Lab2.ipynb

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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Elementary Notions in Statistics\n",
    "\n",
    "* Main ideas behind probability distributions in general: https://youtu.be/oI3hZJqXJuc\n",
    "* The normal distribution, clearly explained: https://youtu.be/rzFX5NWojp0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Random variables"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> A _random variable_ is a function that assigns a real value to each outcome from a sample space $\\Omega$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It usually denoted by capital letters: $X: \\Omega \\to \\mathbb{R}$.\n",
    "\n",
    "In many cases the random variable is effectively just a way to model the output of an experiment as a number.\n",
    "\n",
    "In a mathematical sense, it is *not random* and *not a variable*. The name makes sense when analysing phenomena from a statistical perspective: it is a variable because it can influence the studied outcome (i.e. temperature and humidity are important variables in analysing precipitation levels) and are random because their values are unpredictable (even if we can assign some probabilities to them, they are effectively outside our control)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Several different random variables can be defined on the same sample space."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example\n",
    "\n",
    "Since random variables are just associations between omega and a number, the simplest way to represent a random variable would be through a dictionary. So for the tossing of a coin, for example, we can have:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1\n"
     ]
    }
   ],
   "source": [
    "omega = set(['H', 'T'])\n",
    "X = {'H': 1, 'T': -1}\n",
    "print(X['H'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If two random variables are independent, they can be added or multiplied by adding or multiplying their corresponding values:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Z=X+Y: {'T': 2, 'H': 3}\n",
      "Z=XY: {'T': -3, 'H': 2}\n"
     ]
    }
   ],
   "source": [
    "omega = set(['H', 'T'])\n",
    "X = {'H': 1, 'T': -1}\n",
    "Y = {'H': 2, 'T': 3}\n",
    "Z = {o: X[o]+Y[o] for o in omega} # Z = X + Y\n",
    "print('Z=X+Y:', Z)\n",
    "Z = {o: X[o]*Y[o] for o in omega} # Z = X * Y\n",
    "print('Z=XY:', Z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "**Give it a try!**\n",
    "\n",
    "Create two random variables $A, B$ on the same sample space $\\Omega=\\{red, green, blue\\}$ and add them together."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Your code here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Discrete random variables\n",
    "\n",
    "> Let $\\Omega$ be a sample space. A _discrete random variable_ is a function $X: \\Omega \\to \\mathbb{R}$ that takes on a finite number of values $a_1, a_2,\\dotsc,a_n$ or a countably infinite number of values $a_1, a_2,\\dotsc$.\n",
    "\n",
    "Examples:\n",
    "* Tossing a coin\n",
    "* Number of bees in Europe\n",
    "* How many tries it takes to get a 6 on a die\n",
    "\n",
    "> Two discrete random variables $X$ and $Y$ are *independent* if $P(X=x,Y=y)=P(X=x)P(Y=y)$ for all $x$, $y$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Probability mass function (pmf)\n",
    "\n",
    "> The _probability mass function_ $p$ of a discrete random variable $X$ is the function $p:\\mathbb{R} \\to [0,1]$ defined by\n",
    ">\n",
    ">  $p(a) = P(X = a)=P(\\{ \\omega \\in \\Omega | X(\\omega) = x\\})$ for $-\\infty < a < \\infty$\n",
    "\n",
    "**Important property: The sum of the probabilites of all values must be 1.**\n",
    "\n",
    "<img src=\"img/random_variable.svg\" alt=\"Random Variable\" width=\"500\"/>\n",
    "\n",
    "*Source: Wikipedia. By Niyumard - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=76473622*\n",
    "\n",
    "#### Example\n",
    "\n",
    "In Python, we can create this function using `tools.stats.probability_weighted` from Lab1:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.2"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from tools.stats import probability_weighted, WeightedOutcome\n",
    "from dataclasses import dataclass\n",
    "\n",
    "@dataclass(frozen=True)\n",
    "class Outcome(WeightedOutcome):\n",
    "    weather: str\n",
    "\n",
    "omega = set([Outcome(weather='Rain', weight=1), \n",
    "             Outcome(weather='Sunny', weight=1),\n",
    "             Outcome(weather='Overcast', weight=3)])\n",
    "X = {'Rain': -1, 'Sunny': 2, 'Overcast': 3}\n",
    "\n",
    "def pmf(X, a, omega):\n",
    "    A = set(o for o in omega if X[o.weather] is a)\n",
    "    return probability_weighted(A, omega)\n",
    "\n",
    "pmf(X, 2, omega)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "**Give it a try!**\n",
    "\n",
    "Create a random variable $Y$ that maps the side of a coin to the values 0 and 1 for \"Tail\" and \"Head\", respectively. The face corresponding to \"Head\" is twice as likely to show compared to \"Tail\". Print the pmf for 0 and 1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Your code here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "However, in practice, once a discrete random variable X is introduced, the sample space $\\Omega$ is no longer important, because we can now just operate with the values of $X$ instead (sometimes denoted as $k$ or $x$ instead of $a$).\n",
    "\n",
    "We can plot the pmf for every value of $X$ (or, in other words, for every value of $k$) using `matplotlib.pyplot`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "k = sorted(set(X.values()))\n",
    "pmf_X = [pmf(X, x, omega) for x in k]\n",
    "\n",
    "fig, ax = plt.subplots(1, 1)\n",
    "ax.bar(k, pmf_X)\n",
    "plt.ylabel(\"pmf\")\n",
    "plt.xlabel(\"k\")\n",
    "plt.title(\"Probability mass function\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "**Give it a try!**\n",
    "\n",
    "Plot the pmf of a loaded die for which the side 6 is twice as likely as any other."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Your code here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Bernoulli distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Description\n",
    ": The probability distribution of a single experiment for which\n",
    "the outcome is one of two possible values.\n",
    "\n",
    "Example\n",
    ": Tossing of a (possibly unfair) coin.\n",
    "\n",
    "Notation\n",
    ": Bernoully(p); Ber(p)\n",
    "\n",
    "Parameters\n",
    ": $0 \\leq p \\leq 1$\n",
    "\n",
    "Support (values of X)\n",
    ": $k \\in {0,1}$\n",
    "\n",
    "pmf\n",
    ": $\\begin{cases} p& \\text{if}\\ k=1 \\\\ 1-p& \\text{if}\\ k=0 \\end{cases}$\n",
    "\n",
    "sometimes also written as $p^k \\cdot (1-p)^{1-k}$\n",
    "  \n",
    "Mean\n",
    ": $p$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import bernoulli\n",
    "p = 0.3\n",
    "k = [0, 1]\n",
    "pmf_X = bernoulli.pmf(k, p)\n",
    "\n",
    "fig, ax = plt.subplots(1, 1)\n",
    "ax.bar(k, pmf_X)\n",
    "plt.ylabel(\"pmf\")\n",
    "plt.xlabel(\"k\")\n",
    "plt.title(\"Probability mass function\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Poisson distribution\n",
    "\n",
    "Description\n",
    ": Probability of a given number of events occuring in a fixed interval of time or space.\n",
    "\n",
    "Example\n",
    ": Number of phone calls $k$ received in a day, given that we expect $\\lambda$ calls a day on average.\n",
    "\n",
    "Notation\n",
    ": Pois($\\lambda$)\n",
    "\n",
    "Parameters\n",
    ": $\\lambda \\in \\mathbb{R}^+$\n",
    "\n",
    "Support\n",
    ": $k \\in \\mathbb{N}_0$\n",
    "\n",
    "pmf\n",
    ": $\\frac{\\lambda^ke^{-\\lambda}}{k!}$\n",
    "\n",
    "Mean\n",
    ": $\\lambda$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import poisson\n",
    "lambda_ = 2\n",
    "k=range(10)\n",
    "\n",
    "pmf_X = poisson.pmf(k, lambda_)\n",
    "\n",
    "fig, ax = plt.subplots(1, 1)\n",
    "ax.bar(k, pmf_X)\n",
    "plt.ylabel(\"pmf\")\n",
    "plt.xlabel(\"k\")\n",
    "plt.title(\"Probability mass function\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "**Give it a try!**\n",
    "\n",
    "Calculate the pmf $p(1)$ for a Poisson distribution of parameter $\\lambda = 1$ using `scipy.stats`, then using the formula directly. *Note: constant e is available with `math.e`.*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Your code here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Continuous random variables\n",
    "\n",
    "> A random variable $X$ is _continuous_ if for some function $f:\\mathbb{R} \\to \\mathbb{R}$ and for any numbers $a$ and $b$ with $a \\leq b$,\n",
    ">\n",
    "> $P(a\\leq X \\leq b) = \\int_a^bf(x) \\,dx$.\n",
    "> \n",
    "> The function $f$ has to satisfy $f(x) \\geq 0$ for all $x$ and $\\int_{-\\infty}^{\\infty}f(x) \\,dx=1$\n",
    ">\n",
    "> We call $f$ the probability density function (pdf) of X.\n",
    "\n",
    "Examples:\n",
    "* height of a person\n",
    "* how much weight a bridge can hold\n",
    "* price of a car"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Normal (Gaussian) distribution\n",
    "\n",
    "Description\n",
    ": One of the most important distributions in probability theory and statistics since, among other things, it can model measurement errors and the average of any other distribution.\n",
    "\n",
    "Example\n",
    ": Height of a person can be approximated as a normal distribution.\n",
    "\n",
    "Notation\n",
    ": $N( \\mu, \\sigma^2)$\n",
    "\n",
    "Parameters:\n",
    "* $\\mu \\in \\mathbb{R}$ (mean, location)\n",
    "* $\\sigma^2 > 0$ (variance, squared standard deviation. In scikit, this is the square of the scale parameter.)\n",
    "\n",
    "Support (values of X)\n",
    ": $x \\in \\mathbb{R}$\n",
    "\n",
    "pdf\n",
    ": $\\frac{1}{\\sigma\\sqrt{2\\pi}} e^{-\\frac{1}{2}\\left(\\frac{x - \\mu}{\\sigma}\\right)^2}$\n",
    "  \n",
    "Mean\n",
    ": $\\mu$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import norm\n",
    "import numpy as np\n",
    "mu = 0\n",
    "sigma = 2\n",
    "x = np.linspace(-5, 5)\n",
    "pdf_X = norm.pdf(x, mu, sigma)\n",
    "\n",
    "fig, ax = plt.subplots(1, 1)\n",
    "ax.plot(x, pdf_X)\n",
    "plt.ylabel(\"pdf\")\n",
    "plt.xlabel(\"x\")\n",
    "plt.title(\"Probability density function\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we are interested int the probability that a value is up to and including $x$, we can plot the _cumulative distribution function_:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import norm\n",
    "import numpy as np\n",
    "mu = 0\n",
    "sigma = 2\n",
    "x = np.linspace(-5, 5)\n",
    "cdf_X = norm.cdf(x, mu, sigma)\n",
    "\n",
    "fig, ax = plt.subplots(1, 1)\n",
    "ax.plot(x, cdf_X)\n",
    "plt.ylabel(\"cdf\")\n",
    "plt.xlabel(\"x\")\n",
    "plt.title(\"Cumulative distribution function\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "**Give it a try!**\n",
    "\n",
    "Grades at an exam follow the distribution $N( 5, 1.5^2)$. What percentage of students failed the exam if the minimum grade to pass is 4?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Your code here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Expected result: ~0.252\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Expectation\n",
    "\n",
    "> The _expectation_ of a discrete random variable $X$ taking the values $a_1, a_2,...$ and with probability mass function $p$ is the number\n",
    ">\n",
    "> $E[X] = \\sum_i a_iP(X = a_i) = \\sum_i a_ip(a_i)$\n",
    "\n",
    "Also called _mean_ or _expected value_."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mean is 2.0\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import poisson\n",
    "lambda_ = 2\n",
    "k=range(10)\n",
    "\n",
    "pmf_X = poisson.pmf(k, lambda_)\n",
    "\n",
    "mean = poisson.mean(lambda_)\n",
    "print('mean is', mean)\n",
    "\n",
    "fig, ax = plt.subplots(1, 1)\n",
    "ax.bar(k, pmf_X)\n",
    "ax.vlines(mean, 0, max(pmf_X), colors='red')\n",
    "plt.ylabel(\"pmf\")\n",
    "plt.xlabel(\"k\")\n",
    "plt.title(\"Probability mass function\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "**Give it a try!**\n",
    "\n",
    "What is the mean of the random variable $X$ with the corresponding pmf below?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = {'A': 1, 'B': 2, 'C': 3}\n",
    "k = [1, 2, 3]\n",
    "pmf_X=[0.2, 0.2, 0.6]\n",
    "# Your code here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Expected result: 2.4\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> The _expectation_ of a continuous random variable $X$ with probability density function $f$ is the number\n",
    ">\n",
    "> $E[X] = \\int_{- \\infty}^{\\infty} xf(x) \\,dx$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mean is 7.0\n"
     ]
    },
    {
     "data": {
      "image/png": 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lLjp16oRevXphxowZuHHjBpo1a4ZvvvkGlZWVDb7mxIkT8eWXXyIhIQEHDx5Ev379UFxcjJ07d+KFF17AiBEj4OjoiI4dO2L9+vVo164dmjVrhs6dOxuWZ7hVeHg4Jk2ahBUrViA/Px/9+/fHwYMHsXr1asTHxxu1SBKZhNA5aEQWqHrK8aFDh+563qRJkyRnZ+ca7yssLJRefvllKTAwULK3t5fatm0rvf/++4bpxdUASFOnTpW++uorqW3btpJSqZS6detmNK1ZkiTp5s2b0uTJkyVvb2/JxcVFiouLk86cOSO1bNnSaEp6de179uyRnnvuOcnT01NycXGRxo8fL+Xl5Rlds6HT4M+cOSM98MADkqOjowTgjinx5eXlkqenp+Tu7i6Vlpbe9T281fHjx6X+/ftLKpVKat68uTRnzhzps88+q3X6d1xcnOTu7i6pVCqpdevW0pNPPmm0nEBtfz+3v6bvv/9eevTRRyVfX1/JwcFBatGihfT8889LGRkZRs+H26abFxUVSX/5y18kDw8PCUCNU+IHDx4sAZD27dtX5/ehpmnwkiRJFy9elGJjYyWlUin5+flJb731lrRjx44ap8F36tTpjsdPmjTpjhpLSkqkt99+W2rVqpVkb28v+fv7S6NGjZIuXrxoOGffvn1SZGSk5ODgYDQlvqafjYqKCundd981XC84OFiaMWOGVFZWVqfXePvPJNH9kEkSR5QRkelUVlYiMDAQw4YNw2effSa6HKEee+wxnDhxAhcuXBBdCpHN4RggIjKpjRs3IicnBxMnThRdilAZGRnYvHmz0N3ciWwZW4CIyCQOHDiA48ePY86cOfD29q5xgUZbcPnyZfz+++/49NNPcejQIVy8eLHGKftE1LTYAkREJlG9b5Svr69Nb2q5Z88eTJgwAZcvX8bq1asZfogEYQsQERER2Ry2ABEREZHNYQAiIiIim8OFEGug0+lw/fp1uLq6NmjXYyIiIjI9SZJQWFiIwMDAe+7DyABUg+vXryM4OFh0GURERNQA6enpCAoKuus5DEA1qN6gMj09HW5uboKrISIiorpQq9UIDg422mi6NgxANaju9nJzc2MAIiIisjB1Gb7CQdBERERkcxiAiIiIyOYwABEREZHNYQAiIiIim8MARERERDaHAYiIiIhsDgMQERER2RwGICIiIrI5DEBERERkcxiAiIiIyOYwABEREZHNYQAiIiIim8MAREREjUKSJOSXaKDVSaJLIbon7gZPRET37WjaTcz/+QwOXr4Be4UMzT0c0cLLGS2aOaJlM2e08HJCi2b6m7OSHz0kHn8KiYiowS5kF+H9bWew7VSW4ZhdaSl2zx0CAOjw8vcodVAZPcbHVYkne4dgSv/WkMtlJq2XqBoDEBER1VtmQRmW7DyHbw+nQycBchkwKjIILw1oC0VJCfCh/rzn+7fCxRIZ0vKKkXajBDdLKpBTWI73t51Fcno+Fj8RDleVvdgXQzZJJkkSO2tvo1ar4e7ujoKCAri5uYkuh4jIbBSUVODfey7gi99TUV6pAwA80tEPr8eFoa2fq/6k4mLAxUX/dVER4OxseLy6rAKbj2dg9k+noKnUobWPM1ZMjEJrHxdTvxSyQvX5/GYLEBER3VNZhRar96Vi6a4LUJdVAgB6hHjizUHtEdmyWZ2v46ayx7ieLdAxwA3PrzmCiznFiP/kdyweE4FHOvo1VflEd2ALUA3YAkRE9KeyCi3+snI/jqblAwDC/Fzx+sAwPNzeFzJZDWN47tICdKucwnJMXXsUB1NvAACmx7bFSw+35bggarD6fH5zGjwREdVKp5PwynfHcDQtH+6O9lg0OhxbpvXDgA5+NYefevBxVWLts9GYFNMSALBk53k8t+YI1GUVjVE60V0xABERUa0+3HkOm49nwF4hw38mRGJUZBAUjdhCY6+Q490RnfH+qK5wsJNjZ0oW4pf+jgvZRY32HEQ1YQAiIqIa/XDkKj7+5QIAYO5jXdAr1KvJnmt0VDC+ez4GAe4qXMopRvzS3/HLmax7P5CogRiAiIjoDgcv38Cb/z0OAJjyYGuMjgpu8ucMD/bATy/2Rc9WzVBUXokX1h5lSxA1GQYgIiIykppbjOfXHEaFVsKgzv547dEwkz23j6sSa5+JRt823iir0GH6+j+gqZpuT9SYGICIiMigoKQCT60+hJslFega5I7FT0SYfFaWvUKOD54Ih4eTPU5eU+NfiedM+vxkGxiAiIgIAFCh1WHK2iO4lFOMAHcVPp0YBUcHhZBa/NxUmPtYFwDAv3dfxMHLN4TUQdaLAYiIiCBJEmb9eBL7LubB2UGBzyb1gK+b6t4PbEKDuwRgVGQQJAl4eX0yp8dTo2IAIiIifPrbZXx9MB1yGfDRuG7oGGgei8C+M7wTWjRzwrX8Urzz4ynR5ZAVYQAiIrJx205lYu7PKQCAt4d0xIAO5rMlhYvSDh+OCYdcBvz3j2v437HroksiK8EARERkw7ILy/DKt8cgScD46BZ4qk+I6JLuENmyGV58qA0A4O0NJ3A9v1RwRWQNzCIALV26FCEhIVCpVIiOjsbBgwdrPXflypXo168fPD094enpidjY2DvOf/LJJyGTyYxuAwcObOqXQURkcRZvP4ei8kp0DXLHO8M73ff2Fk3lbwPaIjzYA+qySrzy7THodNzGku6P8AC0fv16JCQkYPbs2Th69CjCw8MRFxeH7OzsGs/fvXs3xo0bh127diEpKQnBwcF49NFHce3aNaPzBg4ciIyMDMPt66+/NsXLISKyGKeuF2D94XQAwOxhHWGvEP6RUCt7hRxLxkTA0V6BpEt5+GzvZdElkYUT/tO+ePFiPPvss5g8eTI6duyI5cuXw8nJCatWrarx/LVr1+KFF15AREQE2rdvj08//RQ6nQ6JiYlG5ymVSvj7+xtunp6epng5REQWQZIkzNl0GpIEDO0agMiWzUSXdE+tvJ0xa1hHAMD7287i9HW14IrIkgkNQBqNBkeOHEFsbKzhmFwuR2xsLJKSkup0jZKSElRUVKBZM+N/vLt374avry/CwsIwZcoU5OXl1XqN8vJyqNVqoxsRkTXbfjoL+y/dgIOdHG8Oai+6nDob2yMYj3T0g0arXyW6rEIruiSyUEIDUG5uLrRaLfz8jGcc+Pn5ITMzs07XeOONNxAYGGgUogYOHIgvv/wSiYmJWLBgAfbs2YNBgwZBq635H8q8efPg7u5uuAUHN/2eN0REopRXajF3i37W13P9QhHk6SS4orqTyWSYP7ILvF2UOJdVhPk/nxFdElko4V1g92P+/Pn45ptvsGHDBqhUfy7YNXbsWAwfPhxdunRBfHw8Nm3ahEOHDmH37t01XmfGjBkoKCgw3NLT0030CoiITO/LfVdwJa8EPq5KTHmwtehy6s3LRYn3R3cFAHyxLxUnrxUIrogskdAA5O3tDYVCgaysLKPjWVlZ8Pf3v+tjFy1ahPnz52P79u3o2rXrXc8NDQ2Ft7c3Lly4UOP9SqUSbm5uRjciImuUV1SOjxLPAwBeiwuDs9JOcEUN81CYL4aFBwIAluw8L7gaskRCA5CDgwMiIyONBjBXD2iOiYmp9XELFy7EnDlzsHXrVkRFRd3zea5evYq8vDwEBAQ0St1ERJZq8Y5zKCyvRKdAN4zqHiS6nPsybUBbyGXAzpQsHL+aL7ocsjDCu8ASEhKwcuVKrF69GikpKZgyZQqKi4sxefJkAMDEiRMxY8YMw/kLFizAzJkzsWrVKoSEhCAzMxOZmZkoKioCABQVFeG1117D/v37kZqaisTERIwYMQJt2rRBXFyckNdIRGQOzmYW4uuDaQCAWUM7mnyX98bWxtcF8RHNAQAf7uCO8VQ/wgPQmDFjsGjRIsyaNQsRERFITk7G1q1bDQOj09LSkJGRYTh/2bJl0Gg0GDVqFAICAgy3RYsWAQAUCgWOHz+O4cOHo127dnj66acRGRmJ3377DUqlUshrJCISrXrau04CBnX2R3Sol+iSGsVLA9pCIZdh19kcHLlyU3Q5ZEFkkiRxOc3bqNVquLu7o6CggOOBiMgqJKZk4enVh+GgkGNnQn+08GrCmV/FxYCLi/7roiLA2bnpngvA698fw7eHr6JfW2+seTq6SZ+LzFt9Pr+FtwAREVHTqtDq8M/N+mnvk/uGNG34EeBvD7eFnVyG387n4uDlG6LLIQvBAEREZOXWJF3BpdxieLs4GDYVtSbBzZzwRA/9+m0cC0R1xQBERGTFbhZrsGSnPhQkPBIGV5W94IqaxtSH2sBBIUfSpTzsu5gruhyyAAxARERW7F+J56Euq0R7f1eM6WG9q9w393DE2J5/tgJxeCvdCwMQEZGVupJXjDX7rwDQT3tXWPi093uZ+lAbONjJcSj1JvZeYCsQ3R0DEBGRlVq19zK0OgkPtPNB7zbeostpcn5uKvw1uiUA/YKPbAWiu2EAIiKyQgUlFfj28FUAwPMPhAquxnT+78FQqOzl+CMtH7vP5oguh8wYAxARkRVadzANpRVatPd3Re/W1rHoYV34uqowKSYEAFuB6O4YgIiIrIymUocv9l0GADzTLxQymXWP/bndcw+EwslBgRPXCrAzJVt0OWSmGICIiKzMlhMZyFKXw8dViWHhtrcJtJeLEk/2DgGgbwXS6dgKRHdiACIisiKSJGHlb5cAAJNiWkJppxBckRjPPRAKF6UdUjLU2HYqU3Q5ZIYYgIiIrMj+Szdw6roaKns5xlfNiLJFHk4OeKpvKwDAhzvZCkR3YgAiIrIin+3Vt/6MigyCp7OD4GrEerpvK7ip7HAuqwjbT2eJLofMDAMQEZGVuJRTZBj0+1SfVoKrEc/d0R5/7aVvBfuqakFIomoMQEREVmLV7/qZX7EdfBHq4yK4GvPwl+gWkMuAvRdycTGnSHQ5ZEYYgIiIrMDNYg2+P6Jf+PDpvraz8OG9BHk64eH2fgDYCkTGGICIiKzAuoNpKKvQoVOgG3qFNhNdjlmZEKPvBvv+yFWUaCoFV0PmggGIiMjClVdq8cW+VADAM/1a2dzCh/fSr403Wno5obCsEj8lXxddDpkJBiAiIgu36VgGcgrL4eemxJAugaLLMTtyucywSeqXSVe4PQYBYAAiIrJokiTh0736wc+TeofAwY6/1msyOioISjs5TmeocTQtX3Q5ZAb4L4WIyILtu5iHlAw1HO0V+EvPFqLLMVseTg4YHq5vHeNgaAIYgIiILNqnVdtePBEVBA8n21748F6qB0NvPp6BvKJywdWQaAxAREQW6kJ2IXadzYFMBkzmwof31DXIA+FB7tBodVh/OF10OSQYAxARkYX6bG8qAOCRDn4I8XYWW4yFmBATAgBYuz8NWu4PZtMYgIiILFBeUTn+e1S/8OEz/bjwYV0N7RoADyd7XMsvxe6z2aLLIYEYgIiILNC3h6+ivFKHLs3d0SPEU3Q5FkNlr8ATUcEA9FPiyXYxABERWRhJkvBd1RiWCb1acuHDehof3QIyGbDnXA6u5BWLLocEYQAiIrIwR67cxKXcYjjaKzC4a4DocixOSy9n9G/nAwBYeyBNcDUkCgMQEZGF+e6wfuzPkK4BcFHaCa7GMk3opZ8S/+3hdJRVaAVXQyIwABERWZDi8kpsOq7fz6p6LAvV34Nhvmju4Yj8kgr87xj3B7NFDEBERBZky4kMFGu0CPFy4uDn+6CQy/DXqlYgrgxtmxiAiIgsSHX31+ioYA5+vk9PRAXBQSHHsasFOJaeL7ocMjEGICIiC3E5txgHU29ALgMe7x4kuhyL5+WixJCqQeRr2ApkcxiAiIgsxPdH9FPfH2jnA393leBqrEN1N9j/jl3HzWKN4GrIlBiAiIgsgFYn4Ycj1wBw8HNj6t7CA50C3VBeqcMPVStrk21gACIisgC/nc9BproMnk72GNDBV3Q5VkMmk2FczxYAgB+OXhNcDZkSAxARkQWoHvw8IqI5lHYKwdVYl6FdA+CgkCMlQ43T19WiyyETYQAiIjJzN4o12H46EwC7v5qCh5MDHm6vb1Xb8Ae7wWwFAxARkZn7MfkaKrQSOjd3Q8dAN9HlWKWR3ZsDADYmX0elVie4GjIFBiAiIjNX3f3F1p+m82CYLzyd7JFTWI7fL+aJLodMgAGIiMiMnbxWgNMZajgo5BgeHii6HKvlYCfHsKr397+cDWYTGICIiMzYd4f1a/882skPHk4OgquxbiOrFpfcdioTReWVgquhpsYARERkpsoqtNiYzI1PTSU8yB2hPs4oq9Dh5xMZosuhJsYARERkpnamZKGgtAKB7ir0aeMtuhyrJ5PJMLKbfjD0f7kmkNVjACIiMlPfVg1+fjwyCAo5Nz41hfiqAJR0KQ/X8ksFV0NNiQGIiMgMXc8vxW/ncwAAoyK58ampBHk6oVdoMwDAxj/YCmTNGICIiMzQD0euQpKAXqHN0NLLWXQ5NmVkN33g/O/Rq5AkSXA11FQYgIiIzIxOJ+G7I1z7R5RBXfyhtJPjYk4xjl8tEF0ONREGICIiM3Mw9QbSbpTARWmHQZ0DRJdjc1xV9ojr5A8A2MBuMKvFAEREZGa+r2r9Gdo1AI4O3PhUhMeqtsb46dh1aCq5NYY1YgAiIjIjZRVabDup3/i0emE+Mr1+bbzh7aLEjWIN9pzLEV0ONQEGICIiM7L7bDYKyysR4K5CVEtP0eXYLDuFHPER+q0xuEO8dTKLALR06VKEhIRApVIhOjoaBw8erPXclStXol+/fvD09ISnpydiY2PvOF+SJMyaNQsBAQFwdHREbGwszp8/39Qvg4jovv10TL/y8/DwQMi59o9Q1d1gO09no6CkQnA11NiEB6D169cjISEBs2fPxtGjRxEeHo64uDhkZ2fXeP7u3bsxbtw47Nq1C0lJSQgODsajjz6Ka9f+HKi2cOFCfPTRR1i+fDkOHDgAZ2dnxMXFoayszFQvi4io3grLKpCYov/dN4wbnwrXMcAN7f1dodHqsOnEddHlUCMTHoAWL16MZ599FpMnT0bHjh2xfPlyODk5YdWqVTWev3btWrzwwguIiIhA+/bt8emnn0Kn0yExMRGAvvVnyZIl+Pvf/44RI0aga9eu+PLLL3H9+nVs3LjRhK+MiKh+dpzOQnmlDqE+zugU6Ca6HJsnk8kwsqoVaAO3xrA6QgOQRqPBkSNHEBsbazgml8sRGxuLpKSkOl2jpKQEFRUVaNZMv3Ln5cuXkZmZaXRNd3d3REdH13rN8vJyqNVqoxsRkan9mPxn95dMxu4vczAiojnkMuDwlZu4klcsuhxqREIDUG5uLrRaLfz8/IyO+/n5ITMzs07XeOONNxAYGGgIPNWPq881582bB3d3d8MtOJgLjxGRaeUVlWPvhVwA+gBE5sHP7c+NaLlBqnUR3gV2P+bPn49vvvkGGzZsgEqlavB1ZsyYgYKCAsMtPT29EaskIrq3LSczodVJ6NLcHaE+LqLLoVs8XrUcwYY/rnFrDCsiNAB5e3tDoVAgKyvL6HhWVhb8/f3v+thFixZh/vz52L59O7p27Wo4Xv24+lxTqVTCzc3N6EZEZEo/JetbF9j6Y34e7eQHJwcF0m6U4MiVm6LLoUYiNAA5ODggMjLSMIAZgGFAc0xMTK2PW7hwIebMmYOtW7ciKirK6L5WrVrB39/f6JpqtRoHDhy46zWJiES5ll+KQ6k3IZMBQ8O59YW5cXL4c0uSH9gNZjWEd4ElJCRg5cqVWL16NVJSUjBlyhQUFxdj8uTJAICJEydixowZhvMXLFiAmTNnYtWqVQgJCUFmZiYyMzNRVFQEQD9qf/r06Xjvvffw008/4cSJE5g4cSICAwMRHx8v4iUSEd3Vpqq1f3qGNEOAu6Pgaqgmj3XTzwb7+WQGKrTcGsMa2IkuYMyYMcjJycGsWbOQmZmJiIgIbN261TCIOS0tDXL5nzlt2bJl0Gg0GDVqlNF1Zs+ejXfeeQcA8Prrr6O4uBjPPfcc8vPz0bdvX2zduvW+xgkRETUVw+KHEez+Mle9QpvB28UBuUUa/H4hFw+G+Youie6TTOKIrjuo1Wq4u7ujoKCA44GIqEldyC5C7OI9sJPLcOjtWHg6O4gu6f4VFwMuVQO5i4oAZ2ex9TSSmRtPYs3+KxgVGYRFo8NFl0M1qM/nt/AuMCIiW1bd+tOvrbd1hB8rNrSrfhzQtlOZKK/UCq6G7hcDEBGRIJIk4X/s/rIYPUKawc9NicKySvx6Lld0OXSfGICIiAQ5eU2Ny7nFUNnL8UjHuy/9QeLJ5TIM6aIPqpuOc28wS8cAREQkyI9Va/8M6OAHF6XwOSlUB9XLFOw4nYVSDbvBLBkDEBGRADqdhE3HMwBw8UNL0i3YA809HFGi0WLX2WzR5dB9YAAiIhLgYOoNZKrL4Kqyw4NhPqLLoTqSyWSGViB2g1k2BiAiIgGqd34f1NkfSjuF4GqoPoZ11bfY/XImG0XllYKroYZiACIiMjFNpQ4/n6zu/mouuBqqr06BbgjxckJZhQ6JKVn3fgCZJQYgIiIT23shB/klFfB2USKmtZfocqieZDIZhlWN2/rfsQzB1VBDMQAREZnYT1XdX0O7BkAhlwmuhhpiaFU32K/nclBQWiG4GmoIBiAiIhMq1Wix/bS+24SLH1quMH9XtPV1gUarw47T7AazRAxAREQmtDMlCyUaLYKbOaJbsIfocug+/NkNxtlglogBiIjIhKr3/hrWNRAyGbu/LFn13mC/X8jFzWKN4GqovhiAiIhMpLCsAnvO5gD4s/WALFeojws6BrihUidh66lM0eVQPTEAERGZyM6ULGi0OrT2cUZ7f1fR5VAjqA6yXBTR8jAAERGZyObj+laCIV0C2P1lJaq7wZIu5iGnsFxwNVQfDEBERCagLqvAr+f03V9DurL7y1oEN3NCeLAHdBIMi1uSZWAAIiIygcRbur/a+bmILoca0bCqVqBNXBTRojAAERGZwOaqnd+HcPaX1RlSFYAOpt5ARkGp4GqorhiAiIiamL77KxeAfvwPWZcAd0f0CPEE8GfQJfPHAERE1MR2ntZ3f7XxdWH3l5Wq3hpjEwOQxWAAIiJqYltO6D8UB3P2l9Ua1MUfchmQnJ6P9BslosuhOmAAIiJqQrd2f1VPmSbr4+uqQq9QLwBsBbIUDEBERE2ouvurra8L2vlx8UNr9mc3GBdFtAQMQERETah6UOxgDn62egM7+0Mhl+HUdTWu5BWLLofugQGIiKiJFJRW4LfzVbO/2P1l9Zo5O6B3a3032OYT7AYzdwxARERNhN1ftqe6pY/T4c0fAxARUROpnv3F1h/bEdeJ3WCWggGIiKgJFJRW4NfzVXt/cfyPzWjm7ICYUHaDWQIGICKiJrDzdBYqtBLa+bmgLbu/bEp1N9gWBiCzxgBERNQEqv/3P6QLd363NXGd/KCQy3DymhppeVwU0VwxABERNTL97K+q7q+u/oKrIVPzclGiV2gzAOwGM2cMQEREjWxHVfdXmJ8r2viy+8sWsRvM/DEAERE1slv3/iLbFNdJvzfYiWsF7AYzUwxARESNiN1fBADeLkrD3mBbTrIVyBwxABERNSJ2f1E1doOZNwYgIqJGtLlqI0wufkgDO+u7wY5fLUD6DXaDmRsGICKiRlJQUoG9F/R7f3H8Dxl1g7EVyOwwABERNZLtpzNRoZXQ3t8VbXxdRJdDZoDdYOaLAYiIqJEY9v5i6w9Vqe4GO8ZuMLPDAERE1AiMur84/oeqeLsoEd2K3WDmiAGIiKgR3Nr91dqH3V/0p+pAzABkXhiAiIgawWZ2f1EtBnZiN5g5YgAiIrpPBSUV+J3dX1QLH1clerbS7w32MxdFNBsMQERE94ndX3Qv1S2Dm09kCq6EqjEAERHdJ3Z/0b3EdfaHTAYcS8/H1ZvsBjMHDEBERPeB3V9UF76uKvQMqeoGYyuQWWAAIiK6D+z+orqq3h5lM2eDmQUGICKi+8DuL6qrgVXdYMnp+biWXyq6HJvHAERE1EDs/qL68HVVoYehG4ytQKIxABERNRC7v6i+/pwNxgAkGgMQEVEDsfuL6mtQVTfYH2nsBhNNeABaunQpQkJCoFKpEB0djYMHD9Z67qlTp/D4448jJCQEMpkMS5YsueOcd955BzKZzOjWvn37JnwFRGSL2P1FDeHrxm4wcyE0AK1fvx4JCQmYPXs2jh49ivDwcMTFxSE7O7vG80tKShAaGor58+fD39+/1ut26tQJGRkZhtvevXub6iUQkY1i9xc11NCqwLzpOAOQSEID0OLFi/Hss89i8uTJ6NixI5YvXw4nJyesWrWqxvN79OiB999/H2PHjoVSqaz1unZ2dvD39zfcvL29m+olEJGNYvcXNdTAzvq9wZLT87k3mEDCApBGo8GRI0cQGxv7ZzFyOWJjY5GUlHRf1z5//jwCAwMRGhqK8ePHIy0t7a7nl5eXQ61WG92IiGpTUFKBvefZ/UUN4+uqQnQrLwDcG0wkYQEoNzcXWq0Wfn5+Rsf9/PyQmdnwVTKjo6PxxRdfYOvWrVi2bBkuX76Mfv36obCwsNbHzJs3D+7u7oZbcHBwg5+fiKzfttOZqNSx+4sabgi7wYQTPgi6sQ0aNAijR49G165dERcXhy1btiA/Px/ffvttrY+ZMWMGCgoKDLf09HQTVkxElmYLu7/oPlV3gx2/WoC0PHaDiSAsAHl7e0OhUCArK8voeFZW1l0HONeXh4cH2rVrhwsXLtR6jlKphJubm9GNiKgm7P6ixuDtokRMa303GNcEEkNYAHJwcEBkZCQSExMNx3Q6HRITExETE9Noz1NUVISLFy8iIIC/qIjo/rH7ixrLkC6BAIDNJ64LrsQ2Ce0CS0hIwMqVK7F69WqkpKRgypQpKC4uxuTJkwEAEydOxIwZMwznazQaJCcnIzk5GRqNBteuXUNycrJR686rr76KPXv2IDU1Ffv27cNjjz0GhUKBcePGmfz1EZH1YfcXNZa4Tn5QyGU4eU2N1Nxi0eXYHDuRTz5mzBjk5ORg1qxZyMzMREREBLZu3WoYGJ2Wlga5/M+Mdv36dXTr1s3w/aJFi7Bo0SL0798fu3fvBgBcvXoV48aNQ15eHnx8fNC3b1/s378fPj4+Jn1tRGR92P1FjcnLRYnerb3w2/lcbD6RgakPtRFdkk2RSZIkiS7C3KjVari7u6OgoIDjgYjI4NvD6Xj9++No7++KrdMfEF2O+SouBlyqugeLigBnZ7H1mLFvDqbhzf+eQMcAN2yZ1k90ORavPp/fVjcLjIioqbD7ixpbXCd/2MllOJ2hxqWcItHl2BQGICKiOmD3FzUFT2cH9Gmj361gC2eDmRQDEBFRHXD2FzUVLoooBgMQEVEdsPuLmkpcR3/YK2Q4k1mIC9nsBjMVBiAionsoKKnA7xfY/UVNw93JHn2rusE2sxXIZBiAiIjuYdupTFRo2f1FTWdIVy6KaGoMQERE9/DTMf2H0rDwQMGVkLV6pKMf7BUynMsqwrms2jfvpsZT5wDUrFkz5Obqm4Cfeuqpu+6uTkRkLXIKy7Hvov5337CuDEDUNNwd7fFAW/2CvewGM406ByCNRgO1Wg0AWL16NcrKypqsKCIic/HzyQzoJCA82AMtvJxEl0NWbGi4fnzZ5hMZ4BrFTa/OW2HExMQgPj4ekZGRkCQJL730EhwdHWs8d9WqVY1WIBGRSP+r7v7i4GdqYrEd/OBgJ8eF7CKcyypCmL+r6JKsWp1bgL766isMHjwYRUVFkMlkKCgowM2bN2u8ERFZg+v5pTiUehMyGTCU3V/UxFxV9ujfrrobjIOhm1qdW4D8/Pwwf/58AECrVq2wZs0aeHl5NVlhRESiVY/F6BHSDP7uKsHVkC0Y2jUAO05nYdOJDLz8SDvIZDLRJVmtBs0Cu3z5MsMPEVm96tlfwzn7i0xkQFU32KWcYpzJ5GSjplTnFqCPPvqozhd96aWXGlQMEZG5uJxbjBPXCqCQyzCos7/ocshGuCjt8FCYD7adysKm49fRIeDuO5pTw9U5AH344YdG3+fk5KCkpAQeHh4AgPz8fDg5OcHX15cBiIgs3qaq1p8+bbzh5aIUXA3ZkiFdA7HtVBY2H8/Aq4+GsRusidS5C+zy5cuG2z//+U9EREQgJSUFN27cwI0bN5CSkoLu3btjzpw5TVkvEZFJ/O84Z3+RGAPa+0JlL0dqXglOXVeLLsdqNWgM0MyZM/Hxxx8jLCzMcCwsLAwffvgh/v73vzdacUREIpzNLMS5rCI4KOR4tBO7v8i0nJV2eLi9L4A/l2GgxtegAJSRkYHKyso7jmu1WmRlZd13UUREIlV/6PQP84G7o73gasgWDQ9vDkA/EF+n46KITaFBAWjAgAF4/vnncfToUcOxI0eOYMqUKYiNjW204oiITE2SJEP3F2d/kSgPhvnAVWWHjIIyHEq9Ibocq9SgALRq1Sr4+/sjKioKSqUSSqUSPXr0gJ+fHz799NPGrpGIyGSOXy3AlbwSONorMKCDr+hyyEap7BWG2Ycbk9kN1hTqPAvsVj4+PtiyZQvOnz+PlJQUAED79u3Rrl27Ri2OiMjUqru/Yjv6wcmhQb8iiRpFfERzfHv4KracyMC7wzvBwa5BbRZUiwa/m5999hkee+wxjB49GqNHj8bIkSPZ+kNEFk2nk7CpavVnzv4i0aJDveDrqkRBaQV+PZcjuhyr06AANGvWLEybNg3Dhg3Dd999h++++w7Dhg3Dyy+/jFmzZjV2jUREJnH4yk1kqsvgqrJD/zAf0eWQjVPIZRhWNQ7tR84Ga3QNat9dtmwZVq5ciXHjxhmODR8+HF27dsXf/vY3/OMf/2i0AomITKW6+yuukz+UdgrB1RABIyIC8dney9hxOhPF5ZVwVrJbtrE0qAWooqICUVFRdxyPjIyscXo8EZG5q9TqsOVEVfcXZ3+RmejS3B2tvJ1RVqHDjtNcZqYxNSgATZgwAcuWLbvj+IoVKzB+/Pj7LoqIyNT2XcxDXrEGzZwd0Kc1N3sm8yCTyQzLMfyYfE1wNdalwW1pn332GbZv345evXoBAA4cOIC0tDRMnDgRCQkJhvMWL158/1USETWx6u6vwV38YafgbBsyH8MjAvGvxPP49Xwu8orKuTddI2lQADp58iS6d+8OALh48SIAwNvbG97e3jh58qThPG7gRkSWoLxSi62nMgEAw7qy+4vMS2sfF3Rp7o4T1wqw5UQGJsSEiC7JKjQoAO3ataux6yAiEubXc7koLKuEn5sSPUKaiS6H6A4jIgJx4loBfky+zgDUSNjOS0Q2r7r7a2jXQMjlbLkm8zMsPBAymX6phvQbJaLLsQoMQERk00o0lYbZNZz9RebKz02FmFD94Pzqvero/jAAEZFNS0zJRmmFFi2aOSE8yF10OUS1GhGhD+g/cW+wRsEAREQ2bcMf+qnFw8MDOXGDzNrATgFwUMhxJrMQZzLVosuxeAxARGSzcgrLsadqj6XHujcXXA3R3bk72ePBqi1afmQr0H1jACIim/XTsevQ6iREBHugtY+L6HKI7mlEhD6o/5R8HTqdJLgay8YAREQ2679HrwIAHmfrD1mIAR184aK0w7X8UhxNuym6HIvGAERENulMphqnrqthr5BhKBc/JAuhslcgrpM/AHaD3S8GICKySRuO6gc/P9zeF57ODoKrIaq76tlgm09koEKrE1yN5WIAIiKbo9VJhtlfI7sHCa6GqH56t/aCt4sDbhRrsPdCruhyLBYDEBHZnN8v5CK7sBweTvZ4KMxXdDlE9WKnkBu6bX/8gzvENxQDEBHZnOrBz8PDA+Fgx1+DZHmGV3WDbT+dhRJNpeBqLBP/5RORTSksqzDs/M7uL7JU3YI9ENzMESUarWErF6ofBiAisik/n8xEWYUOoT7O3PqCLJZMJkN81ZpA/z3KbrCGYAAiIpvy59o/Qdz6giza41UtmL+ez0FGQangaiwPAxAR2YyrN0uw/9INyGRAfDcufkiWLcTbGT1bNYMksRWoIRiAiMhmbKyaMRMT6oXmHo6CqyG6f6Mj9a1A3x1OhyRxa4z6YAAiIpsgSZLhf8kc/EzWYnCXADg5KJCaV4LDV7g1Rn0wABGRTUhOz8el3GI42iswsLO/6HKIGoWz0g5DugQAAL49lC64GsvCAERENqG69WdgZ3+4KO0EV0PUeJ7oEQxAvzVGcTnXBKorBiAisnrllVr877h+48iR3PmdrExUS0+08nZGiUaLLScyRJdjMRiAiMjq7TqTg/ySCvi5KdG7tbfocogalUwmw6jqwdBHrgquxnIwABGR1ate+ye+W3Mo5Fz7h6zPyO7NIZcBBy/fQGpusehyLILwALR06VKEhIRApVIhOjoaBw8erPXcU6dO4fHHH0dISAhkMhmWLFly39ckIut2o1iDXWezAQAju3H2F1mnAHdH9GvrAwD4nq1AdSI0AK1fvx4JCQmYPXs2jh49ivDwcMTFxSE7O7vG80tKShAaGor58+fD37/mWRz1vSYRWbdNx6+jQiuhc3M3hPm7ii6HqMmMjtIH/B+OXoVWxzWB7kVoAFq8eDGeffZZTJ48GR07dsTy5cvh5OSEVatW1Xh+jx498P7772Ps2LFQKpWNck0ism4/VK/9w9YfsnKxHfzg7miPjIIy/H4hV3Q5Zk9YANJoNDhy5AhiY2P/LEYuR2xsLJKSkkx6zfLycqjVaqMbEVm+C9lFOJaeD4VchuERgaLLIWpSKnsF4qt+zr89zDWB7kVYAMrNzYVWq4Wfn5/RcT8/P2RmZpr0mvPmzYO7u7vhFhwc3KDnJyLzUj34+cF2PvB2qbnVmMiajI7Sf35tP52FgpIKwdWYN+GDoM3BjBkzUFBQYLilpzM5E1m6Cq3OMCX48Uh2f5Ft6BTohvb+rtBU6vDTMW6QejfCApC3tzcUCgWysrKMjmdlZdU6wLmprqlUKuHm5mZ0IyLLtvN0FnIKy+HtosQjHf3u/QAiKyCTyQytQN8e5mywuxEWgBwcHBAZGYnExETDMZ1Oh8TERMTExJjNNYnIMq07mAYAeCIqCPYKNnaT7YiPCIS9QoYT1wpwJpNjWmsj9LdCQkICVq5cidWrVyMlJQVTpkxBcXExJk+eDACYOHEiZsyYYThfo9EgOTkZycnJ0Gg0uHbtGpKTk3HhwoU6X5OIrF9qbjF+O58LmQwY17OF6HKITMrLRYkB7fWtnt+xFahWQncEHDNmDHJycjBr1ixkZmYiIiICW7duNQxiTktLg1z+Z0a7fv06unXrZvh+0aJFWLRoEfr374/du3fX6ZpEZP2+PqRv/XmgrQ+CmzkJrobI9EZHBWHrqUxs/OMa3hjYHg52bAW9nUySJK6WdBu1Wg13d3cUFBRwPBCRhdFU6hAzLxF5xRr8Z0Ik4jo1bEwh3YfiYsDFRf91URHg7Cy2HhtUqdUhZv4vyCksx/K/RmJgZ9v4d1Cfz29GQiKyKttOZSKvWAM/NyUGtPcVXQ6REHYKOUZ2bw4A+P4IZzbXhAGIiKzKugP67q8xUcGw4+BnsmGjI/WzwXadzUF2YZngaswPfzsQkdW4mFOEpEt5kMuAMRz8TDauja8LurXwgFYnYcNRrgl0OwYgIrIaX1e1/jwU5ovmHo6CqyES74mqNYG+OZQOHTdINcIARERWoaxCi++rtr74SzRbf4gAYHh4IFyVdricW4zfuEGqEQYgIrIKW09mIr+kAoHuKjwYxsHPRADgrLTDqCj9VjBf7ksVW4yZYQAiIquw9sAVAMCYHi2gkMsEV0NkPib0agkA+OVsNtLySgRXYz4YgIjI4p3LKsSh1JtQyGUY0yNYdDlEZiXUxwUPtPOBJAFfVf1HgRiAiMgKVE99H9DeF/7uKsHVEJmfSTH6VqD1h9JRqtEKrsY8MAARkUUrq9Divxz8THRXD4b5IriZIwpKK/C/Y9dFl2MWGICIyKJtOp4BdVklgjwd8UBbH9HlEJklhVxmGAv0xb5UcBcsBiAisnDVg5/H9WwBOQc/E9XqiahgKO3kOJ2hxtG0m6LLEY4BiIgsVkqGGn+k5cNOLsPoqqm+RFQzDycHxEfo9wdbvY+DoRmAiMhiVQ9+frSTH3xdOfiZ6F4mVA2G3nIiA9lq294fjAGIiCxSiaYSG//Q72/0l54tBVdDZBk6N3dHVEtPVOokfH3QtneJZwAiIov0v2PXUVheiZZeTujd2kt0OUQWY2LvEAD68XMVWp3YYgRiACIiiyNJEtbs5+BnooYY2Mkf3i5KZBeWY9upTNHlCMMAREQWJ+lSHk5eU0NpJzfsdk1EdeNgJzesmfWlDQ+GZgAiIouz8tdLAIDRUUFo5uwguBoiyzM+ugXs5DIcTL2B09fVossRggGIiCzKuaxC7DqbA5kMeKZvqOhyiCySn5sKcZ39AQBr9qeKLUYQBiAisijVrT9xHf0R4u0suBoiyzUpJgQAsOGPaygoqRBbjAAMQERkMbLVZdiYrJ/6/uwDbP0huh89QjzR3t8VZRU6fHfE9qbEMwARkcX4Yl8qKrQSolp6IrKlp+hyiCyaTCbDpKop8V8mXYFOZ1v7gzEAEZFFKCqvxFdVU9/Z+kPUOEZEBMJNZYe0GyXYcy5HdDkmxQBERBbh20PpUJdVopW3Mx7p4Ce6HCKr4ORgZ1hKYnVSqthiTIwBiIjMXqVWh8/2XgYAPNOvFRc+JGpEE2JaQi4Ddp/NQUqG7UyJZwAiIrO35WQmruWXwsvZAY93567vRI2ppZczBncJAAAs3XVBcDWmwwBERGZNkiSs+PUiAGBiTAhU9grBFRFZn6kPtQEAbD6RgUs5RYKrMQ0GICIya9XbXqjs5ZgQw13fiZpChwA3xHbwhSQBy3ZfFF2OSTAAEZFZM2x7ERnMbS+ImlB1K9CGP67h6s0SwdU0PQYgIjJbt2578XTfVqLLIbJq3Vp4ok8bL1TqJKyo+o+HNWMAIiKzVd36M7ATt70gMoXqVqBvDqUju7BMcDVNiwGIiMwSt70gMr2YUC90b+EBTaUOn/12WXQ5TYoBiIjM0ue3bHvRvQW3vSAyBZlMhhcf1rcCfbX/Cm4WawRX1HQYgIjI7BSVV2Jt1bYXz7H1h8ikHgrzRccANxRrtPh8X6rocpoMAxARmZ3qbS9CvZ0Ry20viExKJpMZxgJ98ftlFJZVCK6oaTAAEZFZ0VTeuu1FKLe9IBJgYGd/hPo4Q11Wia/2p4kup0kwABGRWVl/KA3X8kvh7aLEyO7NRZdDZJMUchleeFDfCvTZ3ksoq9AKrqjxMQARkdko1Wjx0S/6vYheGtCG214QCTQiIhBBno7ILdLgm4PW1wrEAEREZuOLfanIKSxHkKcjxvZoIbocIptmr5Dj+f6tAQD/+fUSNJU6wRU1LgYgIjILBaUVWL5HvwfRy7Ht4GDHX09Eoo2ODIKvqxIZBWXY8MdV0eU0Kv6GISKzsOLXiygorUA7PxfEd+PYHyJzoLJX4Nl++qUolu2+iEqt9bQCMQARkXA5heVYtTcVAPDKo2FQcOYXkdn4S3QLeDjZIzWvBJtPZIgup9EwABGRcEt3XUBphRbhwR54tCPX/SEyJ85KOzzVR78Z8b93XYROJwmuqHEwABGRUOk3SrD2gH7V59fjwiCTsfWHyNxM6h0CV6UdzmYV4sdj10SX0ygYgIhIqH8lnkeFVkKfNl7o08ZbdDlEVAN3R3v834P6GWELt55Fqcby1wViACIiYc5nFeK/R/UzS16Lay+4GiK6m6f7tkJzD0dkFJTh098uiS7nvjEAEZEwH2w/B50EPNrRDxHBHqLLIaK7UNkr8PrAMADAsj0XkV1YJrii+8MARERCHEvPx9ZTmZDJgFfjwkSXQ0R1MDw8EOHBHijRaLF4+znR5dwXBiAiEmLR9rMAgMe6NUc7P1fB1RBRXchkMswc0gEAsP5wOk5fVwuuqOEYgIjI5PZdzMVv53Nhr5Dh5dh2osshonqICmmGIV0CIEnA3C0pkCTLnBbPAEREJiVJEhZu1bf+jOvZAsHNnARXRET19cbA9nBQyLH3Qi52n80RXU6DmEUAWrp0KUJCQqBSqRAdHY2DBw/e9fzvvvsO7du3h0qlQpcuXbBlyxaj+5988knIZDKj28CBA5vyJRBRHe1MyUZyej5U9nK8+HAb0eUQUQO08HLCk31CAADvbT6NCgvcIkN4AFq/fj0SEhIwe/ZsHD16FOHh4YiLi0N2dnaN5+/btw/jxo3D008/jT/++APx8fGIj4/HyZMnjc4bOHAgMjIyDLevv/7aFC+HiO5Cq5OwaJu+9Wdyn1bwdVUJroiIGmrqQ23g6WSPiznF+OZgmuhy6k14AFq8eDGeffZZTJ48GR07dsTy5cvh5OSEVatW1Xj+v/71LwwcOBCvvfYaOnTogDlz5qB79+745JNPjM5TKpXw9/c33Dw9PU3xcojoLr4+mIazWYVwU9nh/x5oLbocIroP7o72ePkR/Ri+D3eeh7qsQnBF9SM0AGk0Ghw5cgSxsbGGY3K5HLGxsUhKSqrxMUlJSUbnA0BcXNwd5+/evRu+vr4ICwvDlClTkJeX1/gvgIjqLFtdhgVbzwAAXn6kHdyd7AVXRET3a1zPFmjt44wbxRos3XVBdDn1IjQA5ebmQqvVws/PePNDPz8/ZGZm1viYzMzMe54/cOBAfPnll0hMTMSCBQuwZ88eDBo0CFptzUt3l5eXQ61WG92IqHH9Y9NpFJZVomuQOybGhIguh4gagb1CjrerpsV/vjcV6TdKBFdUd8K7wJrC2LFjMXz4cHTp0gXx8fHYtGkTDh06hN27d9d4/rx58+Du7m64BQcHm7ZgIiu3+2w2Nh3PgFwGzH2sCxRybnhKZC0eCvNFnzZe0Gh1mF/VymsJhAYgb29vKBQKZGVlGR3PysqCv79/jY/x9/ev1/kAEBoaCm9vb1y4UHPz3IwZM1BQUGC4paen1/OVEFFtSjVazPxRP0lhcp9W6NzcXXBFRNSYZDIZ3h7cETIZsPl4Bo5cuSG6pDoRGoAcHBwQGRmJxMREwzGdTofExETExMTU+JiYmBij8wFgx44dtZ4PAFevXkVeXh4CAgJqvF+pVMLNzc3oRkSN46NfziP9RikC3VVIeISLHhJZo46BbngiUt97MmeTZSyOKLwLLCEhAStXrsTq1auRkpKCKVOmoLi4GJMnTwYATJw4ETNmzDCcP23aNGzduhUffPABzpw5g3feeQeHDx/Giy++CAAoKirCa6+9hv379yM1NRWJiYkYMWIE2rRpg7i4OCGvkchWnclUY+Wv+l2j3x3RGc5KO8EVEVFTeeXRdnByUCA5PR/fHb4qupx7Eh6AxowZg0WLFmHWrFmIiIhAcnIytm7dahjonJaWhoyMDMP5vXv3xrp167BixQqEh4fj+++/x8aNG9G5c2cAgEKhwPHjxzF8+HC0a9cOTz/9NCIjI/Hbb79BqVQKeY1Etkink/DWf0+gUichrpMfHunod+8HEZHF8nVTYdqAtgCAOZtO43p+qeCK7k4mWUI7lYmp1Wq4u7ujoKCA3WFEDbTuQBre2nACzg4K7HylPwLcHUWXRKZSXAy4uOi/LioCnJ3F1kMmo9VJGLV8H/5Iy0e/tt748qmekMlMN+mhPp/fwluAiMj6ZBeWYf7PKQCAVx4NY/ghshEKuQyLRodDaSfHb+dz8fVB851UxABERI3uvU0pUJdVoktzd0zqHSK6HCIyodY+LngtLgwA8M/Np812bSAGICJqVHvO5eCnY9e55g+RDZvcpxV6hHiiWKPFGz8ch05nfqNtGICIqNGUVWgxc6N+zZ9JvUPQJYhr/hDZIoVchvdHhUNlL8e+i3lYe+CK6JLuwABERI3m41/OI+1GCfzdVHjl0TDR5RCRQCHeznhzYHsAwNwtZ3Alr1hwRcYYgIioUZzJVOM/e6rX/OkEF675Q2TzJsaEoFdoM5RWaPHad+bVFcYARET3rbi8Ei+u+wOVOgmxHfwQ16n2rWmIyHbIq7rCnBwUOJh6A1/sSxVdkgEDEBHdF0mS8PaGE7iQXQRfVyXmP95FdElEZEaCmznhrcH6HeMXbjuDSzlFgivSYwAiovvyzaF0bEy+DoVcho/HdYO3C1dcJyJj46NboG8bb5RV6PDa98ehNYOuMAYgImqwU9cLMPunUwCAVx8NQ3Sol+CKiMgcyWQyLBjVFS5KOxy5chOf7b0kuiQGICJqGHVZBaauPQpNpQ4Pt/fF8w+Eii6JiMxYcw9HzByq7wpbtP0cLmQXCq2HAYiI6k2SJLz5w3Gk5pWguYcjPhgdDjkXPCSie3giKhj92/lAU6nDvC1nhNbCAERE9bZ6Xyq2nMiEvUKGT/7SDZ7ODqJLIiILIJPJMP/xLngiKgjvjw4XWgsX6iCieklOz8c/t+g3On1rcAd0a+EpuCIisiQB7o5YOEps+AHYAkRE9ZBfosHUtUdRoZUwqLM/nuRGp0RkoRiAiKhOdDoJr3x7DNfyS9HSywkLRnWFTMZxP0RkmRiAiKhOVv52CYlnsuFgJ8fSv3SHm8pedElERA3GAERE93TgUh4WbjsLAHhnWCd0bs5d3onIsjEAEdFdnbpegGe/PAytTkJ8RCDG9QwWXRIR0X1jACKiWl3MKcLEzw5CXVaJqJaemDuyC8f9EJFVYAAiohpdvVmCv356AHnFGnQKdMOqyT3g5MCVM4jIOjAAEdEdsgvL8NdPDyCjoAytfZzx5VM9OeiZiKwKAxARGckv0WDiZwcN21x89Uw0vLjDOxFZGQYgIjIoKq/Ek58fwpnMQvi4KrH2mWgEuDuKLouIqNExABERAKCsQovnvjyM5PR8eDjZ46unoxHi7Sy6LCKiJsEARESo0Orw4rqj2HcxD84OCqye3BNh/q6iyyIiajIMQEQ2TquT8Op3x7AzJRtKOzk+e7IHwoM9RJdFRNSkGICIbFiFVoc3fjiOH5Ovw04uw/K/RqJXqJfosoiImhwX9SCyUTeLNZiy9gj2X7oBmQz4cEwEHmrvK7osIiKTYAAiskEXsovw9OpDuJJXAmcHBT4a1w0DOviJLouIyGQYgIhszK/ncjB13VEUllUiyNMRn03qwQHPRGRzGICIbIQkSfgy6Qr+sek0tDoJUS09sXxCJLy5yCER2SAGICIbUKHV4d3/ncJX+9MAAI93D8LckZ2htFMIroyISAwGICIrl1+iwdR1R/H7hTzIZMCbA9vjuQdCuas7Edk0BiAiK3YppwhPrz6My7nFcHJQ4F9ju+GRjhzsTETEAERkhXQ6CV8duIIFP59BsUaL5h6O+HRSFDoEuIkujYjILDAAEVmZC9mFeOOHEzhy5SYAoGdIMywd3x0+rhzsTERUjQGIyEpoKnVYtvsilu66AI1WB2cHBd4c1B7jo1tCLud4HyKiWzEAEVmBo2k38eYPx3EuqwgA8HB7X7wX3xmBHo6CKyMiMk8MQEQWrLi8Eou2n8UX+1IhSYCXswNmD++EYV0DOMuLiOguGICILJAkSdh9Lgd/33AS1/JLAQAjuzfHzCEd4ensILg6IiLzxwBEZEEkScKv53PxyS/ncShVP8g5yNMRcx/rggfa+QiujojIcjAAEVkAnU7CzpQsfLLrAo5fLQAAOCjkmBjTEi8/0g7OSv5TJiKqD/7WJDJjWp2ELScysHTXBZzJLAQAONorMD66BZ59IBR+birBFRIRWSYGICIzVKHV4cfk6/j3rgu4lFsMAHBR2mFS75Z4qk8reHEDUyKi+8IARGRGUnOLsTH5Gr47fNUwuNnDyR5P9WmFSTEhcHeyF1whEZF1YAAiEiy3qBybjl3HxuTrSE7PNxz3dnHAs/1CMb5XS7hwjA8RUaPib1UiAUo0ldhxOgsb/riG387nQquTAAByGdCnjTfiI5pjSNcAqOwVgislIrJODEBEJpJTWI79l/KQmJKF7aezUKLRGu4LD3LHiIjmGBoeAF9XDmwmImpqDEBETaSgpAL7L+ch6aL+djar0Oj+ll5OGBHRHPERgQj1cRFUJRGRbWIAImokBaUV+CPtJpIu5mHfxTycvF4ASTI+p2OAG3q39sLgrgHoFuzB7SqIiARhACKqJ61OwpW8YqRkFCIlQ40zmWqkZBQaZm3dqrWPM3q39kbv1l6IDvVCM25TQURkFhiAiGpRUFqBqzdLcPVmKdJvlOBCdhFSMgtxLrMQpRXaGh8T5OmImFAv9GnjjZjWXlyokIjITJlFAFq6dCnef/99ZGZmIjw8HB9//DF69uxZ6/nfffcdZs6cidTUVLRt2xYLFizA4MGDDfdLkoTZs2dj5cqVyM/PR58+fbBs2TK0bdvWFC+HzJwkSSgqr0RekQZ5xeXILdLgen6pIehcvVmKqzdLoC6rrPUaSjs52vu7or2/GzoEuKJ9gBs6+LtxnR4iIgshPACtX78eCQkJWL58OaKjo7FkyRLExcXh7Nmz8PX1veP8ffv2Ydy4cZg3bx6GDh2KdevWIT4+HkePHkXnzp0BAAsXLsRHH32E1atXo1WrVpg5cybi4uJw+vRpqFT8H7k1kCQJGq0OhWWVKCqrRGFZJQrLK/R/llWiqKzq61uCTl6RBnlF5cgt1kBTqavT83g5OyDI0xFBnk4I8XZChwA3tPd3QytvZyjkHL9DRGSpZJJ0+zBN04qOjkaPHj3wySefAAB0Oh2Cg4Pxt7/9DW+++eYd548ZMwbFxcXYtGmT4VivXr0QERGB5cuXQ5IkBAYG4pVXXsGrr74KACgoKICfnx+++OILjB079p41qdVquLu7o6CgAG5ubo30SvVdKurSCqNjNb37Eu48ePt5kuG4dNv3xmdIkv6r6uMSJP2x274HAJ2k/1onSYbHSFVf63QSdFXfayX91zpJMhzX6iTDfVqdhErtLV/rJGi1OmglQKvToUIroUKrq7pJ0FTq7vi+vFKH8kotSjValFX/WaFDWYUWZRValFZoobvPn1xnBwW8XJRo5uyAQA8VgjydqsKOo+FrJwfh/0cgsjzFxYBL1czGoiLA2VlsPWQz6vP5LfS3u0ajwZEjRzBjxgzDMblcjtjYWCQlJdX4mKSkJCQkJBgdi4uLw8aNGwEAly9fRmZmJmJjYw33u7u7Izo6GklJSTUGoPLycpSXlxu+V6vV9/OyarX2wBUs3Hq2Sa5ty1yUdnBV2f35p8oerio7uFZ938xZCS8XB3i7OMCr6msvZyUcHbjIIBGRrRIagHJzc6HVauHn52d03M/PD2fOnKnxMZmZmTWen5mZabi/+lht59xu3rx5ePfddxv0GurDXi6HUx0/dG/tXLl1qrSslpNkt50ru+0+mUxW9af+SPX91cfkhvv19916TC6TAVXfK6rul8tkUMhlkMv0j5HfcsxOIYNCLodCBijkctjJZVAoZPo/Zfr7HezksFfI4aDQ/2mvkMPeTmb0vaODHI72Cqiqbo5Gf8qhclDAxcEOcnZFEZkXZ+eam7eJzAjb9wHMmDHDqFVJrVYjODi40Z/n2QdC8ewDoY1+XSIiIqofucgn9/b2hkKhQFZWltHxrKws+Pv71/gYf3//u55f/Wd9rqlUKuHm5mZ0IyIiIuslNAA5ODggMjISiYmJhmM6nQ6JiYmIiYmp8TExMTFG5wPAjh07DOe3atUK/v7+Rueo1WocOHCg1msSERGRbRHeBZaQkIBJkyYhKioKPXv2xJIlS1BcXIzJkycDACZOnIjmzZtj3rx5AIBp06ahf//++OCDDzBkyBB88803OHz4MFasWAFAPx5l+vTpeO+999C2bVvDNPjAwEDEx8eLeplERERkRoQHoDFjxiAnJwezZs1CZmYmIiIisHXrVsMg5rS0NMjlfzZU9e7dG+vWrcPf//53vPXWW2jbti02btxoWAMIAF5//XUUFxfjueeeQ35+Pvr27YutW7dyDSAiIiICYAbrAJmjploHiIiIiJpOfT6/hY4BIiIiIhKBAYiIiIhsDgMQERER2RwGICIiIrI5DEBERERkcxiAiIiIyOYwABEREZHNYQAiIiIim8MARERERDZH+FYY5qh6cWy1Wi24EiIiIqqr6s/tumxywQBUg8LCQgBAcHCw4EqIiIiovgoLC+Hu7n7Xc7gXWA10Oh2uX78OV1dXyGSyRr22Wq1GcHAw0tPTuc9YE+L7bBp8n02D77Np8H02jaZ8nyVJQmFhIQIDA402Uq8JW4BqIJfLERQU1KTP4ebmxn9gJsD32TT4PpsG32fT4PtsGk31Pt+r5acaB0ETERGRzWEAIiIiIpvDAGRiSqUSs2fPhlKpFF2KVeP7bBp8n02D77Np8H02DXN5nzkImoiIiGwOW4CIiIjI5jAAERERkc1hACIiIiKbwwBERERENocByISWLl2KkJAQqFQqREdH4+DBg6JLsirz5s1Djx494OrqCl9fX8THx+Ps2bOiy7J68+fPh0wmw/Tp00WXYpWuXbuGv/71r/Dy8oKjoyO6dOmCw4cPiy7Lqmi1WsycOROtWrWCo6MjWrdujTlz5tRpPymq3a+//ophw4YhMDAQMpkMGzduNLpfkiTMmjULAQEBcHR0RGxsLM6fP2+y+hiATGT9+vVISEjA7NmzcfToUYSHhyMuLg7Z2dmiS7Mae/bswdSpU7F//37s2LEDFRUVePTRR1FcXCy6NKt16NAh/Oc//0HXrl1Fl2KVbt68iT59+sDe3h4///wzTp8+jQ8++ACenp6iS7MqCxYswLJly/DJJ58gJSUFCxYswMKFC/Hxxx+LLs2iFRcXIzw8HEuXLq3x/oULF+Kjjz7C8uXLceDAATg7OyMuLg5lZWWmKVAik+jZs6c0depUw/darVYKDAyU5s2bJ7Aq65adnS0BkPbs2SO6FKtUWFgotW3bVtqxY4fUv39/adq0aaJLsjpvvPGG1LdvX9FlWL0hQ4ZITz31lNGxkSNHSuPHjxdUkfUBIG3YsMHwvU6nk/z9/aX333/fcCw/P19SKpXS119/bZKa2AJkAhqNBkeOHEFsbKzhmFwuR2xsLJKSkgRWZt0KCgoAAM2aNRNciXWaOnUqhgwZYvRzTY3rp59+QlRUFEaPHg1fX19069YNK1euFF2W1enduzcSExNx7tw5AMCxY8ewd+9eDBo0SHBl1uvy5cvIzMw0+v3h7u6O6Ohok30ucjNUE8jNzYVWq4Wfn5/RcT8/P5w5c0ZQVdZNp9Nh+vTp6NOnDzp37iy6HKvzzTff4OjRozh06JDoUqzapUuXsGzZMiQkJOCtt97CoUOH8NJLL8HBwQGTJk0SXZ7VePPNN6FWq9G+fXsoFApotVr885//xPjx40WXZrUyMzMBoMbPxer7mhoDEFmlqVOn4uTJk9i7d6/oUqxOeno6pk2bhh07dkClUokux6rpdDpERUVh7ty5AIBu3brh5MmTWL58OQNQI/r222+xdu1arFu3Dp06dUJycjKmT5+OwMBAvs9WjF1gJuDt7Q2FQoGsrCyj41lZWfD39xdUlfV68cUXsWnTJuzatQtBQUGiy7E6R44cQXZ2Nrp37w47OzvY2dlhz549+Oijj2BnZwetViu6RKsREBCAjh07Gh3r0KED0tLSBFVknV577TW8+eabGDt2LLp06YIJEybg5Zdfxrx580SXZrWqP/tEfi4yAJmAg4MDIiMjkZiYaDim0+mQmJiImJgYgZVZF0mS8OKLL2LDhg345Zdf0KpVK9ElWaUBAwbgxIkTSE5ONtyioqIwfvx4JCcnQ6FQiC7RavTp0+eOpRzOnTuHli1bCqrIOpWUlEAuN/44VCgU0Ol0giqyfq1atYK/v7/R56JarcaBAwdM9rnILjATSUhIwKRJkxAVFYWePXtiyZIlKC4uxuTJk0WXZjWmTp2KdevW4ccff4Srq6uhH9nd3R2Ojo6Cq7Merq6ud4yrcnZ2hpeXF8dbNbKXX34ZvXv3xty5c/HEE0/g4MGDWLFiBVasWCG6NKsybNgw/POf/0SLFi3QqVMn/PHHH1i8eDGeeuop0aVZtKKiIly4cMHw/eXLl5GcnIxmzZqhRYsWmD59Ot577z20bdsWrVq1wsyZMxEYGIj4+HjTFGiSuWYkSZIkffzxx1KLFi0kBwcHqWfPntL+/ftFl2RVANR4+/zzz0WXZvU4Db7p/O9//5M6d+4sKZVKqX379tKKFStEl2R11Gq1NG3aNKlFixaSSqWSQkNDpbffflsqLy8XXZpF27VrV42/kydNmiRJkn4q/MyZMyU/Pz9JqVRKAwYMkM6ePWuy+mSSxKUuiYiIyLZwDBARERHZHAYgIiIisjkMQERERGRzGICIiIjI5jAAERERkc1hACIiIiKbwwBERERENocBiIiIiGwOAxARERHZHAYgIiIisjkMQERk9XJycuDv74+5c+caju3btw8ODg5Gu1ETke3gXmBEZBO2bNmC+Ph47Nu3D2FhYYiIiMCIESOwePFi0aURkQAMQERkM6ZOnYqdO3ciKioKJ06cwKFDh6BUKkWXRUQCMAARkc0oLS1F586dkZ6ejiNHjqBLly6iSyIiQTgGiIhsxsWLF3H9+nXodDqkpqaKLoeIBGILEBHZBI1Gg549eyIiIgJhYWFYsmQJTpw4AV9fX9GlEZEADEBEZBNee+01fP/99zh27BhcXFzQv39/uLu7Y9OmTaJLIyIB2AVGRFZv9+7dWLJkCdasWQM3NzfI5XKsWbMGv/32G5YtWya6PCISgC1AREREZHPYAkREREQ2hwGIiIiIbA4DEBEREdkcBiAiIiKyOQxAREREZHMYgIiIiMjmMAARERGRzWEAIiIiIpvDAEREREQ2hwGIiIiIbA4DEBEREdkcBiAiIiKyOf8P9CMtL+5NAOUAAAAASUVORK5CYII=",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import norm\n",
    "import numpy as np\n",
    "mu = 7\n",
    "sigma = 1.5\n",
    "x = np.linspace(0, 10)\n",
    "pdf_X = norm.pdf(x, mu, sigma)\n",
    "\n",
    "mean = norm.mean(mu, sigma)\n",
    "print('mean is', mean)\n",
    "\n",
    "fig, ax = plt.subplots(1, 1)\n",
    "ax.plot(x, pdf_X)\n",
    "ax.vlines(mean, 0, max(pdf_X), colors='red')\n",
    "plt.ylabel(\"pdf\")\n",
    "plt.xlabel(\"x\")\n",
    "plt.title(\"Probability density function\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Properties:\n",
    "1. $E[aX+b] = aE[X]+b$ with $a, b \\in \\mathbb{R}$ (\"change of unit\" formula)\n",
    "2. $E[X+Y] = E[X] + E[Y]$\n",
    "3. $E[g(X)] = \\sum_ig(a_i)P(X=a_i)$ where X is a discrete variable taking the values $a_1, a_2,...$ and $g$ is a function $g:\\mathbb{R} \\to \\mathbb{R}$ (\"change of variable\" formula)\n",
    "4. $E[XY]=E[X]E[Y]$, if X,Y independent."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Variance and standard deviation\n",
    "\n",
    "> The _variance_ Var(X) of a random variable $X$ is the number\n",
    ">\n",
    "> $Var(X) = E[(X-E[X])^2]$\n",
    "\n",
    "Variance is a measure of the spread of the data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "variance is 2.0\n"
     ]
    }
   ],
   "source": [
    "from scipy.stats import poisson\n",
    "lambda_ = 2\n",
    "\n",
    "var = poisson.var(lambda_)\n",
    "print('variance is', var)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The _standard deviation_ is simply the square root of variance, representing the spread of data in normal units:\n",
    "\n",
    "$\\sigma = \\sqrt{Var(X)}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Properties:\n",
    "* $Var(aX +b) = a^2Var(X)$\n",
    "* $Var(X) = E[X^2] - E^2[X]$\n",
    "* $Var(X+Y) = Var(X) + Var(Y)$ - only for independent variables."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "\n",
    "**Give it a try!**\n",
    "\n",
    "Given a random variable $X$ with $E[X]=1$ and $Var(X)=1$, calculate:\n",
    "\n",
    "1. $E[3X]$\n",
    "2. $Var(3X)$\n",
    "3. $Var(X+3)$\n",
    "\n",
    "Expected results: 3, 9, 1.\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Covariance\n",
    "\n",
    "> For two jointly distributed real-valued random variables $X$ and $Y$ with finite second moments, the covariance is defined as the expected value (or mean) of the product of their deviations from their individual expected values:\n",
    "> \n",
    "> $Cov(X, Y) = E[(X-E[X])(Y-E[Y])]$\n",
    "\n",
    "By using the linearity properties of the expectation, it can be shown that:\n",
    "\n",
    "$Cov(X, Y) = E[XY] - E[X]E[Y]$\n",
    "\n",
    "For two independent variables $X$ and $Y$, $Cov(X, Y) = 0$, but the inverse is not necessarilty true, since covariance only measures linear dependence."
   ]
  }
 ],
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  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
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