCIX8I3m6tRP43REseCK\nNnoMpwNdjojMQKH9XVJERMatue8Ilph+bB0ZZFquC3Q5UxYZZiGO2wHInvs7HM7+AFckIjONgrKI\nyCzQ1deGyfIOAN1tKzAYDAGuyDfmJayksSWa2Fgb56xHA12OiMwwCsoiIrPAqaZfYjT2c+JMMinR\nVwa6HJ+ZkxTPoWNXAPBR00H6Hb0BrUdEZhYFZRGRGc7W08A561GcTgOHjl1BZlJo7nQxkriYCMor\nMqipN9PVb6W89XCgSxKRGURBWURkhjvR+FOcDFB6NpuGZjMZSTGBLslnDAYD6UkWfvXGFYDrEJK+\nge7AFiUiM8aYQbmmpoaioiLMZjOFhYWcOnVqzEGPHDnCkiVLiI+PJy0tjc997nN0dmrrHhERf7N2\nVVLd8b+YDOG8etR1Al9mkiXAVflWZpKZj8qSCHfm0DNg42zr/wS6JBGZIcYMyps2bWLJkiW0trZS\nXFxMcXHxmIMuXryY1157jfb2ds6dO0dDQwNPPvmkTwoWEZHxO9X0SwCyY4u40BxGTGQYsdHhAa7K\ntzISYwADPe3LADjTcojeAR1AIiJT5zUod3R0UFJSwo4dO4iMjGTbtm1UVlZy8uRJr4NmZGSQlZUF\nQF9fHw6Hg7i4ON9VLSIiY7J2V1Jrew+TIZzoAVeIzEq2zJgdL9yykl0r5HX1qaTGLKJ3wM7HLb8K\ncFUiMhN4DcplZWVERUVhNptZtmwZFRUV5OXlcfr02Bu7V1VVER8fT3JyMomJiTz66KM+K1pERMb2\nUaNrNTkvcSWNra5v91kpM6vtAobeU22LnWvS7gXgTMtr9PTbAlmWiMwAYd7utNvtWCwWbDYbpaWl\nWK1WYmNjsdvtYw48d+5c2tvbqaio4DOf+Qx79+5ly5YtIz42OTl5ctWLR3i4659SNZe+ofn0Lc2n\n74x3Lps7z1Fjew+TMYJbrvwrXqw4D8D8rOQZ9/eweN4cABrbe1g091bOthdSbf09lRePcMu8v/b6\nXH1t+pbm03c0l77lns+J8hqUzWYznZ2dZGdn09zcDIDNZsNiGf+KxLx589ixYwfPPPPMqEF5165d\nnj8vX76cFStWjHt8ERG53HtVPwbg6sy1mCOTqG76EICc1JnXBud+TzVNrhXkm3IfoNr6e07WvcL1\n2fcSFR4byPJEJECOHTvGG2+8AYDJZGL58uUTHsNrUM7Pz6erq4va2lqysrLo7e2lvLychQsXTuhF\nHA4HTqdz1Pu3bt067POWlpYJjS9Dv3Fq7nxD8+lbmk/fGc9ctnVXca75LUyGcK4w305LSwvlNU0A\nJMYYZ9zfQ4ypD4Dqxnaam5sJM6SSbi6gwX6S/y17iYK0z4z6XH1t+pbm03c0l1NXUFBAQUEB4JrP\n48ePT3gMrz3KcXFxrFmzht27d9Pd3c2zzz5Lbm6u50UBioqK2LFjx7DnvfTSS5w4cQKHw0FVVRXP\nPPMMd95554SLExGRiXPvdDE/cSXR4QkA1La4tuh0X/g2k1iiI0gwR9LdN0CrzbWH8lWp6wH4uOXX\n9A10BbI8EQlhY24Pt3fvXk6cOEFSUhIHDhxg//79w+6vrKyksbFx2G0tLS3ce++9xMfHc8stt7B8\n+XK+8pWv+LZyERG5TFt3NTUd72IyhLM4ZZ3n9tpm17Ulc2ZgUAbITHadNuj+hSDNvIjUmIX0OS5S\n1vqbQJYmIiHMa+sFQHZ2NkePHh31/oqKistue+ihh3jooYemVJiIiExcafPLAMxPLPKsJvcPOKi3\nuoLyTDq++lJZyRZKq1qpbe5kybxUwLWqfKzy65xpeY0rkz9FmDEqwFWKSKjREdYiIjNEZ28j1e2/\nw4CJhclrPbc3tF1kwOEkLSGayHBTACucPu6WktqWoV2Z0s0FJEXPp2fARnnrkUCVJiIhTEFZRGSG\nON18CCdOchNuwRyR4rm9rnnm9ie7ZaUMtl4MvlcAg8HA1amfBuB0yyEGHL0BqU1EQpeCsojIDNDd\n305Fm2sbpEXJ64bd5+7bnan9yXDpinLnsNszLdeREDWX7v42zg3Oj4jIeCkoi4jMAB+3/AqHs4+s\n2ELio7KG3Vc32I4wo1eU3cdYtww/EMtgMHBVimsHjNPNr+Bw9vu9NhEJXQrKIiIhrnfgImWthwFY\nlLLusvs9W8PNwOOr3eakuINy52X3ZcfdSGxEJhf7Wqhs/52/SxOREKagLCIS4spbX6fP0UVazGJS\nYvIvu39oD+WZueMFQHpCDCajgYa2i/T0DQy7z2AwerbKO938Mk6nIxAlikgIUlAWEQlh/Y5ePm79\nFQCLU+8e8THuC9xm8opymMlIRqLrFwH3VniXyk24lZjwZDp66qi1ve/v8kQkRCkoi4iEsPNtv6W7\nv52EqFzSzQUjPsbdtzsnaeYGZYA5yZfvfOFmNIR5tsz7qOkgTqfTr7WJSGhSUBYRCVEO5wCnm18F\nYHHK3RgMhsse09nVS5u9h8hwE8lxM/vAjdF2vnCbn7iCSFMs1u4KGuwn/VmaiIQoBWURkRBV3fEO\n9r4mLBHpZMd9YsTHDG0NZx4xSM8k7taSmhFWlAHCjJEsSL4DGDrBUETEGwVlEZEQ5HQ6KW1yhb1F\nyXdhNIz87byq0QbA3NRYv9UWKDmD77G6yTbqY/KTbifcGE2jvZTmi2X+Kk1EQpSCsohICLrQ+Ufa\ne6qJCkvgioSloz7OHRpzZkFQdv8y4P7lYCQRJjP5SbcDWlUWkbEpKIuIhKDTza8AsDD5DkzG8FEf\nVzUYlOemzfygnJM29ooywILkOzAZwqmzvU+L/bwfKhORUKWgLCISYi60n6Lp4hnCjTHkJa70+lhP\n60VanD9KC6jsFAsGg2uXj77+0fdKjgqLZ15iEQDvV//UT9WJSChSUBYRCTHucHdl0irCTdFeH+tZ\nUZ4FrRcRYSYyk8w4nM5Rd75wW5S8FgMmyhqP0d51wU8VikioUVAWEQkhLfYKKlvfwWSI4MrkNV4f\n63Q6qW6cPT3KcEmf8hjtF+aIFHITbsGJgw9q/tsfpYlICFJQFhEJIR9Uu0LdvMQVRIV5b6ewdvbQ\n2d2HJSqcREukP8oLOM/OF14u6HNzHWtt4HR9CV19bdNcmYiEIgVlEZEQ0dnbyNnGYxgwsmjwlDlv\nPDtepMXO+D2U3ca7ogwQF5nF/JRbcDj7OdPy2nSXJiIhSEFZRCREnGl5DScOrkwrwhyRMubjZ1N/\nstt4d75wuyHnfgDKra/T0++9r1lEZh8FZRGRENDd306F9RgAN+TcN67nzLb+ZBjfXsqXSotdQHbC\n9fQ7uilrLZnO0kQkBCkoi4iEgI9b/ocBZx9XJN9Mkjl3XM+pbOwAIHcW7KHs5t4Gr6qpY9zPKZxb\nDMDHrf9D30D3tNQlIqFJQVlEJMj1DXRR1vobAK7Pvnfcz5tNp/K5pSfEEBluoqWjG3t337ieMyf+\nGpKj8+kd6OSc9cg0VygioURBWUQkyJW1HqbPcZHUmEVkxl817ufNplP53IxGA1kpFmD8fcoGg4HF\nqXcDrj7wAcf4AraIzHwKyiIiQazf0cvHrb8C8IS58XA4nNQ2uy5Oy0mZPUEZJrbzhdscy3XER2bT\n1W/lfPub01WaiIQYBWURkSB2vu0NuvvbSYjKJcN8zbifV2+109vvICUumpio8GmsMPhMZC9lN4PB\nyOIU1y8ip5tfweEc/QhsEZk9FJRFRIKUwznA6eZDAFyVcveE9kKumoU7Xri5L16snMCKMkBO/Ccx\nh6fR2dtATcc701GaiIQYBWURkSBV1f477H1NxEZkkBX3iQk9t6KhHYB5Gd5P75uJrkh3veeK+vYJ\nPc9oMLEo5S4ASptfxul0+rw2EQktCsoiIkHI6XRwuvkVABal3IXRMLFv1+cuuELi/Mx4n9cW7Nzv\n2T0HEzEvYSlRYQm0dVdxofOPvi5NRELMuL7z1tTUUFRUhNlsprCwkFOnTo35nEOHDvGJT3yC+Ph4\n5s6dy9e+9rUpFysiMlvUdf6B9p4aosMSyY1fOuHnV9S79hGenzH7gnJuWhwGg2vXi77+ifUam4wR\nLEy+A4CPmg5qVVlklhtXUN60aRNLliyhtbWV4uJiiouLx3yO3W7nqaeeorm5md/97ne8+OKL/OhH\nP5pywSIiM53T6aS06WUAFibfickYNuEx3G0Hs7H1IioijKxkCwMOJ9XNE+tTBshLXEmEyUxL11ma\nLp6ZhgpFJFSMGZQ7OjooKSlhx44dREZGsm3bNiorKzl58qTX5913332sXLmS8PBw5syZwx133MHb\nb7/ts8JFRGaqpounaekqI8JkYX7ibRN+vsPh5HyDa0V5XvrsW1EGmJcx+faLcFM0VyZ9CnD1KovI\n7DXmMkVZWRlRUVGYzWaWLVvGt7/9bfLy8jh9+jQFBQXjfqG3336bv/mbvxnxvuTk5PFXLCMKD3dt\n/6S59A3Np29pPifmrbpfA3Bt9noy0rKG3TeeuaxqbKe7b4D0RDNX5GROX6FB7Kor0vntyVoabf1e\n52q0+fxk3J9zpuVX1Hd+iCPCSmps/rTWO1Po/3Xf0Vz6lns+J2rMoGy327FYLNhsNkpLS7FarcTG\nxmK328f9Inv27KG3t5fPf/7zI96/a9cuz5+XL1/OihUrxj22iMhM0mQro9r6e8KMUVwz555JjXG2\nxgpAflaiL0sLKe73fra2dVLPjwqP4+rMO/hj7S95v/oAa676J1+WJyJ+cOzYMd544w0ATCYTy5cv\nn/AYYwZls9lMZ2cn2dnZNDc3A2Cz2bBYLON6gUOHDvHMM8/w29/+dtQ0v3Xr1mGft7S0jGtsGeL+\njVNz5xuaT9/SfI7f76pfBGB+4m3YO3qxM3zOxjOXfzxbDUBOcsysnfO0WNePt9LzjV7nwNt8zjXf\nxjxA2uEAACAASURBVAnDy5Q3v0lF3UniImfn6vxE6P9139FcTl1BQYGn+yE5OZnjx49PeIwxe5Tz\n8/Pp6uqitrYWgN7eXsrLy1m4cOGYg7/11lts3ryZV199lezs7AkXJyIym3T0XKC6412MBpNn54XJ\nGLqQb3b2J8PQFnET3Uv5UjHhSVyRsAxwerbqE5HZZcygHBcXx5o1a9i9ezfd3d08++yz5ObmDutP\nLioqYseOHcOe9+GHH3Lfffdx4MABrrrqKt9XLiIyw3zU9H8BJ/MSVhATnjTpcWbzHspuOSmxmIwG\nals66e7tn/Q4i5LXYcDA+bY3udinlT2R2WZc28Pt3buXEydOkJSUxIEDB9i/f/+w+ysrK2lsbBx2\n23PPPUdTUxOrV68mNjaW2NhY7rrrLt9VLiIyg9h66qlqfwsDJhanrJvSWBWzfMcLgPAwI3PTYnE6\nobKxY9LjxEamkxP3SZwMcLr5NR9WKCKhYFybc2ZnZ3P06NFR76+oqLjstu9+97t897vfnXRhIiKz\nyUfNB3HiZF7CUswRqZMep3/AQdVgMHQf5TxbzcuIp6K+g3MX2lmYPfkV+kWp66jq+B3nrEe4KvUe\nosJm97yKzCY6wlpEJMA6exupbHsTA0auSp3cThdu1U02+geczEk2Ex058YNKZhL3qYTuUwonKzEq\nl0zLdQw4eznb8mtflCYiIWJ2fxcVEQkCHzUdxImDKxKWYYlIm9JY56bxQj5nTze0t8JFO3RdhK6L\nOLsvQlcXdF+Evl5wOsHpAIdz6M8GA0REXvZhiIwESzzExUNsAkRFYzAYfFav59CRKVzQ57Y49W4u\ndP6Bs62/YVHKOsJN0VMeU0SCn4KyiEgA2XubON92HAMGrkqZ2moywNnaNgDyMxMm9Dyn0wntVmio\nw9lQC831YG3F2d4KbYMfXePfP39cr/mnN4RHQGy86yMpBUNqBqRkYEhNh5QMSE7DMIFDA/LnuILy\nx7XWKdeaGrOA1JiFNF08Q1nrYRanTq2PXERCg4KyiEgAlTa/gpMBcuP/jNjIjCmPd2bwsJEF2SMf\nNuJ0OsHaAtXncFafg7pqVzBuuAA9Xd4HDwuH+ESwxEFUNETHYIiKgehoiIpx3W80ulaQ3f81GF2r\nyr290NsDvd2D/+3F2d0FnR3Q0Qa2NtdjWptcH5VlniDtCdQGAySlQlYuhqxc13+zr4D0LAxhl/84\nWzg4Bx/XWHE6nVNerV6ceg9NlU/zceuvuDJ5NWHGiCmNJyLBT0FZRCRA7L3NVLQdAwxclbreJ2N+\nPBiU3SHR2dqMs7zUFTyrzkH1/9/ence1dZ0JH/8dLUiAWIXB7AZswDZgvO/GW+I4SbPacZNmnSQz\naSZ5pzPTNl3eeTvvtHXdujOtk/ZN4zhJm7Zp0qTZGide4n3D4N0YMDb7DmbfQdJ5/7iAjYMxYEBg\nn+/ncz+S7pWuHh2Q7qOj556TqyWnvbF4gH8QIiAI/APBxw/hbQVvX21xswxpacTVZFurljTX1yKr\nK+FSOVSWITsvqa6EqgqoqkCeSdUeA6A3QGAoImISRMYgImIgMAQ/T1d8LCZqGtsorW4iyNq/ibKu\nZbx7PN7mcGpb88mt2cck6203/qIVRRnVVKKsKIriJOmVn+CQdsK85g/JrG8Ouw1RnMdjhkoS97yN\n/fdZWnJ5NTcLhEUiwiK1XtmAYAgIQlicO5qDMJlh3HgYNx4RFfuV7dJm0xLnojwozkMW50NxvpZE\nF+Uii3LhwA4teXZ1gwmT+D8WHR82m8jOq7jhRFkI7QvN4cKXybj0dyJ8klSvsqLc5FSirCiK4gQN\nbWXk1u5HoCNu3AOD2oeUEsqLkemnkOmncGSe5RN9C+iBk513cnWHqBhERLSWGIdGgu+4Ye0ZHi7C\nYIDAEERgCMxe1L1etrZoiXJOFuScR+aeh+pLkHGaB4EHzWB/LRX7pFhETAIiNgEioxGG/tc7dwnx\nmNndq3yx+kti/e4cwleoKMpooxJlRVEUJ0ir/BCJgwjvpAHVJjsa6mk/nYLj6AFkxiktIeykAwoc\nruR7BLPkvpWIiVMgKAyhu7lHAhVmV5g4RXu9nWRNFeSe5+yuQ9gzzhCnr4Osc8isc8i//wVcXCA6\njuZ5SZhmLgCDqX/PJXTE+6/hQMF/k3npM6J8lqkRMBTlJqYSZUVRlBFW21pIQV0yOmFg6rj7rnt/\nWV6CPH0UeTqVSxczwGG/vNHiiZg8DSZP40+lJr7/SRZPLZnC0qULh/EVjH7Cxwo+C2gwTeChU94s\njvTiLw+EIc+fRWae0Uo20k7QmHaCxi2/gvEhiIRZiPhZWtLdy8mBXQIt07C6TqKq5QJZVduZ6n/9\nv6GiKGOTSpQVRVFGWFrF3wBJlM8y3F38vrJdOhyQm4U8mYw8nQJlRZc36vUY42dii45DTEmEkIju\nHuMTv9sLQHRw7yNe3Iq6Tmo8VdoMiXPRTZ8HgKyvQaadxOX8GdpPpSDLirRlx8fg6o6YNhsxYwFM\nnY5w6dnbLIQgIWAte/LWc77qcyb6rsRkuLH6Z0VRRieVKCuKooygquZsihuOoxcuTL5i3GQpJeRd\nQKYeQB4/1KOkAjd3RNwsSJyDdfFKdO4WqqqqvrLvrvGCY64xNNytyM/LFV8PM9UNrZRUNxHceUKf\n8PRBLFiO19fWIm02qlIOIc+mIs8cg9JCZPJeZPJeMJkRcTNh5gJE/CytzAPwd59MgPtUypvOcb5q\nKwkB65z4KhVFGS4qUVYURRlBZys+AGCS9XbMBi9kQTYy9SDy2EFtOLQuPn6ImQsQiXMhanJ3KYDO\nvfeeS4dDktU52ci1xlC+VcWE+HAko5SsopruRPlKwmBAxMQhYuJgzVNaqcuJw8jjh7Vh9Y4fguOH\nkEYXSJiFbt5SiJtJvP9aynPPkVW1g0m+q3A1DmySF0VRRj+VKCuKooyQiqYMypvS8K4zMuV8E47j\n34SKkst38PJFzFqImLVIGw94ACfhFV1qoKXNhr+3Kz4W8zBEP3ZFB2uJ8vmiGpZNC73u/UVAEGL1\nGli9BllVgTxxBHniMGRnwvHDOI4fBjcLPrMWEhMRyXnvbDIufcqMwMdH4NUoijKSVKKsKIoyAhyN\n9VRue4VlpxuxltmBj7UNHl6ImQsRsxdpJ5ENcoSK7hn5VH3yV3T1sA9mKmth9Ufcdi/cdq82eUvq\nfmTyPm04uv3bid8PkR6CotjPaLwtDkvEjKEOX1EUJ1KJsqIoyjCRdjukn0Ie3oXj1BFibdpoFdJk\nRjdrIWLuUoiOQ+j1N/xcmYWqPvlaYjvbJKOg+ob2I3z9EKsegFUPIIvzkUf3Io/ux726kpjUVkj9\nT+xhkYiFKxFzlyKuUSajKMrYoRJlRVGUISZLC5EHv0Qe3Qt1l3sxy0P16Beuwn/Rk9osdEPobJ52\n8l/chK+OonGrmxJuBSCzsJoOmwOj4cbHlRbB4YgHnkDe9xitmUcp/vJXBGe14VKQgyzYjHz/LcSM\n+YhFt0FM/E0/lrWi3KxUoqwoijIEZEc78vhh5P5tcCH98gb/IC4lBnM0JBOXceHcHvWPCDH0SVNa\nZ6IcrxLlr/B0c2FCgCd55fVkFdcwtTNxHgpCp8N1ynxa/Ur4rPQjogt9icvyhszTyJT9yJT94BeA\nWLgCsWAFwnfckD23oijDTyXKiqIoN0CWFiH3b0ce2Q1NDdpKkxkxZwli4UrawgI5dPE7dDh0zBn/\nMLphSJJrm9rIr2jAbNQzKViNvNCb+Al+5JXXk5Z3aUgT5S4x1rvIrtlDZkQNPosfJrTjn5GHdyEP\n7YJL5chP3kF++heYOh1d0h0QP3tISm4URRleKlFWFEUZINnRro2EsH8bZJ27vCEsErHkDsTcJQiz\nGwDnSv9Ah6OFQEsC4y3xwxLPuTxtTOXJYb4Y9Oon/t7ER1j5+9EczuZdYl1SzJDv36g3E++/htSS\nNzhd/leCJ/4c/T2PIO9eBxlnkAd3Ik8lQ9oJHGknwNcPsXgVYvHtCC9VV64oo5VKlBVFUfpJll3R\ne9x4Ve/xklWICZN63L++rYTs6t0IBNMCHh62uFR98vV1laSczf3qRC1DZYL3ErKqdlDXVsiF6p3E\n+t2F0Om12f2mTkc21COP7ELu2wYVpchP/oz87F1E4jzE0tVaLbMQwxafoigDpxJlRVGUPkiHHc4e\nx7H7M0g/dXlDaERn73ESwtXtq4+TklNlf0biINJnGV7mkGGLUdUnX1/Xl4hzBVXYHQ70w3BynU7o\nSBz/MPvyf0F65SeEey3sMQmJ8PBE3H4/cuW9kHkax94v4HQK8vghbVKT8SGIpDsQC5Yj3NSIGYoy\nGqhEWVEUpReyqRF5aCdyz+eXZ8xzcUHMSUIsuQMmTOyz96+44TiljWcw6tyI939wWGM921l6ER8x\n9LW3NwtfDzPBVgvFVY3klNYxaZjGmx5viSfIkkhJ4ylOl7/LvJDnvnIfodPBlOnop0xH1lQhD2xH\nHtgBZUXI97YgP3obMXsJYtldiPCoYYlTUZT+UYmyoijKFWRRHnL3Z9rQbu3t2kq/AMSyO7Xxcd09\nrrsPm6OVk2V/AiDefw1mg9ewxdvU2kF2aS0GvSAmxHfYnudmEB9hpbiqkbN5VcOWKANMD3yM8ovn\nyK87RKRPEv7uk695X+FjRdzzCPLOh+BMitbLnHEaeehL5KEvYeJkxPK7EdPnd09jrijKyFHvOkVR\nbnnSbodTR7Xyiqy0yxumJKJbfjfEz9RqTfspvfITmjuq8DaHE+W7YhgivuK58quQEmJCfDEZ1SgK\nfYmb4Me2Y/mczb3EAwsnDtvzWFz8mez3NdIqP+R46R9YFfUTdKLvw60wGGDGAvQzFiDLipH7tmmJ\n8sUM5MUMpLcvImm1VgvvqUY2UZSRohJlRVFuWbKhTjs5b982qNHqfDG5ajWiy+5CBA68rri+rYTz\nVV8AMDPwyWEZDu5KZ7vrk1XZxfV0n9DX2WbDKdbvLvLqDlLfVkxW1XZi/e7q92PF+GDEuqeR9z6C\nTN6D3L0VSgu1k/+2voeYvVjrZb7q5FFFUYaeSpQVRbnlyPyLyF2fIVMPgK1DWxkQrCXHC5b3enJe\nv/YrJcdL/4BD2on0Xoqf2/D1WnY5fqECgIRINZHF9UyL1BLl0zmV2OyOYR1KT69zYcb4J9hfsJFz\nlR8R5jUfN+PASmOE2RWx9E5k0mrIOK394nEmFXlkD/LIHoiK1f5nZy5AGIzD9EoU5damEmVFUW4J\n0tahzZy3ZytkZ2orhYD4WVp5xZTEG55muLD+KBVN6bjoLSQEPDQEUV9fapZ2ouGc6PEj8nxj2Tgv\nt+4Z+tILqkiIGN4vF4EeCYR4zKKo4Rinyv7MgtAXB7UfIQRMSUQ/JRFZWYbc+zny4E7IzkRmZ2rT\nZSfdgUhahfBUYzIrylC67lGhqKiIpUuX4u7uzsyZMzl37tz1HoKUknXr1hEaGopOp6OgoGBIglUU\nRRkoWVuN49N3cHzvGeSW/9aSZFd3xMp70f3kd+j/1/9BxM244SS5zdbQfQJfgv9DmAzXP+nvRhVX\nNVJc1YinmwsxISpB6o/Z0QEApJwvH5HnSxz/DfTChcL6FIobTtzw/sS48ejW/gO6X7yFePR5CAqD\numrkp+/geOlpHG/8DzL3whBErigK9CNR/sd//EcSEhKorq5m3bp1rFu3rl87XrRoER988MENB6go\nijJQUkpkdiaO13+J43tPI//+LtTVQFAY4tHn0W18C926pxH+gUP2nCfK/kirrY5xbjFE+iQN2X77\ncqyzN3nmRH90OjVRRX/M7ux5T80qG5Hnc3fxI95/DQDHSt6i3d40JPsVJjO6pDvQ/ecr6P7tx5A4\nD+wOZPJeHOv/Hfv6b+NI3ovsKi1SFGVQ+iy9qK+vZ+fOnWzZsgWTycS3vvUtfvzjH5OWlkZcXNw1\nHyeE4MUXX8Rmsw15wIqiKNciO9qRKQeQuz+DgmxtpdDB9HlaecUwzXxWVH+Mgroj6IULs4OeRQzz\nCXxdupK9WZ29pMr1dfUoH8sqR0o5Is85ybqKooZjXGrO4kTpH3sdW3mwhBAweRr6ydOQl8q1sowD\nOyA3C/nG/yA/eEsbKWPJHQhvNXygogxUn4nyxYsXMZvNuLu7s3jxYrZs2UJUVBSZmZl9JsqKoigj\nSVZVIvd9oSUIjfXaSncPxOLbEUtXI6z+w/bcbbYGjpf+HoCEgIfwMI1c0tpdnxyj6pP7a2KQN94W\nE2U1zRRWNuDnN/yzGeqEjjlBz7A9+4fk1x0i1GsOwR4zhvx5hF8AYs1TyK89jDy6Vxstozgf+fd3\nkZ+/j5ixALH8bu0kQDVVtqL0S5+JclNTExaLhYaGBjIyMqipqcHDw4OmpqH56aiL1aqGNbpRRqN2\nxrNqy6Gh2nNoDUd7SinpOHeSls8/oC3lADgcABgionG9cw3mRSsRJtOQPd+17Mx4g1ZbHYGeU5k3\n6evD3pvc1ZYurhbSC6ox6HWsmB2Lm1mNetBfC6aG8PnRbDJKmpiTMDLvdStW5jme5FDO65wo+wPR\nwfMwG4exjv3+byDve0R7j2x9n7bUg8jUA8jUA53vkQcxL7ptyN8j6rNz6Ki2HFpd7TlQfSbK7u7u\nNDY2EhISwqVL2riTDQ0NWCxDOwf9j3/84+7rS5YsISlpZOr7FEUZe2RrC637ttP8xd+wF+RoK/V6\nTItW4nbnGgwxcSPWW5Zz6TAXKvdi0JlYHvOvI1ZyAZCSWYrDIZkRPV4lyQM0f4qWKB9JL+aJO6aP\n2PPGB99D9qVDlNWnczD7NVbGfntYn08IgUvcDFziZmCvLKNlxye07PwUW24WDb/9GY1/+C2uK+/G\nddX96AOChjUWRXGGffv2sX//fgD0ej1LliwZ8D76TJQnTpxIS0sLxcXFBAcH097eTnZ2NjExMYOL\n+Bqef/75HrerqqqGdP+3gq5vnKrthoZqz6E1FO0pK0q1+stDX0Jz569ant7asFhLVmHztlIPUF19\n4wH3Q0tHLXuyXwYg3n8ttmYTVc3D///S1Za7jmUBkBhpVf+nAzQ1ROvJPXA6j44O7WS3kWrDGf5P\nsr3hh2RV7MbPFEeo5+wReV50RrhjDWLFPZB6UJumPf8izR+/Q/Mnf4GE2eiW3wWx025oBBj12Tl0\nVFveuLi4uO5SYavVysGDBwe8jz4TZU9PT1atWsWGDRvYuHEjmzZtIjw8vEd98tKlS5k3bx4bNmzo\n8di2trbuD6DW1lZaW1sxm80DDlBRlFuXdDgg/ZQ20ULaceg6+SoyRpuZzEkTLTikg+TiV2mz1+Pv\nPplJvreNeAwHzxUDMC9W1ScP1LTIcZiNejKLaiivaSLAx33EntvDFEhCwNc5WfZHUou34GMOx+Iy\nfDX0VxNGF8SC5bBgOTI3C7l7K/LYATidguN0CowPRiy9sYl3FOVmct0JR1577TUeffRRfH19mTx5\nMu+9916P7fn5+URGRn7lcTExMRQUFCCEIDZWO3HAbrcPXeSKoty0ZGM98vAu5L7tUFGirTQYELOX\nIJbfhbOn7s2o/ISKpnRMek/mBT8/oiUXAHVNrRy/UIFeJ1g4NXhEn/tmYHYxMH9yIHvOFPHl8Vy+\nsXJkT06f5HsbFU3pFDcc50jRb1k+4T/Q60Z+/i8REY14Ohq59inkgR3aVO5lxch3NyM/+iNi/jLE\nsjsRQWEjHpuijBbXfWeGhISwd+/ea27Pzc3tdX1eXt5gY1IU5RYkpYTsDOS+bchjhy5PLe3j111e\nITy8nBskUNGUwbnKjwDBvJDncDV6j3gMe07mY3dI5saMx9PNZcSf/2aQlBDCnjNF7DieM+KJshCC\nOcHPsj07n+qWHM5UvMf08d8Y0Rh6xOPpjbjrIeQdD8Lpozh2b4XzZ7VSp72fw6QpiKTV2qgZgzwh\nSlHGKjWFtaIoTiVbmpHJe5H7voDifG2lEBA3E13SHRA/C6HXOzfITq22Oo4U/T8kksl+9zDeEu+U\nOHYe1zookhJCnPL8N4OlnW2360QuDsfIjKd8JRe9O/ND/pnduT8hq2ob/m6TCfYc+iHjBkLo9TBj\nAfoZC5DF+cg9W5HJ++BCOvJCOtKyGbFgBWLxKsR49UuGcmtQibKiKE4hC7KRe79ApuyHtlZtpYcX\nYtFt2vjH40ZX7a1DOjha/BqttlrGucUQ5/+AU+KQUrLjuDbax1KVKA/axCBvgq0WiqsaOZVdTrjv\nyPeU+rlNJCFgLafL3yWlZDO3m3+Cu8vwj+vcHyI4HPHo88g1TyKP7kfu3wYFOcgdHyN3fKxN3pO0\nGjF9rlPOE1CUkaISZUVRRoxsbkIeO4A8sBPyLlzeEBOvlVdMnzdqD7pny9+jrPEsLnoL80KeRyec\n08udVVRNYUU9vh5m4ieMjqRqLBJCsDQhhD/vyWTn8RyeuW1oR3PqrxjraiqaMiltPMWBgv9hRcR/\nYNS7OiWW3gizGyLpDuSSVZB3QSuNSj2glWacP4v08EIsXKmVRo2yL7eKMhRUoqwoyrCSUtJ+7iSt\nX36G48huaG/XNri5az/jLrkDETi6e0ZzavaSWfU5Aj0LQl7Azei8qYB3dvYmJ8UHo9Op2dVuRFJ3\nopzrtERZCB1zg/+JL3P/L3VthSQXv8rC0G+hG+ETRK9HCAER0YiIaORDT2sz/+3bps38t+1vyG1/\ng9gEWlfdh2memgtBuXmoRFlRlGEha6q0kSsOfUltZdnlDTHxiEUrEdMXjMjMeTeqoimDYyW/B2Bm\n4BMEWKY6NZ7Pj2YDqj55KCyaGoReJ0hOL6amsRUfi3OGMDUZLCwJ+ze+zP2/lDSc5Ez5uySOf8Qp\nsfSHcHNHLLsLufROyM5E7u88ATfzDPWZZxCvu8PMhYiFK7WhHNV02coYphJlRVGGjLR1aOOxHvwS\nzp0EqU0rrbP6Y162mtbpCxD+gU6Osv8a2so4VPgyEjvR1tVE+S5zajxV9S3sO52PQa9j5XQ1ZNeN\n8nI3sTQxnF0n8th2LI+Hl8Y6LRYPUyALQv8X+/J+wfmqL/AwBRHls9Rp8fSHEAImTkZMnIz8+rPI\nlAPoU/Zhu5AOB3YgD+yA8SHaL0fzlyK81VTMytijEmVFUW6IlFKrXUzeq52Y11ivbdAbEInzEYtW\nYl20AqHX0zaGZphqszVyoOC/abc3EmRJZFrA150dEltTcrE7JKtmRzit9/Nms2bJZHadyOPTIzlO\nTZQBAtynMDPoSY6VvMHxkt9jMY5z+i8Y/SXcLIilq/F98FFshbnUbP0bMnkPlBUhP/wD8qM/QtwM\ndAtXQMIcNcycMmaoRFlRlEGRFSXI5H3Io3uhovTyhuBwbeSKuUsRHp4Ao2Z4t/5qtzezL/8XNLSX\n4WUK7Tx5z/k1o58ma/XJa5ZMdnIkN497F0bz4ivbOZReQlV9C1ZP555IF+WzlIa2Us5Xfc7Bwl+R\nFP5d/NyinRrTQBlCI9CtfQp5/2Nw7iSOw1/C6VQ4ewzH2WPgZtFm1Zy7VBuj+QamzFaU4aYSZUVR\n+k021CFTDyCP7oOc85c3ePlos+bNS4KwqDFdk9hhb2F//kZqWnNxN/qzJPzbo2IUgvKaZpIzS3Ex\n6rlnwSRsrU3ODumm4OvhysoZEWxLzWZrSi6Pr5zi7JBICFhHq62O/LpD7M//JUsnfA9f16/OgDva\nCYMBps1GP2229tlxdB/y0C4oytVmAjywA7ytiDlLEHOXQGjkmP7sUG5OKlFWFKVPsqkReeoo8vgh\nOHcCHFrdMSYzYsZ8rVcoNmHM9Rr3xuZo40DBf1PVchE3o5VlE77v1BEurrQ1JQcp4faZEXi5m6lS\nifKQeXBJLNtSs/k0OWdUJMo6oWNO8LM4ZAeF9Snsy/8FSyd8Hx9zuLNDGzTh4YVYeQ+svEebzOTo\nPq1Uq6oCueMj5I6PtHrmuUla4jyGzmVQbm4qUVYU5Su6k+NjByHjNNht2gadTpspb24SInEuwnTz\n1MnaHe0cLPgVlc3ncTX4sGzC90fN5A8AHx7SRrtYk6TKLobaPQsm8c+bdCRnllJc1Uiw1eLskNAJ\nPfNCvom90EZJwwn25f2cZRN+iJd57M+IJ4LDEQ88rpVmZGciU/YhUw9q9cyf/Bn5yZ+1X6ZmzEfM\nXKhmAVScSiXKiqIAXclxsjbM05XJsdDB5GnaAWv6PISnt3MDHQbt9iYOFWyiojkDk96TpRO+h8Ul\nwNlhdTuTW8nJ7Ao83Vz42vxJzg7npuPlbuaOWRP4NDmHP+3K4KWHZjs7JAB0wsCCkBc4WPhryhrP\nsCfvpywO+3esblHODm1I9Bg146FnIPO01tN88igUZGuzd378J+28h86kmaAwVZ6hjCiVKCvKLUxe\nKkeeTkGeToGsNLDbtQ1dyfGshYjp8xEeXs4NdBg1d1SzP38jdW1FmA1eJIW/hKcpyNlh9fD7nekA\nrEuKxt3s4uRobk5P3T6VT5Nz+POeTL51/wxMxtFRSqTXGVkY+i8cKtzUmSyvZ0HoiwR5JDo7tCEl\nDAaIm4mIm4nsaIdzJ5EnDiNPpWiTmhTnI//+LgQEa0nz9PkQHqVOBFSGnUqUFeUWIh0OyM9Gnj6K\nPHUUivMvb9TdOslxl7rWIvblb6TFVo2HSxBJ4d/G3WWcs8PqobqhlY8Pa2UXT4yC+tmb1ezoAKaE\n+ZJeUM1nR3N4cNHo6bk36FxYHPavpJa8SV7tAQ4W/IqZQU+N+nGWB0sYXSBxLiJxrjY2e+YZ5PHD\nyFPJUF6M/OID5BcfaCcRx89CTJsNkxNvqlIwZfRQibKi3ORkazNknkWePYY8nQp11Zc3ml0RcTNh\n2hxE/EyEu4fzAh1h5U3pHCrYRIejGT+3aBaF/ismg/NrU6/27t7ztHXYWT4tlIjxN/+XF2cRQvDU\n7VP5zpYDvLUjfVQlyqCVYcwJehY3o5X0yo85VvIGzR1VxI27HzEKhi4cLsJgvNzT/OjzkJWmYiwB\nFwAAGoZJREFU9TSfSYXqS8iDO5EHd4LBqJ1UnDBbW6yj6wuvMnapRFlRbjLS4YDCXOS5E8hzJyE7\n43JJBYCvH2LaHMS0uRATpx2IbiFSOsi49BlpFR8gkQR7zGJeyDcx6EZfSUNru423dp4D4InbVG/y\ncLt/wUR+8s5RTmZXkHK+jDkx450dUg9CCOL9H8TV4MOJ0t+TXvkx1S05zAt+DpPh5v+SK/R67Vev\nydOQjzwHRXnIM6la6VjeBUg7jkw7jnznd1pd89TpiMmJMGkqwmRydvjKGKUSZUW5Cci6GmT6KTh3\nQrtsqLu8UeggKhYxZToicc4tPVZpm62B5OLfUdZ4BoDJfl8jzn/NqJhMpDd/3JVBSVUTk0N9WT4t\n1Nnh3PRcTQaeWjWVX390kp+9l8KH//G1Uflemei7HHejH8nFr1LWeIbt2f+bBaH/POYmJrkRQggI\njUCERsBdDyHra5Bnj2tJc/qpy3XNOz4GgwEmTkFMSURMSdQ+A1Vts9JPKlFWlDFI1tUgs9Lg/Fnk\n+TQoK+p5B18/xNQZiKnTIXYawn30lRSMtMrmLI4U/pYWWzUuegvzgp8j0GOas8O6psaWdl7+5BQA\nLz00C51u9CVsN6N/ujOBP+xMJ+V8ObtPF7IiMczZIfUq0COBVVE/4XDhb6hqucju3PUkBKwlxrr6\npi7FuBbh6YNYuBIWrkR2dMDFdGTGKWT6aSjI1uqcM88gP3wbLB6I2GkQE4+IngqBoaPyC5EyOqhE\nWVHGAFlb3ZkYp2mXVyfGLiaInqr91Dh1hjZwv/rgB7SZ9s5UvM/F6i8BidV1IgtCX8DNaHV2aH3a\n/PlZqhtamTUpgJXTR2eydjPydHPhhXsS+fE7R9nwXirLEkJH7ZcUN6OV5RE/5Ez5Xzlf9QWny9+l\nqP4Ys4Oexssc4uzwnEYYjd0lGjwAsqEemXka0k9pv7hVV2pjxB87iASweGpTaUfHaYlzyASEbnSM\neqI4n0qUFWWUkTabNsVr9nnIOY/MyYRL5T3vZDJD1GRETBwiJl4bJukWqzXuj+KGExwv+QMttmoE\nOmL97ibO/wF0YnR/9OVX1PPqVq085Adfn62+9IywJ26bwpZtaaQXVPOn3RmjYra+a9EJA4njH2Gc\nWyzHSt+kquUiO3L+N7F+X2OK3z3odepzQXh4ImYvhtmLkVJCeYmWOGedQ2ad005wPpmMPJmsJc6u\nblqpRlQsIjIGJkxCuLo5+2UoTjK6jxaKcpOTUkJtNeRmIXMyteQ4/yJ0tPe8o8kVJsYiYuIR0XEQ\nPlEbd1TpVUNbGWcq/kpRfSoAvq6RzAr6hzExBbDDIfm31/bR3GbjnnmRzI1VU/mONFcXAz96dB7P\nvbyLH79zlKSEEML9PZ0dVp+CPWcwzj2GM+V/JbtmN+mVH1NYd5TE8Q8TaElUX7Y6CSFgfLA229/S\nO7XP4Moy5IVz2i92F85pHRNnj2kjBWkP0n6li4yGiBhExCQInqCdXKjc9NSRVlFGiJRS+wAuyEbm\na7NOUZDT88S7LgHBWk9GV49GcJj6KbAfmjuqOVf5Ebk1+5E4MOhMxPuvZaLvbaP2hL2rvbE9jeTM\nMsZ5ufLTJxc6O5xb1tfmRrJ1bi5/P5rDv2/ez19/cNeoLcHo4qJ3Z1bQU4R7LSC15E0a2ks4UPA/\nWF0nkRCwFn93Nf351YQQ4B+I8A+EhSsBkNWXtIQ55zwyN0v7nC4tRJYWwqFdWvLs4gJhExFhkZ0n\nFUZqswYaVQ/+zUYlyooyDGRri/bBWlKgnX1dkAOFOdDc9NU7u1m00onIGERULEREIyyju/dqtGnu\nqOJ81XYuVn+JQ3Yg0BHpvZSp/veN+lrkKx27UM6G97Re8F88vRhfDzWBgjOtf2ohRzJKOZJRyi//\ndpzvrp3l7JD6ZZx7DKuifsLFml1kVH5KVcsF9uStZ7wlnil+9+LnFq16mPsgfP0Qc5NgbhKAdnJg\nYY6WNOdkIXPPQ2WZdsLgRW3WTAmg12snBoZGaCNrhEVCUDjCQ32ej2UqUVaUGyDbWrWZokoKoLig\nOzGmqqL3B3h4aWUTYVGI8EgIiwKrvzpoDYKUkkstF7hQtZ2i+mNIHACEes4lzv9BPE1jq2Qhp6yO\nJ3+5ndYOO4+tmMztM0d/mcjNztfDzK+fS+LxjdvZ9PFJQsdZeHhprLPD6he9zkiM9Q4ivZPIqtpG\nZtXnlDWepazxLD7mCUyy3k6Y5zxVw9wPwmiEyBjt170V2jrZUA/5F5GFuVoSXZgL5cXa2M5FeXBk\nj5Y8g/a5HxiKCArVLgNDISgMPL3VZ/8YoBJlRbkO2dEBl8q0E0DKS6Ci87K8BGqren+Q3qDVwQWF\naT/HhUZCeCR4+aoPxhvUaqujsC6F3NoD1LTmAiDQE+Y5jxi/u/B1neDcAAehsLKBx36xjZrGNpYn\nhvKTJxY4OySl07Jpoax/aiHfe/MgL71xEE83E3fNiXB2WP1m1Lsy1f9+JvquJKtqG9k1e6hpzSOl\neDOny94lwnsxYd7z8TaFqc+mARAenhA3AxE3o3udbGvVEuXCXG3Sp8IcKCnUyusa6rQRi+ByAu3m\nrtU++weBf+DlEhD/wFtqltTRTiXKyi1PdnRATSVUVSKrKqCqgvrGOuwVZdjLS6D6EkhH7w/WG2Dc\neC0ZDg7TEuPgcBgXqE62G0JttkZKGk9RUHeY8sZz3b3HLnoLUT7Lmei7Ajejr5OjHJzTOZU88cvt\nVNa1ED/Bj9+9uAKDfmzUU98qHlsxmaJLjfzm01P808tf8qNvzOPZ1fHODmtATAYP4gPWMmXcveTX\nJXOheju1rQVkVm0ls2ornqZgwr3mE+o5B4vLeJU0D4IwmbXzSqIu/+ogpYSaS1BahCwtgJLOWueS\nAq0UL+c8Muf85ft3XXGzUB0Uin58MA6LF1jHIXzHQeci3NxH9sXdwtSRXLmpydZmbVSJuhpkbbU2\nDFBdDdRWdyfF1NWAlD0e13rlDaHTkmH/QERAMPgHIQKCICBI+8BSZz4POYd0UNOSS1njGUobz1Dd\nko3sPIQI9ARaEgnzmkeI5+xROfV0fzgckrd2nGP9eym0tttZNDWI1791G+5m9VP4aPS9h2ZhMRvZ\n8NdU/vNPyZzOqeS/Hl8w5urI9ToXIn2WEOG9mEstF8ivPUxh/VHq24o5W/EBZys+wN3oT6AlnvEe\nCfi7TcGoH1uvcTQRQlxObqdO714vpdSOPeUlyIoSqCxFVpRCRSlUlEFzI7aLGdguZlx+zJU7dnW7\nvF9rZwLtbUV4eYOXL3j6gLtFzUA4BK6bKBcVFfHoo4+SmppKbGwsb7/9NlOnTr3ujl9++WXWr19P\ne3s7zz33HOvXrx+SgJVbm7TboakeGhqgsR4a67RascbOpb4WWVcNtTVaUtzWev2dCh34+mnf2K0B\nYB2HJSwCvX8g9Uaz9kGkzmQeVm22BqpasqlqvkhVy0WqWrKxOS7/7XRCzzi3aEI95xHqORuTYWz/\nLJlyvoyf/iWFYxe08bG/nhTNz/5hES4G9aVrtBJC8OK9iYSOs/Dvr+/no8PZ7E8r5qW1s1m7ZNKY\n+9sJIRjnFs04t2hmBD5KWWMaBXVHKG08Q1NHBRdrdnGxZhcCHV7mUPxcJ2J1m4TVdSIWl3G35Ox/\nQ0kIAd6+4O2LiInrsU1KCQ21eLY2Yy8rprEgV5skpaoSqjuXlmbtfJji/B4JdI9kWm8AT29t8fJB\nePmAl4922+KplXdYPMHiAe6eCJNpJF76mCOkvKor7Sp33nknEydOZOPGjWzatIm3336btLS0Pnd6\n9OhR7rzzTg4ePIiXlxeLFi3i5z//OWvXrv3KfXft2sXkyWrImhtltVrJyMjA39/f2aH0SUoJ7W3a\nT04tTd2Xsut2S3OPbbKlGZobobEzMW5uHNgTurho3669fLUPCW/fzts+CKs/WDu/hV9VJjFW2nOs\nsFqtpKefw+JjoKmjksb2curaiqlrLaSurYhW21eHyHM3jmO8JYFASwL+7pMx6l2dEPnQaWrtYPvx\nfN7Zk8mRjFIAArzd+NlTC1k1a0K/96P+N4fWYNozt6yO72w50P13DLK688TKKdw3P4qQcWP7S5z2\na04OpZ2/5tS05HaXOnUx6Ex4moLxMoXiZQ7B4uKPu3Gc9p71D1b/n0PkWv+bUkpoauhOmmXVJaiu\ngNoaraOovlbrrR7M8dLdE9w9wKMzkXZz13qvXTsvzW4Itytuu7qD2Q1c3UZ9uaHVauXgwYOsWLFi\nQI/rM1Gur6/HarWSn59PUFAQ7e3tWK1Wjhw5Qlxc3LUexne+8x3q6urYvHkzAOvXr+f48eP87W9/\n+8p9VaI8NKxWKx9++CFJSUkDfqx02KHDBrZ26OgAW+fSceVle+d9OpC2Dm1CjLY2aGvRLttbtd7b\ntjZk9/XOpb2texvtrV8pcxgQIbQ3scUTPDy1b8UWz87bXtqb28v3ckLs6jaoWrsbac9biZQOOhwt\ntNub6LA302ZvpNVWS4utltaOWlptdbTYammX9dQ1l4HovdbboDPjbQ7D6joRP7eJWF0n4Wr0HuFX\nM7SqG1rJKKjmdE4lh9NLSD5fRkubDdCmSX56VRzPro7Dy31gvTjqf3NoDbY9HQ7Jp8nZvPzJKc4X\n1XSvT4jwY8GUIGZN8mdKuJVQP49RP/5yX2yOVqpbcqlqvsillotUt+TQaqu95v3NBk9sLS6Mt0bg\navDBbPDEpPfAZPDsvO6Ji96CUW8e9TNkOtuNvtdlR/vlpLmuBllXA/U1UFeLbKrXOqCaun6dbdCO\n9zfCxUVLmk1mcDFdXkxmhIsJTF3rzN3ru9YJkxkMLmA0gMF4eTEar7rduV1vGPCxfbCJcp//pRcv\nXsRsNuPu7s7ixYvZsmULUVFRZGZm9pkoZ2VlsWTJEjZt2kRhYSGLFi3inXfeGVBgo0Xe8X20lhch\npNQSPCm168ge63B03XZo96HrvvR8nHR85bGix365an8SIR0IhwNht1++3rnQeVkrBJG1tVTve7/H\n9msudjs6uw2dzXb5eUeI3WDAZjJhN5mwmczYXUzYzCZsLi7abZMJm8mEzeTSfb8OV1dtcTHDNWuu\nutqvFllTCzU5141FXuMkPTc3Ny7UX8R2rud2yfXaqn9tqTV5X/eVA9gb1/nyIZHYAQcSe2fvkKN7\nnXbdAd3b7EhsSNq1S9GOpKN7gQ4ctCFpRdKG9k/eDwJ00gOd9EGPL3rpj0EGYJAB6PBGtOmoqpNo\n44jUdy6DeLk97tfPv0f/dgcS2mx2mltttLR10Nxm61w6qGlso6y6ibKaJkqrm2ho+epBZ3Z0APfN\nj+KBRZPwdBubtdWKRqcT3LdgIvfMi2L36UI+PHSR7cfzOJN7iTO5l7rv52LQMd7HnfG+bgT6WvDz\ncsXNZOhcjN3XXV0M6PU6dEKg04FOCIQQ6HWic512WycY1sRb0Nu+/QA/PJiHhwHs+ibaZCltjlLa\nZTkdsrp7abXVgxGK6i/1sp+rn8uADhM6YUKHufPShE6YERgQ6C9fit5uGzonEtJ1Ri4AAeiuuC4Q\nXdvFtbZf/Zqv1759bRd93BrYYy1lHuQ1FGO+eKJ/YfXF0wU8A4CA3rdLia6jA0NLc+fSgr6lGUNb\nK/q2NvTdl1de77lOtLdDe3vvu79OeIPJQhx6PVKvx6E3IA2dl3o9UqdD6vRIve7ydZ0O759tGcSz\nXCdRbmpqwmKx0NDQQEZGBjU1NXh4eNDU1MukCb08Lj09nfz8fFavXk1j47V/ArBaR++EABnb3iI6\nr9rZYfTLYH/wk4BDD3YDOPQCux4chs5LvdC2dV6/8j52o8BmFNiNYDMKbIbL62zGXq4btOt0f8jb\ngabOZYABD7dm8IiES+wdnv0P9XHOiR1WLa16WtoMtLRqS32jC/WNLjR0XtY3aZc1dWbaO66s42wF\n8juXm4+72cjUCeNIiPRnUXwoSQnhBFotN7xfo9HI8uXL8fYe2z3uo8VQtOe6lX6sWzmd5tYODqcX\nse90AScvlJGWW0FZTRMFlQ0UVDYA5UMX+KhhAPwBfwQSi3s7Vu9WfL3a8LC0Y3H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       "text": [
        "<matplotlib.figure.Figure at 0x7fd655902860>"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So what is this telling us? The blue gaussian is very narrow. It is saying that we believe $x=23$, and that we are very sure about that. In contrast, the red gaussian also believes that $x=23$, but we are much less sure about that. Our believe that $x=23$ is lower, and so our belief about the likely possible values for $x$ is spread out - we think it is quite likely that $x=20$ or $x=26$, for example. The blue gaussian has almost completely eliminated $22$ or $24$ as possible value - their probability is almost $0\\%$, whereas the red curve considers them nearly as likely as $23$.\n",
      "\n",
      "If we think back to the thermometer, we can consider these three curves as representing the readings from three different thermometers. The blue curve represents a very accurate thermometer, and the red one represents a fairly inaccurate one. Green of course represents one in between the two others. Note the very powerful property the Gaussian distribution affords us - we can entirely represent both the reading and the error of a thermometer with only two numbers - the mean and the variance.\n",
      "\n",
      "The standard notation for a normal distribution for a random variable $X$ is just $X \\sim\\ \\mathcal{N}(\\mu,\\sigma^2)$ where $\\mu$ is the mean and $\\sigma^2$ is the variance. It may seem odd to use $\\sigma$ squared - why not just $\\sigma$? We will not go into great detail about the math at this point, but in statistics $\\sigma$ is the *standard deviation* of a normal distribution. *Variance* is defined as the square of the standard deviation, hence $\\sigma^2$.\n",
      "\n",
      "It is worth spending a few words on standard deviation now. The standard deviation is a measure of how much variation from the mean exists. For Gaussian distributions, 68% of all the data falls within one standard deviation($1\\sigma$) of the mean, 95% falls within two standard deviations ($2\\sigma$), and 99.7% within three ($3\\sigma$). This is often called the 68-95-99.7 rule. So if you were told that the average test score in a class was 71 with a standard deviation of 9.4, you could conclude that 95% of the students received a score between 52.2 and 89.8 if the distribution is normal (that is calculated with $71 \\pm (2 * 9.4)$). "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The following graph depicts the relationship between the standard deviation and the normal distribution. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from gaussian_internal import display_stddev_plot\n",
      "display_stddev_plot()\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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5CxqsIg7qaGo6k5clYLHAC4Oam84RcUkjLl/LOmnpXo6fzTBcIyJ/RoNVxEGN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\niIjI1/4FdO2S7HSiY/gAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7fd65572d940>"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<p>\n",
      "<p>\n",
      "<div style=\"border:1px dotted black;background-color:#DCDCDC#\">**Sidebar**: An equivalent formation for a Gaussian is $\\mathcal{N}(\\mu,1/\\tau)$ where $\\mu$ is the *mean* and $tau$ the *precision*. Here $1/\\tau = \\sigma^2$; it is the reciprocal of the variance. While we do not use this formulation in this book, it underscores that the variance is a measure of how precise our data is. A small variance yields large precision - our measurement is very precise. Conversely, a large variance yields low precision - our belief is spread out across a large area. You should become comfortable with thinking about Gaussians in these equivalent forms. Gaussians reflect our *belief* about a measurement, they express the *precision* of the measurement, and they express how much *variance* there is in the measurements. These are all different ways of stating the same fact.</div>"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Interactive Gaussians"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For those that are reading this in IPython Notebook, here is an interactive version of the Gaussian plots. Use the sliders to modify $\\mu$ and $\\sigma^2$. Adjusting $\\mu$ will move the graph to the left and right because you are adjusting the mean, and adjusting $\\sigma^2$ will make the bell curve thicker and thinner."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "from IPython.html.widgets import interact, interactive, fixed\n",
      "import IPython.html.widgets as widgets\n",
      "\n",
      "def plt_g(mu,variance):\n",
      "    xs = np.arange(2, 8, 0.1)\n",
      "    ys = [gaussian (x, mu, variance) for x in xs]\n",
      "    plt.plot(xs, ys)\n",
      "    plt.ylim((0, 1))\n",
      "    plt.show()\n",
      "\n",
      "interact (plt_g, mu=(0, 10), \n",
      "          variance=widgets.FloatSliderWidget(value=0.6, min=0.2, max=4.5))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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eFi3bWrwyxexnB8jP6m50JFRhfdvXP7uCxaiPVivqULLBiQBUJJRjAJfVgvWx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\ndpuXMPd/IhEAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7fd65574eb38>"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 9,
       "text": [
        "<function __main__.plt_g>"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Finally, if you are reading this in an IPython Notebook, here is an animation of a Gaussian. First, the mean is being shifted to the right. Then the mean is centered at $\\mu=5$ and the variance is modified.\n",
      "\n",
      "<img src='gaussian_animate.gif'>"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Computational Properties of the Gaussian"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Recall how our discrete Bayesian filter worked. We had a vector implemented as a numpy array representing our belief at a certain moment in time. When we performed another measurement using the `update()` function we had to multiply probabilities together, and when we performed the motion step using the `predict()` function we had to shift and add probabilities. I've promised you that the Kalman filter uses essentially the same process, and that  it uses Gaussians instead of histograms, so you might reasonable expect that we will be multiplying, adding, and shifting Gaussians in the Kalman filter.\n",
      "\n",
      "A typical textbook would directly launch into a multipage proof of the behavior of Gaussians under these operations, but I don't see the value in that right now. I think the math will be much more intuitive and clear if we just start developing a Kalman filter using Gaussians. I will provide the equations for multiplying and shifting Gaussians at the appropriate time. You will then be able to develop a physical intuition for what these operations do, rather than be forced to digest a lot of fairly abstract math.\n",
      "\n",
      "The key point, which I will only assert for now, is that all the operations are very simple, and that they preserve the properties of the Gaussian. This is somewhat remarkable, in that the Gaussian is a nonlinear function, and typically if you multiply a nonlinear equation with itself you end up with a different equation. For example, the shape of `sin(x)sin(x)` is very different from `sin(x)`. But the result of multiplying two Gaussians is yet another Gaussian. This is a fundamental property, and a key reason why Kalman filters are possible."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Computing Probabilities with scipy.stats"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In this chapter I have used by custom written code for computing Gaussians, plotting, and so on. I chose to do that to give you a chance to look at the code and see how these functions are implemented.  However, Python comes with \"batteries included\" as the saying goes, and it comes with a wide range of statistics functions in the module scipy.stats. I find the performance of some of the functions rather slow (the scipy.stats documentation contains a warning to this effect), but this is offset by the fact that this is standard code available to everyone, and it is well tested. So let's walk through how to use scipy.stats to compute various things.\n",
      "\n",
      "The scipy.stats module contains a number of objects which you can use to compute attributes of various probability distributions. The full documentation for this module is here: http://http://docs.scipy.org/doc/scipy/reference/stats.html. However, we will focus on the  norm variable, which implements the normal distribution. Let's look at some code that uses `scipy.stats.norm` to compute a Gaussian, and compare its value to the value returned by the `gaussian()` function."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from scipy.stats import norm\n",
      "print(norm(2, 3).pdf(1.5))\n",
      "print(gaussian(x=1.5, mean=2, var=3*3))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "0.131146572034\n",
        "0.13114657203397997\n"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The call `norm(2,3)` creates what scipy calls a 'frozen' distribution - it creates and returns an object with a mean of 2 and a standard deviation of 3. You can then use this object multiple times to get the probability density of various values, like so:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "n23 = norm(2,3)\n",
      "print('probability density of 1.5 is %.4f' % n23.pdf(1.5))\n",
      "print('probability density of 2.5 is also %.4f' % n23.pdf(2.5))\n",
      "print('whereas probability density of 2 is %.4f' % n23.pdf(2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "probability density of 1.5 is 0.1311\n",
        "probability density of 2.5 is also 0.1311\n",
        "whereas probability density of 2 is 0.1330\n"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "If we look at the documentation for `scipy.stats.norm` <a href=\"http://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.norm.html#scipy.stats.normfor\">here</a>[1] we see that there are many other functions that norm provides.\n",
      "\n",
      "For example, we can generate $n$ samples from the distribution with the `rvs()` function."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print(n23.rvs(size=15))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[ 0.8890914  -1.21963858 -0.36342745 -1.69175021  1.04580929 -4.03699521\n",
        "  0.62282703  2.86612581  3.95733631  0.29548548  4.90591126  1.03687335\n",
        " -0.75739343 -3.71850042  1.8192876 ]\n"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can get the *cumulative distribution function (CDF)*, which is the probability that a randomly drawn value from the distribution is less than or equal to $x$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# probability that a random value is less than the mean 2\n",
      "print(n23.cdf(2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "0.5\n"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can get various properties of the distribution:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print('variance is', n23.var())\n",
      "print('standard deviation is', n23.std())\n",
      "print('mean is', n23.mean())"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "variance is 9.0\n",
        "standard deviation is 3.0\n",
        "mean is 2.0\n"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "There are many other functions available, and if you are interested I urge you to peruse the documentation. I find the documentation to be excessively terse, but with a bit of googling you can find out what a function does and some examples of how to use it. Most of this functionality is not of immediate interest to the book, so I will leave the topic in your hands to explore. The <a href=\"http://docs.scipy.org/doc/scipy/reference/tutorial/stats.html\">SciPy tutorial</a> [2] is quite approachable, and I suggest starting there. "
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Summary and Key Points"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The following points **must** be understood by you before we continue:\n",
      "\n",
      "* Normal distributions occur throughout nature\n",
      "* They express a continuous probability distribution\n",
      "* They are completely described by two parameters: the mean ($\\mu$) and variance ($\\sigma^2$)\n",
      "* $\\mu$ is the average of all possible values\n",
      "* The variance $\\sigma^2$ represents how much our measurements vary from the mean\n",
      "* The standard deviation ($\\sigma$) is the square root of the variance ($\\sigma^2$)"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "References"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "[1] http://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.norm.html#scipy.stats.normfor\n",
      "\n",
      "[2] http://docs.scipy.org/doc/scipy/reference/tutorial/stats.html"
     ]
    }
   ],
   "metadata": {}
  }
 ]
}