{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "20fce00f",
   "metadata": {},
   "source": [
    "### ML, Data Analysis\n",
    "#### Discrete random variables and their distribution functions\n",
    "In this notebook, we talk about discrete random variables with an example from *Bernoulli probability function*.\n",
    "<br>**Definition:** *Random variables* are *discrete* when their range is **finitely countable** or **infinitely countable**. \n",
    "<br> **Reminder:** A set is *countable* if the number of its elements is not greater than that of the *natural numbers*. As a result, any finite set is a countable set.  \n",
    "<br>**Contents:**\n",
    "- Defining *sample space* of Bernoulli\n",
    "- Defining the *random variable* of Bernoulli\n",
    "- Bernoulli *probability (mass) function*\n",
    "- Displaying Bernoulli probability function\n",
    "- Computing and displaying *(cumulative) distribution function*\n",
    "- Compute and then displaying *entropy* for Bernoulli with different values of parameter $p$\n",
    "<hr>\n",
    "https://github.com/ostad-ai/Machine-Learning\n",
    "<br> Explanation: https://www.pinterest.com/HamedShahHosseini/Machine-Learning/background-knowledge"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6c88da5f",
   "metadata": {},
   "source": [
    "For **enumerations** in Python, see our repository for Python\n",
    "<br> https://github.com/ostad-ai/Python-Everything"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9b279ce8",
   "metadata": {},
   "source": [
    "Here, we review discrete random variables for **Bernoulli probability (mass) function**.<br>\n",
    "**Bernoulli** probability mass function with parameter $p$ is defined on the set of random variables {0,1} such that:<br>\n",
    "$Bernoulli(0)=1-p$  <br>\n",
    "$Bernoulli(1)=p$ \n",
    "<br> Bernoulli function appears in tossing a coin in which we get *heads* denoted by $X=1$ with probability $p$ and *tails* denoted by $X=0$ with probability $1-p$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "84c6d039",
   "metadata": {},
   "outputs": [],
   "source": [
    "# importing the required modules\n",
    "from matplotlib import pyplot as plt\n",
    "import numpy as np\n",
    "from enum import Enum,auto"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "c47b5d52",
   "metadata": {},
   "outputs": [],
   "source": [
    "# defining the sample space for tossing a coin with two outcomes: tails and heads\n",
    "class SampleSpace(Enum):\n",
    "    TAILS=auto()\n",
    "    HEADS=auto()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "03ea179b",
   "metadata": {},
   "outputs": [],
   "source": [
    "# defining a random variable for the outcomes of tossing a coin\n",
    "def X(outcome):\n",
    "    if outcome==SampleSpace.TAILS:\n",
    "        return 0\n",
    "    elif outcome==SampleSpace.HEADS:\n",
    "        return 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "80bb7ecd",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Bernoulli probability (mass) function for random variable x and parameter p   \n",
    "def Bernoulli(x,p=.5):\n",
    "    if x==0: return 1-p\n",
    "    elif x==1: return p\n",
    "    else: return 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "7ccffb38",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Bernoulli probabilities with parameter p=0.7:\n",
      "Probability of TAILS is 0.30000\n",
      "Probability of HEADS is 0.70000\n"
     ]
    }
   ],
   "source": [
    "# display the probability of outcomes based on Bernoulli with parameter p\n",
    "p=.7\n",
    "print(f'Bernoulli probabilities with parameter p={p}:')\n",
    "for outcome in [SampleSpace.TAILS,SampleSpace.HEADS]:\n",
    "    prob=Bernoulli(X(outcome),p)\n",
    "    print(f'Probability of {outcome.name} is {prob:.5f}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "eb758abd",
   "metadata": {},
   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# displaying Bernoulli mass function with parameter p\n",
    "xs,p=[0,1],.7\n",
    "ys=[Bernoulli(x,p) for x in xs]\n",
    "plt.bar(xs,ys,.01,color='orange')\n",
    "plt.scatter(xs,ys,s=50,c='k')\n",
    "plt.title(f'Bernoulli probability (mass) function with p={p}')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "bb8aa9ae",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#compute, then display (cumulative) distribution function of Bernoulli with parameter p\n",
    "xs,p,pmf=[0,1],.7,Bernoulli\n",
    "#-------computing cdf\n",
    "cdf=[]\n",
    "for x in xs:\n",
    "    if cdf:\n",
    "        cdf.append(cdf[-1]+pmf(x,p))\n",
    "    else:\n",
    "        cdf.append(pmf(x,p))\n",
    "#--- displaying cdf\n",
    "for i in range(len(xs)+1):\n",
    "    if i==0:\n",
    "        x0=min(xs)-1; y=0\n",
    "    else:\n",
    "        x0=xs[i-1]; y=cdf[i-1]\n",
    "    if i<len(xs):\n",
    "        x1=xs[i]\n",
    "    else:\n",
    "        x1=max(xs)+1\n",
    "    plt.plot([x0,x1],[y,y],'-',c='b')\n",
    "    if i>0:\n",
    "        plt.scatter(x0,y,c='k')\n",
    "plt.title(f'(Cumulative) Distribution function of Bernoulli with p={p}')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6759e0f8",
   "metadata": {},
   "source": [
    "Compute the entropy of Bernoulli with parameter $p$:<br>\n",
    "$H(X)=-(1-p) \\cdot log(1-p)- p\\cdot log(p)$<br>\n",
    "The base of logarithm is usually *e* or *2*."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "14836fc5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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mm5gwYQL++OOPRq6UiIiIDIVebNDZv39/9O/fv87nr1y5Es2bN8cXX3wBAAgODsbRo0fx5ZdfIiIiorHKJCI9VaFSo7hchRKlCuUq9T3/38pcBmsLM1iaS+u0qR8RGQe9CDnaioyMRHh4eLVjERERePPNN+97TVlZGcrKyjTfKxSKxiqPiBqAIAjIKixDSk4xUnJKkJJTjIyCUtwuVOJ2oRLZRWXILVKiqEwFZQ3BpiYSCWBtLoOdpTmcbS3gbCuHi40FnG0t4OVohWZO1vBxsoZPE2tYWcga+RkSUWMzyJCTnp4Od3f3asfc3d2hUChQUlICKyure65ZuHAhPvzwQ12VSERayFSU4mKaAlcyCnApvQBXMgpwLbMIJeUqrR5HKgHMZdV74QUBmhAkCECRUoUipQrpitJaH8vdXo5Adzu0crdDK4/Kr0B3O1iaM/wQGQqDDDn1MXPmTEyfPl3zvUKhgI+Pj4gVEZmm0nIVzqfm40xyHs6k5CI2OQ+38msOHFIJ4OlgBR8nK/g0sYangyVc7ORwtpHD2dYCTjYWsJGbwdpcBisLGeRmNXdHqdQCSspVKFZWoESpQn5JOW4XKe+0CpUhq6AMN3NLkJJbjOScYhSUViBDUYYMRRn+ScjWPI65TIIQLwd08HFEh2aO6ODTBD5OVuwCI9JTBhlyPDw8kJGRUe1YRkYG7O3ta2zFAQC5XA65XK6L8ojoLiq1gLjUfBy7mo3j17JxOikXZRXVu5ekEsDf1bayxcT931YTb0crWJg9/PwImVQCW7kZbOV1+5OXX1yOa9mFuJJegMsZBbicXtnClFOkxNmUPJxNycP645XnejtaoUcLZ/QMcEGPAGe42Vk+dL1E1DAMMuSEhYVhz5491Y7t378fYWFhIlVERHdTlJbjyOUsHIjPwKFLmVCUVlT7/y62cnRs5ogOzZqgQzNHtPV2gE0dA4guOFibo2OzJujYrInmmCAIuJlbgjMpeTiTnIszyXm4cCsfqXkl+Dn6Jn6OvgkACPa0x+PBbggPcUdbbwe28hCJSC/+qhQWFuLq1aua7xMTExEbGwsnJyc0a9YMM2fORGpqKn744QcAwGuvvYZvvvkGM2bMwMsvv4y//voLP/30E3bv3i3WUyAyeTlFSuyJS8MfF9Jx4vptlKsEzf+zszRDmL+zpsUjwM3W4N78JRJJ5aBkJ2s8HeoFAChWViAqMQfHr93GsavZuJimQPydr6/+ugp3eznCg90xsJ0XujV3glRqWM+ZyNBJBEEQHnxa4zp8+DD69et3z/ExY8Zg/fr1GDt2LJKSknD48OFq17z11lu4ePEimjZtijlz5mDs2LF1/pkKhQIODg7Iz8+Hvb19AzwLItNTVFaB/Rcz8GtsKv5JyEaF+t8/Jy1cbRAe4o7Hg93RoVkTyEzgDT6nSIlDlzJxID4DR65koVj578BpD3tLDAr1xDPtvdHay97gQh6RvtDm/VsvQo4YGHKI6kcQBMQk52JLVAp2n0urNgOqjbc9BrbzwhMh7vB3tRWxSvGVlqsQef029sWlY+/5tGpddgFuthjexQfPdWwKJxsLEaskMjwMOXXAkEOkndwiJXacScXWqGQkZBZqjjd3scHToV54ur0XWph4sLmfsgoVjlzOwm9nb+FAfAZKyysHXlvIpHiitTtGdG2GMH9ndmcR1QFDTh0w5BDVTUJGAdYeS8KOmJuaWVGW5lIMaueF4V2boWMzR3a9aKGgtBy/n03DlqhkxKXma44HuNliXE8/PNehKRciJKoFQ04dMOQQ3Z8gCPg7IRvfH03E31eyNMeDPe0xslszPNPeC/aW5iJWaBzOp+ZjS1Qydp1JRdGd8TtNrM0xslszjA7zg7s9p6MT/RdDTh0w5BDdS60W8MeFdHz911VcTKvc+kQiAR4Pdsf4Xs3RtbkTW20agaK0HD+dSsH640m4mVsCoLIra0jnppjUpwV8nKxFrpBIfzDk1AFDDtG/KlRq/O9cGpYfuqoZb2NtIcOwLj4Y28MPvs42IldoGlRqAfsvpmPNP4k4fSMXQOVChs928Mbkvi1MfjA3EcCQUycMOUSVLTe749KwZP8VJGYXAahc02Zcz+YY18MPTTjzRzQnr9/GN4euaraVkEqAwe298dbjgWzZIZPGkFMHDDlkygRBwD8J2Vj8xyWcT63slmpibY4Jvf0xKsyX4230yJnkXCw/dBUH4jMBVO6f9WI3X0x9NAAuttyqhkwPQ04dMOSQqYq7mY8Fe+IRef02AMBWboZXHvHHy72a13lvJ9K9czfz8NkflzUtO9YWMkzo7Y/X+vjD2oK/NzIdDDl1wJBDpiazoBSf7buM7TE3IQiVA1tHhfliSr8ALkhnQI5dzcbifZdw9mbl9HMPe0u81z8Iz7T34qBwMgkMOXXAkEOmoqxChfXHkvD1X1dRWFa56u6zHbzxdkQreDtaiVwd1YcgCNh7Ph0L98YjJadyNlbHZo6YN6g1Qn0cxS2OqJEx5NQBQw6Zgn8SsjBn13kk3S4GAIT6OGLeoJBqu2uT4SotV+H7o4lYfuiqZp+sYZ198F7/IA4aJ6PFkFMHDDlkzLIKyvDx7ov4NfYWAMDVTo53nwzCcx28uXWAEcpQlOLTfZewIyYVAOBkY4HZTwXj2Q7e7MIio8OQUwcMOWSM1GoB206nYOGeeChKKyCVAKPD/PB/TwTCjjOmjN7ppBy8vzMOVzIq1zrq0cIZHw9uw/V1yKgw5NQBQw4Zmxu3i/DO9nOISswBALT2ssfC59qiXVNHcQsjnVJWqLHm6HUsO5CAsgo1LMykmP54ICb29oeMrXhkBBhy6oAhh4yFWi1g08kbWLDnEkrKVbC2kGH644EY28MPZjKp2OWRSG7cLsLsXec1U847NHPE50NDuVM8GTyGnDpgyCFjkJJTjHd/OYfj1yrXvOnu74TPhoRyRVwCUDkL6+fom5j/+0UUlFVAbibFOxGt8HLP5hybRQaLIacOGHLIkAmCgB0xqZj32wUUllXAylyG9/oHYVR3X7550T1S80rw3i/nNK06XZs74cth7bmEABkkhpw6YMghQ6UoLcfsnefx29nKmVOdfZvg86Gh8HPhJpp0f4IgYEtUCj7ZfRFFShXsLc3w6fPt0L+tp9ilEWmFIacOGHLIEEXfyMW0rWdwM7cEMqkEb4W3xKS+ARxQSnV243YR3tgai7MpeQCAEV19MGdgCLeGIIPBkFMHDDlkSNRqAd8evoovDyRApRbQtIkVlg3vgE6+XNSPtFeuUuPL/Vew4sg1CALg72qD5SM7ItiTfwtJ/2nz/s2pF0R6LrdIiZc3nMLnf16BSi3gmfZe2DOtNwMO1Zu5TIoZTwZh0/hucLeX43pWEZ799hi2R98UuzSiBsWQQ6THzt3Mw8Cvj+Lw5SzIzaRY/Hw7LB3WHvZc2I8aQI8AF+yd9gj6BLqitFyNt38+i5k7zqG0XCV2aUQNgiGHSA8JgoCNJ25gyIpIpOaVwNfZGjsm98ALXXy4TD81KCcbC6wb2wVvhQdCIgG2RKVgyMrjSMkpFrs0oofGkEOkZ8oqVJix/Rxm7zoPpUqNJ0Lc8dvUXmjt5SB2aWSkpFIJpoW3xIZxXdHE2hznUxUY+PVRHL0z5ZzIUDHkEOmRzIJSjFh1Aj9H34RUArzXPwjfjeoEByt2T1HjeyTQFbvf6I1QH0fkl5RjzLoorDuWCBOdn0JGgCGHSE/E3czHM98cQ0xyHuwszbBuXFe81qcFu6dIp7wcrbDtle54roM3VGoBH/5+Ee/9EoeyCo7TIcPDkEOkB34/ewtDVh5HWn4p/F1t8OuUnugT6Cp2WWSiLM1l+OKFUMwaEAypBNh2OgUjV59EVkGZ2KURaYUhh0hEgiBg+aGreH3LGZRVqNG3lSt2TekJf26iSCKTSCSY+Ig/1o7tAjtLM0TfyMVzK47hamah2KUR1RlDDpFIylVqzNwRh8/+uAwAmNCrOb4f04XTw0mv9G3lhl1TesLX2RopOSV47ttjOHH9tthlEdUJQw6RCApKyzF+w2lsPZUCqQT46JnWmD0whNszkF5q4WqLHZN6oEMzRyhKKzD6+yj8GpsqdllED8SQQ6Rj6fmlGLoyEn9fyYKVuQyrRnXG6DA/scsiqpWzrRxbJnZH/zYeUKrUmLY1FssPXeXMK9JrDDlEOnQ9qxDPrziOS+kFcLGVY9ur3REe4i52WUR1Ymkuw/KRHTGxd3MAwGd/XMaHv1+EWs2gQ/qJIYdIR86n5mPoysoVjP1dbLBzcg+0a+oodllEWpFKJZj1VAjmDQoBAKw/noTpP8WiXKUWuTKiezHkEOlA5LXbGL7qBG4XKdHG2x4/vRYGHydrscsiqrdxPZtj6bD2MJNKsCv2Fl754TRKlFxLh/QLQw5RI/vjQjrGrItCYVkFuvs7YcvE7nCxlYtdFtFDG9zBG6tHd4aluRSHLmdh1PcnkV9cLnZZRBoMOUSN6NfYVEzaGA1lReUeVOvHdYUdp4iTEekX5IaN47vB3tIMp2/kYsTqE8gpUopdFhEAhhyiRrM9+ibe3BYLtQA837Epvn2xIyzNZWKXRdTgOvs54afXwuBiK8fFNAVGrDrB1ZFJLzDkEDWCLVHJeGf7WQgCMLJbM3w2pB3MZPznRsYryMMeW1/pDjc7OS5nFGD4qkhkKErFLotMHP/qEjWwHyKTMHNHHAQBGNvDD58MbgMpF/kjExDgZoufXg2Dl4MlrmUVYdh3kbiVVyJ2WWTCGHKIGtDao4mY++sFAMDE3s0xb1AIdxEnk+LnYoNtr4ahaRMrJN0uxrBVlcsmEImBIYeogfwYmYSP/ncRADC5bwu8PyCYAYdMko+TNba9Gga/O/tdjVx9Aun57Loi3WPIIWoAW6OSMedOC86kvi3wTkQrBhwyad6OVtjySnf4OFnhxu1ijFxzApkFDDqkWww5RA/pl+ibmLkzDgAwvldzzGDAIQIAeDpYYfOE7vB2tML1rCK8tOYkbhdy1hXpDkMO0UP47ewtzSyq0WG+mP0Uu6iI7ubjZI3NE7vB3V6OKxmFeOn7KOQVcx0d0g2GHKJ62n8xA2/dWQdneBcffDCoNQMOUQ18nW2w+c5K3/FpCoxZW7kCOFFjY8ghqocT129jyuYYqNQCnuvgjQXPtuU0caJatHC1xeaJ3dDE2hxnb+bj1R9Po6yCe11R42LIIdLS+dR8TNhwGsoKNcKD3bF4SDsGHKI6CHS3w/pxXWFjIcOxq7cxbUssKrh7OTUihhwiLVzPKtQ0tXdr7oRvRnbgSsZEWgj1ccSq0Z1hIZNi34V0zNp5HoIgiF0WGSn+dSaqo7T8Eoz6Pgq3i5Ro422PNWM6cy8qonroGeCCr0Z0gFQCbDudgkX7LoldEhkphhyiOsgvKceYtVFIzSuBv4sNdxMnekhPtvHAoufaAQC+O3Ida/65LnJFZIwYcogeoKxChVd/PI0rGYVwt5fjh/Fd4WIrF7ssIoP3QhcfzOwfBAD4ZE889sSliVwRGRuGHKJaqNUCZmw/hxPXc2ArN8O6sV3RtIm12GURGY1XHvHHmDBfCALw5rZYnErKEbskMiIMOUS1+OzPy/g19hbMpBKseKkjQrzsxS6JyKhIJBLMHdQaj4e4Q1mhxsQfTuNaVqHYZZGRYMghuo+NJ25gxeFrAICFz7VF75auIldEZJxkUgm+Gt4B7X0ckVdcjrHropBVwO0f6OEx5BDV4NDlTMz99TwA4K3wQAzt7CNyRUTGzcpChu/HdIbvnZ3LJ2w4hRIlFwukh8OQQ/QfVzIK8PrmM1ALwAudm+KNxwLELonIJDjbyrF+XFfNqshvbz/LNXTooehNyFm+fDn8/PxgaWmJbt26ISoqqtbzly5dilatWsHKygo+Pj546623UFpaqqNqyVjdLizD+A2nNIv9fTy4LfejItKh5i42WPlSJ5jLJNh9Lg3LDiaIXRIZML0IOdu2bcP06dMxb948xMTEIDQ0FBEREcjMzKzx/M2bN+O9997DvHnzEB8fj++//x7btm3D+++/r+PKyZiUVajw2sZopOSUwNfZGitf6gQLM734J0JkUrr5O+PjwW0AAEsPJOD3s7dErogMlV78BV+yZAkmTpyIcePGISQkBCtXroS1tTXWrl1b4/nHjx9Hz549MXLkSPj5+eGJJ57AiBEjHtj6Q3Q/giBg9s7zOJWUCzu5Gb4f0xlNbCzELovIZA3r0gwTezcHALz981mcTckTtyAySKKHHKVSiejoaISHh2uOSaVShIeHIzIyssZrevTogejoaE2ouX79Ovbs2YMBAwbc9+eUlZVBoVBU+yKqsvqf6/g5+iakEuCbFzsiwM1O7JKITN57/YPxaJAbyu5MLU/LLxG7JDIwooec7OxsqFQquLu7Vzvu7u6O9PT0Gq8ZOXIkPvroI/Tq1Qvm5uZo0aIF+vbtW2t31cKFC+Hg4KD58vHhbBmqdORKFhburdw7Z+7AEPQJ5FRxIn0gk0qwbHh7tHK3Q2ZBGV7bGIPScs64oroTPeTUx+HDh7FgwQJ8++23iImJwY4dO7B7927Mnz//vtfMnDkT+fn5mq+UlBQdVkz6Kvl2Md7YcgaCAAzr7IMxPfzELomI7mJnaY41YzrD0docZ1PyMPdX7lpOdWcmdgEuLi6QyWTIyMiodjwjIwMeHh41XjNnzhyMGjUKEyZMAAC0bdsWRUVFeOWVVzBr1ixIpfdmN7lcDrmc+w3Rv4qVFXjlx9PILylHex9HfDS4NWdSEekhHydrfD2iA8asjcJPp2+iXVNHvNTdV+yyyACI3pJjYWGBTp064eDBg5pjarUaBw8eRFhYWI3XFBcX3xNkZDIZADDhU50IgoB3f4nDpfQCuNjKsfKlTpCbycQui4juo3dLV8x4snIzzw9/v4DT3OOK6kD0kAMA06dPx+rVq7FhwwbEx8dj0qRJKCoqwrhx4wAAo0ePxsyZMzXnDxo0CCtWrMDWrVuRmJiI/fv3Y86cORg0aJAm7BDVZs0/ifj9bOWeVN++2BEeDpZil0RED/DqI/54qq0nylUCJm2KQYaCa6NR7UTvrgKAYcOGISsrC3PnzkV6ejrat2+Pffv2aQYjJycnV2u5mT17NiQSCWbPno3U1FS4urpi0KBB+OSTT8R6CmRAjl/NxsK98QCAuYNC0LW5k8gVEVFdSCQSLB7SDlczC3E5owCTNkZj6ythXM+K7ksimGj/jkKhgIODA/Lz82Fvz52lTUWGohRPffUPsguVeL5jU3w+tB3H4RAZmKTsIjz9zVEoSivwcs/mmDsoROySSIe0ef9m/CWTUaFS4/UtZ5BdqESQhx0+ebYNAw6RAfJzscEXL7QHAKw9loi9cWniFkR6iyGHTMYX+68gKjEHtnIzfPtiR1iac/wWkaF6PMQdrz7iDwCYsf0ckrKLRK6I9BFDDpmEg/EZWHH4GgBg0fNt4e9qK3JFRPSw3o5ohS5+TVBQVoHJm7hQIN2LIYeMXkpOMab/dBYAMCbMFwPbeYlcERE1BHOZFF+P6AhnGwtcTFPgw98viF0S6RmGHDJqygo1pm45g/yScoQ2dcD7TwWLXRIRNSAPB0ssHd4eEgmwJSoFO8/cFLsk0iMMOWTUvth/GWdT8mBvaYZvRnbkgn9ERqh3S1e88WhLAMDsnec5Poc0GHLIaP2TkIXvjlwHACwe0g4+TtYiV0REjeWNx1qia3MnFClVmLb1DJQVarFLIj3AkENG6XZhmWYczshuzfBkG0+RKyKixiSTSrB0WHs4WJnj7M18fLH/stglkR5gyCGjIwgC3v75LLIKytDSzRZznuJCYUSmwMvRCp8+3w4A8N2R6/gnIUvkikhsDDlkdNYdS8Khy1mwMJPi65EdYGXBcThEpuLJNh54sVszAMD0n87idmGZyBWRmBhyyKhcuJWPRXsvAQDmPBWMIA9u2UFkauYMDEGguy2yCsrw9s9nYaK7FxEYcsiIlJar8ObWWChVajwe4o6XuvuKXRIRicDSXIavRnSAhZkUhy5nYePJZLFLIpEw5JDR+OyPy0jILISLrRyfPs+NN4lMWZCHPWb2DwIAfLL7Iq5nFYpcEYmBIYeMwvFr2fj+aCIAYPGQtnCysRC5IiIS25gwP/QMcEZpuRpv/XQWFSpOKzc1DDlk8BSl5Xj7znTxEV2b4dEgd5ErIiJ9IJVK8NmQUNhZmuFsSh6WH7omdkmkYww5ZPA++PUCbuWXwtfZGrO5bQMR3cXL0QofD24DAPjqrwScTckTtyDSKYYcMmh749Kw40wqpBJgyQvtYSM3E7skItIzT4d64al2nlCpBbz1UyxKlNyt3FQw5JDByioow/s74wAAk/q2QCffJiJXRET6SCKR4JPBbeBmJ8f1rCJ8uu+S2CWRjjDkkEESBAFzdp1HbnE5QjztMe2xQLFLIiI95mhtgcVDKldDXn88CVGJOSJXRLrAkEMGaXdcGvZdSIeZVILPh4bCwowvZSKqXd9WbhjW2QcAMGP7WXZbmQC+M5DBuV1Yhrm/XgAATOkXgBAvrmpMRHUza2AwPOwtkXS7GJ//yU08jR1DDhmcub9dQE6REkEedpjSL0DscojIgNhbmmPhc20BAGuPJSL6BrutjBlDDhmUvXFp2H0uDTJ2UxFRPfULcsPzHZtCEIB3fj6H0nJ2WxkrvkOQwcgpUmLOr+cBAJP7tkAbbweRKyIiQzV3YEjlbKvsIny5/4rY5VAjYcghg/HR7xeQXahEoLstpj7Kbioiqj8Ha3MseLay22r1P9cRy0UCjRJDDhmEI1eysCv2FqQS4LMhoZCbycQuiYgMXHiIO55p7wW1AMzcEYdy7m1ldBhySO8VKysw686if2N7NEeoj6O4BRGR0ZgzMASO1uaIT1NoNvkl48GQQ3pv2YEE3MwtgbejFf7vCS76R0QNx8VWjlkDKve8W3rgCm7cLhK5ImpIDDmk1y7cyseaO5+u5g9uzb2piKjBDenUFGH+zigtV2P2rvMQBEHskqiBMOSQ3lKpBczcEQeVWsBT7TzxaJC72CURkRGSSCRY8FxbWJhJ8U9CNn6NvSV2SdRAGHJIb204noRzN/NhZ2mGeYNCxC6HiIxYcxcbTHusJQDgo/9dRG6RUuSKqCEw5JBeupVXolly/f0BwXCzsxS5IiIydhN7+6OVux1yipT4ZE+82OVQA2DIIb300e8XUaxUoYtfE82GekREjcnCTIoFd7Z82B59E6eSuOWDoWPIIb1z+HIm9l1Ih0wqwfzBbSCVSsQuiYhMRCffJhjepfKD1Zxd51HBtXMMGkMO6ZXSchXm/Va5w/jYHn4I8uAO40SkWzOeDIKjtTkupRdgQ+QNscuhh8CQQ3pl1d/XceN2Mdzs5HgzvKXY5RCRCXKyscCMiCAAwJf7ryBDUSpyRVRfDDmkN1JyirH80FUAwOyBIbCzNBe5IiIyVcO7+CDUxxGFZRVYwEHIBoshh/TGB79dQFmFGj1aOGNQO0+xyyEiEyaVSvDxM20gkQC/xt7C8WvZYpdE9cCQQ3ph/8UMHLyUCXOZBB890xoSCQcbE5G42jZ1wIvdmgEA5v56AcoKDkI2NAw5JLrSchU++l/lYOPxvfwR4GYnckVERJXeeSIIzjYWuJpZiA3Hk8Quh7TEkEOi+/5oIlJySuBuL8frjwaIXQ4RkYaDtTnefbJyEPJXBxOQXVgmckWkDYYcElWGolQz2Hhm/2BuwElEemdIp6Zo6+2AgrIKfHFnJXYyDAw5JKpP911CsVKFjs0c8Ux7L7HLISK6h1Qq0eyft/VUCs6n5otcEdUVQw6J5kxyLnbEpAIA5g3iYGMi0l+d/ZwwKNQLglC57YwgCGKXRHXAkEOiUKsFfPj7RQDA8x2bItTHUdyCiIge4L3+QbA0lyIqKQe749LELofqgCGHRLErNhWxKXmwsZDh3SdbiV0OEdEDeTta4bU+LQAAC/dcQolSJXJF9CAMOaRzRWUVWLT3EgBgyqMBcLO3FLkiIqK6efWRFvBysERqXgm++/ua2OXQAzDkkM599/d1ZBaUoZmTNV7u2VzscoiI6szKQoaZA4IBAN8duc59rfQcQw7pVIaiFKv/vg6gqn9bJnJFRETaGdjOEx2bOaKkXIUlf14RuxyqBUMO6dSSP6+gpLxyynj/Nh5il0NEpDWJRIJZT1W25vwcnYLL6QUiV0T3w5BDOnMpXYGfo1MAALOeCuaUcSIyWJ18ndC/jQfUArBwL3cp11cMOaQzC/dcgloABrT1QCdfJ7HLISJ6KO8+GQQzqQSHL2fhaAJ3KddHDDmkE/8kZOHIlSyYyySYEREkdjlERA/Nz8UGL3X3BQB8siceKjUXCNQ3DDnU6FRqAQv2VE4Zf6m7L/xcbESuiIioYbzxWEvYWZohPk2BnWdSxS6H/oMhhxrdzjOpiE9TwM7SDG882lLscoiIGoyTjQWm9AsAAHzx52WUlnOBQH3CkEONqrRcpdm1d0q/ADSxsRC5IiKihjW2hx+8Ha2Qll+KtccSxS6H7qI3IWf58uXw8/ODpaUlunXrhqioqFrPz8vLw5QpU+Dp6Qm5XI7AwEDs2bNHR9VSXW08cQNp+aXwdLDE2B5+YpdDRNTgLM1l+L8nAgEAKw9fQ35xucgVURW9CDnbtm3D9OnTMW/ePMTExCA0NBQRERHIzMys8XylUonHH38cSUlJ2L59Oy5fvozVq1fD29tbx5VTbQpKy7H80FUAwJvhLbnwHxEZrWfaeyPQ3RaK0gpu96BH9CLkLFmyBBMnTsS4ceMQEhKClStXwtraGmvXrq3x/LVr1yInJwe7du1Cz5494efnhz59+iA0NFTHlVNt1vyTiNzicvi72uD5jk3FLoeIqNHIpBK8/UTlZsPrjiUhs4DbPegD0UOOUqlEdHQ0wsPDNcekUinCw8MRGRlZ4zW//fYbwsLCMGXKFLi7u6NNmzZYsGABVKr7D/gqKyuDQqGo9kWN53ZhGdb8U7l9w/893gpmMtFfakREjerxEHd0uLPdwzd/XRW7HIIehJzs7GyoVCq4u7tXO+7u7o709PQar7l+/Tq2b98OlUqFPXv2YM6cOfjiiy/w8ccf3/fnLFy4EA4ODpovHx+fBn0eVN23h6+hSKlCG297bt9ARCZBIvl3HbAtUclIySkWuSISPeTUh1qthpubG1atWoVOnTph2LBhmDVrFlauXHnfa2bOnIn8/HzNV0pKig4rNi238krw44kbAIB3IoIglXL7BiIyDWEtnNG7pQvKVQK+3M/NO8UmeshxcXGBTCZDRkZGteMZGRnw8Ki5BcDT0xOBgYGQyf4dyBocHIz09HQolcoar5HL5bC3t6/2RY1j2YEEKCvU6NbcCY+0dBG7HCIinapqzdkZm4pL6RwaISbRQ46FhQU6deqEgwcPao6p1WocPHgQYWFhNV7Ts2dPXL16FWq1WnPsypUr8PT0hIUF12ER07WsQs0mnDOeDOImnERkcto2dcCAth4QBODzP9iaIybRQw4ATJ8+HatXr8aGDRsQHx+PSZMmoaioCOPGjQMAjB49GjNnztScP2nSJOTk5GDatGm4cuUKdu/ejQULFmDKlCliPQW6Y9mBBKgFIDzYDZ18m4hdDhGRKKY/3gpSCXAgPgNnU/LELsdkmYldAAAMGzYMWVlZmDt3LtLT09G+fXvs27dPMxg5OTkZUum/eczHxwd//PEH3nrrLbRr1w7e3t6YNm0a3n33XbGeAgFIyCjA7+duAQDeejxQ5GqIiMQT4GaLwR28sSMmFUsPXMG6cV3FLskkSQRBMMltUxUKBRwcHJCfn8/xOQ1k6uYY/O9cGiJau+O7UZ3FLoeISFRJ2UV4bMkRqNQCdk7ugQ7N2LrdELR5/9aL7ioyfJfTC7A7Lg0A8GY4W3GIiPxcbPBsh8qV+JceSBC5GtPEkEMNYtnBKxAEoH8bDwR7smWMiAgAXn80ADKpBEeuZCH6Rq7Y5Zgchhx6aPFpCuyJq1y4cVp4S5GrISLSH77ONni+Y1VrDmda6RpDDj20ZXeaYZ9q64kgD7biEBHdbWq/ljCTSvBPQjaib+SIXY5JYcihh3LhVj72XUiHRMJWHCKimjRzttZsUvzlfo7N0SWGHHood7fiBLrbiVwNEZF+mvpoAMykEhy9mo2oRLbm6ApDDtVbfJoCf17MqGzFeYytOERE9+PjZI2hnStbc77+i605usKQQ/W2/NBVAMCANp5oyVYcIqJaTe5bOdPqn4RsroKsIww5VC/Xswo16+JM7tdC5GqIiPSfj5M1ngn1AgB8c+dDIjUuhhyqlxWHr0EQgMeC3NDay0HscoiIDMLkfi0gkQD7L2Zwh3IdYMghrd3MLcbOM6kAgCmPBohcDRGR4Qhws0P/Nh4AgG8PXRO5GuPHkENaW/X3dVSoBfRo4YyO3IuFiEgrk/tWfjj837lbSMouErka48aQQ1rJLCjF1lMpAICp/diKQ0SkrTbeDujXyhVqobLrnxrPQ4Wc8vJypKSk4PLly8jJ4bx/U/D9P4lQVqjRoZkjwlo4i10OEZFBmnqnq3/HmZu4lVcicjXGS+uQU1BQgBUrVqBPnz6wt7eHn58fgoOD4erqCl9fX0ycOBGnTp1qjFpJZHnFSmw8cQNAZSuORCIRuSIiIsPUydcJ3f2dUK4SsOrv62KXY7S0CjlLliyBn58f1q1bh/DwcOzatQuxsbG4cuUKIiMjMW/ePFRUVOCJJ57Ak08+iYQELnhkTNYfT0KRUoVgT3s8GuQmdjlERAZtar/KRVS3RCUju7BM5GqMk1Yh59SpU/j7778RFRWFOXPmICIiAm3btkVAQAC6du2Kl19+GevWrUN6ejoGDx6Mf/75p7HqJh0rUaqw4XgSAGBS3xZsxTFCffv2xZtvvil2GVoztLoFQcArr7wCJycnSCQSxMbGil0SiaRngDNCmzqgrEKNH+78faWGJREEQRC7CDEoFAo4ODggPz8f9vbcOftBfoxMwpxfL8DHyQqH/q8vzGSmM2Z97Nix2LBhwz3HIyIisG/fvjo9Rt++fdG+fXssXbq0gatrODk5OTA3N4ednWGtXm1ode/duxfPPPMMDh8+DH9/f7i4uMDMzEzsshrMBx98gA8//LDasVatWuHSpUsiVaTf9sSlYfKmGDham+P4e4/C2sJ4XguNRZv3b63v5q5duzB48OD61kYGSKUWsPqfRADAhF7+JhVwqjz55JNYt25dtWNyubzBf45SqYSFhUWDP25dODk5ifJzH5ah1X3t2jV4enqiR48e9X4MMV8nddG6dWscOHBA870xhbiGFtHaA77O1rhxuxg/nUrB2J7NxS7JqGj9bjVs2DB89dVXtZ5joo1DRmvf+XQk5xTD0dpcs8GcqZHL5fDw8Kj21aRJ5RpBffv2xRtvvIEZM2bAyckJHh4e+OCDDzTXjh07FkeOHMGyZcsgkUggkUiQlJSkuXbq1Kl488034eLigoiICJSVleGNN96Am5sbLC0t0atXr3sG81ddN3XqVDg4OMDFxQVz5szR/Nv74Ycf4OzsjLKy6v38gwcPxqhRo2p8jv/t9tm+fTvatm0LKysrODs7Izw8HEVF91/To2/fvnj99dfx5ptvokmTJnB3d8fq1atRVFSEcePGwc7ODgEBAdi7d6/mmn379qFXr15wdHSEs7MzBg4ciGvX/p1Sm5WVBQ8PDyxYsEBz7Pjx47CwsMDBgwdrrLs+dfj5+d3Tyta+fftqv8f6PO5/jR07Fq+//jqSk5MhkUjg5+cHAA/8ndf0OqlJeno6JBIJli1bhg4dOsDS0hKtW7fG0aNH71tTYzAzM6v2b8XFxUWnP9+QyKQSTOhVGWzWHE1EhUotckXGReuQs2PHDsyaNavGPnCVSoX169cjODi4IWojPSAIAlb9XfmmMzrMj02p97FhwwbY2Njg5MmTWLx4MT766CPs378fALBs2TKEhYVh4sSJSEtLQ1paGnx8fKpda2FhgWPHjmHlypWYMWMGfvnlF2zYsAExMTEICAhARETEPcs0bNiwAWZmZoiKisKyZcuwZMkSrFmzBgAwdOhQqFQq/Pbbb5rzMzMzsXv3brz88ssPfD5paWkYMWIEXn75ZcTHx+Pw4cN47rnnHvgBZsOGDXBxcUFUVBRef/11TJo0CUOHDkWPHj0QExODJ554AqNGjUJxcTEAoKioCNOnT8fp06dx8OBBSKVSPPvss1CrK//Qu7q6Yu3atfjggw9w+vRpFBQUYNSoUZg6dSoee+yxBqujrh72cZctW4aPPvoITZs2RVpamibI1OV3/t/XSU2qxvesXbsWS5cuRWxsLJo1a4YXX3xRc0/vtmDBAtja2tb6lZycrNU9AoCEhAR4eXnB398fL774Yr0ew5QM6eQDJxsL3Mwtwd7z6WKXY1yEeoiOjha8vLyEZ599VigpKRHKysqEb7/9VvDz8xMcHR2FuXPn1udhdSo/P18AIOTn54tdil6LvJYt+L77PyFw1h4hu6BU7HJEMWbMGEEmkwk2NjbVvj755BNBEAShT58+Qq9evapd06VLF+Hdd9/VfN+nTx9h2rRp9zx2nz59hA4dOmi+LywsFMzNzYVNmzZpjimVSsHLy0tYvHhxteuCg4MFtVqtOfbuu+8KwcHBmu8nTZok9O/fX/P9F198Ifj7+1e75r+1VNUYHR0tABCSkpJquzX3XH/3faioqBBsbGyEUaNGaY6lpaUJAITIyMgaHyMrK0sAIMTFxVU7PnnyZCEwMFAYOXKk0LZtW6G09N/X4n/vbX3q8PX1Fb788stqPzM0NFSYN29egz4/QRCEL7/8UvD19dV8X5ff+X9fJ/ezaNEiwdzcXEhMTNQcO336tABASE5Ovuf827dvCwkJCbV+lZeXP/Dn3m3Pnj3CTz/9JJw9e1bYt2+fEBYWJjRr1kxQKBRaPY6p+XL/ZcH33f8JA7/6577/RqmSNu/f9fpY3rFjR5w4cQIDBgxAt27dkJ2djfLycrz55puYOnUqB/Iake+OVLbiDOnUFM62DT8GxVD069cPK1asqHbs7rEg7dq1q/b/PD09kZmZWafH7tSpk+a/r127hvLycvTs2VNzzNzcHF27dkV8fHy167p3715tlltYWBi++OILqFQqyGQyTJw4EV26dEFqaiq8vb2xfv16jB07tk4z40JDQ/HYY4+hbdu2iIiIwBNPPIEhQ4Zouuju5+77IJPJ4OzsjLZt22qOubu7A4Dm3iQkJGDu3Lk4efIksrOzNa0NycnJaNOmjea6zz//HG3atMHPP/+M6OjoB46H0raOumqMx63r7/zu18n9xMbG4rnnntN0gwGo9e+xk5NTg49p6t+/v+a/27Vrh27dusHX1xc//fQTxo8f36A/y5iMDvPDyiPXEJeaj8hrt9EjgF18DaFeI0gLCgqwceNGZGRkICEhAXl5eTh48CDef/99BhwjciWjAIcuZ0EiASb09he7HFHZ2NggICCg2tfdbw7m5ubVzpdIJDV2D9zvsRtDhw4dEBoaih9++AHR0dG4cOECxo4dW6drZTIZ9u/fj7179yIkJARff/01WrVqhcTExFqvq+k+3H2sKmBV3ZtBgwYhJycHq1evxsmTJ3Hy5EkAlQNr73bt2jXcunULarVaM56pIeuQSqX3dMWVl5c/9OM2pLq8TmJjY9G+fftqxyIjI+Hi4gJvb+97zm+s7qq7OTo6IjAwEFevXn2oxzF2TjYWGNqpshv7Oy4O2GC0Djlz5syBr68v1qxZg08++QRZWVkYOnQowsPDudKxkalahfPJ1h5o7tI4b8SmwsLCAiqV6oHntWjRQjPuokp5eTlOnTqFkJCQaudWBYIqJ06cQMuWLSGTyTTHJkyYgPXr12sW8Lx7LNCDSCQS9OzZEx9++CHOnDkDCwsL7Ny5s87XP8jt27dx+fJlzJ49G4899hiCg4ORm5t7z3lKpRIvvfQShg0bhvnz52PChAlat8A8iKurK9LS0jTfKxSKBwa6hqLN77w2JSUlSEhIqPY6U6vVWLp0KcaMGQOp9N4/96+99hpiY2Nr/fLy8nqo51dYWKiZUUa1m9C7OaQS4MiVLMSnKcQuxyho3V21fft2LF26FC+++KLmj+n69esxZ84c9OvXD5s3b8bTTz/d4IWSbmUoSvFrbCoA4JVHTLsVB6ic/ZKeXn1AoJmZWZ1njfj5+eHkyZNISkqCra0tnJycanzTsbGxwaRJk/DOO+/AyckJzZo1w+LFi1FcXHxPU39ycjKmT5+OV199FTExMfj666/xxRdfVDtn5MiRePvtt7F69Wr88MMPdX6+J0+exMGDB/HEE0/Azc0NJ0+eRFZWVoNOKmjSpAmcnZ2xatUqeHp6Ijk5Ge+99949582aNQv5+fn46quvYGtriz179uDll1/G//73vwar5dFHH8X69esxaNAgODo6Yu7cudXCYmPS5ndem7i4OEgkEmzcuBGPPvqo5nnk5eVh9uzZNV5T3+6qb775Bjt37tTMcLvb22+/jUGDBsHX1xe3bt3CvHnzIJPJMGLECK1/jqnxdbZB/zae2B2XhtX/XMeSF9qLXZLB0zrkXLx4scY+/fnz56NZs2Z44YUX8Pnnn2Pq1KkNUiCJ48fIGyhXCejs2wQdmtU+DsMU7Nu3755PotoscPb2229jzJgxCAkJQUlJCRITE6uNm7jbokWLoFarMWrUKBQUFKBz5874448/7hkPM3r0aJSUlKBr166QyWSYNm0aXnnllWrnODg44Pnnn8fu3bu1Wt/K3t4ef//9N5YuXQqFQgFfX1988cUX1cZbPCypVIqtW7fijTfeQJs2bdCqVSt89dVX6Nu3r+acw4cPY+nSpTh06JCmK/zHH39EaGgoVqxYgUmTJjVILTNnzkRiYiIGDhwIBwcHzJ8/X2ctOUDdf+e1iY2NRVBQEGbMmIHnn38e+fn5iIiIwJEjR+Do6Nig9WZnZ1eb6n+3mzdvYsSIEbh9+zZcXV3Rq1cvnDhxAq6urg1ag7Ga+Ig/dsel4fezt/Be/yC42VmKXZJBa/AVj/fs2YPhw4dDodDvpjaueHx/peUq9Fj0F3KKlFjxYkf0b8tmZn2jzQrKjz32GFq3bv3A9a3IsE2ZMgW5ubnYvHmz2KXQQ3ru22OISc7DtMda4q3HA8UuR+9o8/6t1ZicugxAGzBgAA4fPgwASE1N1ebhSU/8GpuKnCIlvB2t8HiIu9jlUD3l5uZi586dOHz4MKZMmSJ2OdTIYmNj75nlR4bp5TuLA246eQOl5Q8ey0f3p1XI6dKlC1599dVaBxjn5+cjOjoabdq0wS+//PLQBZJuCYKAtUeTAABjevia5BYOxqJDhw4YO3YsPv30U7Rq1UrscqgRCYKAuLg4hhwjEdHaA54OlsguVOL3s7fELsegaTUm5+LFi/jkk0/w+OOPw9LSEp06dYKXlxcsLS2Rm5uLixcv4sKFC+jYsSMWL16MAQMGNFbd1Egir93G5YwCWFvIMKxzM7HLofuoai2tTV2mWpNxkEgkej9EgOrOXCbF6DA/fLrvEtYdS8KQTk3rtL4V3Uurj+nOzs5YsmQJ0tLS8M0336Bly5bIzs5GQkICAODFF19EdHQ0IiMjGXAM1NpjlYMth3RqCgdr8wecTUREjWFEVx9YmktxMU2Bk4k5D76AalSvFY+trKwwZMgQDBkypKHrIRElZRfh4KXK9UfG9PATtxgiIhPmaG2B5zo2xeaTyVh7NBHd/Z3FLskgaT3gws7ODo888gjeeustbNy4ERcuXOCu40Zi/fEkCALQr5UrWrjail0OEZFJG3fnw+b++Awk39ZuM1mqpHXI+fTTT9GyZUv89ddfGDduHNq1awc7Ozv06NEDr7/+OtatW4ezZ882Rq3UiBSl5fj5dAqAf0f2ExGReFq62+GRQFcIArAhMknscgyS1t1VkydP1vx3SUkJbGxs8PrrryMnJwcnTpzAmjVroFQq67SEPemPn06loEipQks3W/TixnBERHphXE8//H0lC9tOpeCtxwNhK6/XKBOT9VB3y8rKCgAwYsQIzdTFiooKXLx48eErI51RqwX8eOIGAGBsTz+O4ici0hN9WrrC39UG17OKsDPmJkaF+YldkkFp8EVQzMzMuFaDgfnnajZu3C6GndwMz3a4d6diIiISh1QqwajuvgCAH0/c4BhYLXGlN8KPkZWtOM93agprCzaFEhHpk+c6NoWVuQxXMgoRxenkWtE65EyYMAErVqzAqVOnUFZWBgDs3jBgN3OL8delDADAS3c+LRARkf5wsDLH4A5eAKAZWkB1o/XH9oSEBPz8888oKCiAmVnl5R9++CH69u2Ljh07on379rC2tm7wQqlxbIlKhloAegY4I8CN08aJiPTRS919sSUqBfvOpyOzoJS7k9eR1iHnyJEjACrDTnR0NGJiYhATE4O5c+ciLy8PMpkMgYGBuHDhQoMXSw2rrEKFrVGV08ZHsRWHiEhvtfZyQCffJoi+kYutUSl447GWYpdkEOo9AKNly5Zo2bIlhg8frjmWmJiI06dP48yZMw1SHDWufefTcbtICXd7OcKDuds4EZE+G9XdF9E3crH5ZDIm923BDZTroEHvUPPmzTF06FAsWLCgIR+WGknVgOORXbnbOBGRvuvf1gNONhZIV5TiQHym2OUYBL6zmaiLtxQ4fSMXZlIJhnf1EbscIiJ6ALmZDMO6VP693sgByHXCkGOiNp6s/AcS0doD7vYcwEZEZAhGdm0GiQQ4ejUb17IKxS5H7zHkmCBFaTl2nUkFAIwK44BjIiJD4eNkjceC3ACwNacuGHJM0K+xt1CsVCHAzRbdmjuJXQ4REWnhxTuzYXfEpKK0nPtE1oYhxwRtjUoGAIzo2owLORIRGZhHWrrC29EK+SXl2Hc+Xexy9BpDjok5n5qPC7cUsJBJuU8VEZEBkkklGNq5KQBg66lkkavRbww5JmbLnVaciDaVUxGJiMjwvNDZB1IJcOJ6DhKzi8QuR28x5JiQYmUFfou9BQAY0YXTxomIDJWXoxX6BLoCYGtObRhyTMjuc2koKKtAMydrdPd3FrscIiJ6CMO6NAMA/BJ9E8oKtcjV6CeGHBOy9VTlPlXDuvhAKuWAYyIiQ/ZYsBtcbOXILlTir0sZYpejlxhyTERCRgGib+RWDljr1FTscoiI6CGZy6QYcufv+ZY7my1TdQw5JqKqFefRIDe4cYVjIiKjMPzO+Mq/E7JwM7dY5Gr0j16FnOXLl8PPzw+Wlpbo1q0boqKi6nTd1q1bIZFIMHjw4MYt0ECVVaiwI+YmAGAE96kiIjIafi42CPN3hiAAP5++KXY5ekdvQs62bdswffp0zJs3DzExMQgNDUVERAQyM2vfaTUpKQlvv/02evfuraNKDc+fFzKQW1wOD3tLPNLSVexyiIioAVVtsvzz6RSo1ILI1egXvQk5S5YswcSJEzFu3DiEhIRg5cqVsLa2xtq1a+97jUqlwosvvogPP/wQ/v7+OqzWsGy701X1QuemMJPpza+ciIgaQERrDzhYmeNWfin+TsgSuxy9ohfveEqlEtHR0QgPD9cck0qlCA8PR2Rk5H2v++ijj+Dm5obx48frokyDdCuvBMeuZQMAhnZmVxURkbGxNJdpVrDfHs0uq7vpRcjJzs6GSqWCu7t7tePu7u5IT695X46jR4/i+++/x+rVq+v0M8rKyqBQKKp9mYKdZ1IhCEC35k7wcbIWuxwiImoEz3esnGW1/2IG8ovLRa5Gf+hFyNFWQUEBRo0ahdWrV8PFxaVO1yxcuBAODg6aLx8f42/VEAQBv9xJ9c9z2jgRkdFq422PQHdbKCvU+F/cLbHL0Rt6EXJcXFwgk8mQkVF9MaOMjAx4eHjcc/61a9eQlJSEQYMGwczMDGZmZvjhhx/w22+/wczMDNeuXbvnmpkzZyI/P1/zlZJi/GsKnEnJw/XsIliZyzCgrafY5RARUSORSCSa1pxf2GWloRchx8LCAp06dcLBgwc1x9RqNQ4ePIiwsLB7zg8KCkJcXBxiY2M1X08//TT69euH2NjYGltp5HI57O3tq30Zu6oX+pNtPGArNxO5GiIiakzPdvCGVALEJOfhelah2OXoBb1555s+fTrGjBmDzp07o2vXrli6dCmKioowbtw4AMDo0aPh7e2NhQsXwtLSEm3atKl2vaOjIwDcc9xUlZar8PvZyibLqnRPRETGy83eEo8EuuLw5SzsiEnF2xGtxC5JdHoTcoYNG4asrCzMnTsX6enpaN++Pfbt26cZjJycnAypVC8angzCgfgMKEor4OVgibAW3IyTiMgUPN+xKQ5fzsLOM6mY/nigye9TKBEEwSRXDlIoFHBwcEB+fr5Rdl2NWxeFQ5ezMKVfC7wTESR2OUREpAOl5Sp0+eQACkorsHlCN/QIqNvkHEOizfs3m0aMUGZBKf5OqFwb5zl2VRERmQxLcxkGhXoBALbHcAAyQ44R+vXMLajUAjo0c0QLV1uxyyEiIh2qGoe5Ny4dhWUVIlcjLoYcIyMIgmbFSw44JiIyPR2bOaK5iw1KylXYG5cmdjmiYsgxMhduKXA5owAWZlIMaucldjlERKRjlWvmVG7z8IuJd1kx5BiZX2NTAQDhwW5wsDYXuRoiIhLDs3da8k8m5iAtv0TkasTDkGNEVGoBv91ZG+eZ9t4iV0NERGLxdrRCVz8nCALwv7Om22XFkGNEohJzkKEog52lGfq2chW7HCIiEtGg9pVDFn49mypyJeJhyDEiVa04/dt4QG4mE7kaIiIS01NtPWEmleB8qgLXTHSbB4YcI6GsUGPPnVH07KoiIiInGwv0blm5GOBvsaa5MzlDjpH4+0oW8kvK4WonR3d/buNARETA03e6rH47ewumuMEBQ46RqOqqGtjOEzIT36uEiIgqPR7iAUtzKRKzi3A+VSF2OTrHkGMEisoqsP9iBgB2VRER0b9s5WYID67c6LpqiRFTwpBjBA7EZ6CkXAVfZ2uENnUQuxwiItIjT9/Zy+r3c5Vb/pgShhwjUDWg7JlQL0gk7KoiIqJ/9WnlCntLM2QoyhCVmCN2OTrFkGPgcouUOHIlC8C/A8yIiIiqyM1k6N/GEwDwm4mtmcOQY+D2nE9DhVpAiKc9AtzsxC6HiIj00DN3PgTviUuHskItcjW6w5Bj4DRdVWzFISKi++jm7ww3OznyS8o1rf+mgCHHgGUqShGVVNm/OjCUIYeIiGomk0rwVLvKLquqhWNNAUOOAfvjQjoEAWjv4whvRyuxyyEiIj32VNvKkHPgYgbKKlQiV6MbDDkGbE9cOgBgQFsPkSshIiJ917FZE7jZyVFQVoFjV7PFLkcnGHIMVHZhGU4m3gYAzah5IiKi+5FKJejfpvJDcdWHZGPHkGOg/ryQAbUAtPV2gI+TtdjlEBGRAeh/p8vqzwumMcuKIcdA7T1fOXCsP7uqiIiojrr4OcHF1gKK0gpEXr8tdjmNjiHHAOUWKXH8GruqiIhIOzKpBBGtKz8c7zWBWVYMOQZo/8UMqNQCgj3t0dzFRuxyiIjIgAy402X1x4V0VKiMu8uKIccA7bnTVTWgDbuqiIhIO92aO6GJtTlyi8tx0sj3smLIMTD5xeWaqX9VA8iIiIjqykwm1XRZGfvCgAw5BuZAfAbKVQIC3W0R4GYrdjlERGSA+t/VZaVSCyJX03gYcgyMZlYVBxwTEVE99WjhDAcrc2QXKnEqyXi7rBhyDEhBaTn+vlLZVTWAXVVERFRP5jIpHg9xB2Dcs6wYcgzIX5cyoVSp4e9qg0B3dlUREVH9VW0JtPd8OtRG2mXFkGNA/ryYAQB4srUHJBKJyNUQEZEh6xngAlu5GTILynAuNV/schoFQ46BUFaoceRyFgBomhiJiIjqS24mQ59AVwCVO5MbI4YcA3Ey8TYKyyrgYitHaFNHscshIiIjEB7iBqBy5q4xYsgxEFUpOzzYDVIpu6qIiOjh9WvlBplUgkvpBUjJKRa7nAbHkGMABEHAgfhMAEB4MLuqiIioYThaW6CzbxMAxtmaw5BjAOLTCpCaVwJLcyl6BriIXQ4RERmRqnGeDDkkiv13uqp6BbjCykImcjVERGRMqkLOyes5yC8pF7mahsWQYwCq0vUTnFVFREQNzNfZBi3dbFGhFnD4cqbY5TQohhw9l5ZfgrjUfEgkQL8gN7HLISIiIxSu6bJiyCEdOnjnBdfBxxGudnKRqyEiImNUNanl8OVMKCvUIlfTcBhy9FxVV1U4u6qIiKiRdPBxhIutBQpKK4xqw06GHD1WVFaB41dvAwAe59RxIiJqJFKpBI8FVb7P7Dei1Y8ZcvTYPwlZUKrU8HW2RoAbN+QkIqLGE37XVHJBMI4NOxly9Nj+i/8uAMgNOYmIqDH1CnCBpbkUN3NLcDmjQOxyGgRDjp5SqQX8damyyZAbchIRUWOzspChV0Dlhp37LxhHlxVDjp46dzMPucXlsLM00yy5TURE1JgeC65cquTIlSyRK2kYDDl6quoF1rulC8xk/DUREVHjeySwsiUnJjkX+cWGv/ox3z31VFXI6XPnBUdERNTYvB2t0NLNFmoBOHYtW+xyHhpDjh7KLVLibEoegH9TNRERkS5Ufbg+ctnwu6wYcvTQ0avZUAtAK3c7eDpYiV0OERGZkKoP10euZBn8VHKGHD2k6apqxVYcIiLSra7NnWBpLkW6ohRXMgrFLuehMOToGUEQOB6HiIhEY2kuQ3d/ZwDAkSuGvWEnQ46eiU8rQFZBGazMZejsx6njRESke33u6rIyZAw5eqbqBdWjhTPkZjKRqyEiIlNUFXJOJeaiqKxC5GrqjyFHz1Q1DXI8DhERiaW5iw18nKygVKlx4vptscupN70KOcuXL4efnx8sLS3RrVs3REVF3ffc1atXo3fv3mjSpAmaNGmC8PDwWs83BIVlFYi+kQuA43GIiEg8EolE8z70twF3WelNyNm2bRumT5+OefPmISYmBqGhoYiIiEBmZs2Dng4fPowRI0bg0KFDiIyMhI+PD5544gmkpqbquPKGE3ntNspVAvycreHrbCN2OUREZML6BBr+Fg96E3KWLFmCiRMnYty4cQgJCcHKlSthbW2NtWvX1nj+pk2bMHnyZLRv3x5BQUFYs2YN1Go1Dh48qOPKG46mq4qtOEREJLKwFs4wl0mQdLsYSdlFYpdTL3oRcpRKJaKjoxEeHq45JpVKER4ejsjIyDo9RnFxMcrLy+Hk5NRYZTYqQRBw+DLXxyEiIv1gKzdDZ9/K99S/EwyzNUcvQk52djZUKhXc3d2rHXd3d0d6enqdHuPdd9+Fl5dXtaB0t7KyMigUimpf+iQxuwg3c0tgIZNq1icgIiISU9WHbkPd4kEvQs7DWrRoEbZu3YqdO3fC0tKyxnMWLlwIBwcHzZePj4+Oq6xdVZ9n1+ZOsLYwE7kaIiKif4dPHL92G2UVKpGr0Z5ehBwXFxfIZDJkZGRUO56RkQEPD49ar/3888+xaNEi/Pnnn2jXrt19z5s5cyby8/M1XykpKQ1Se0M5drVyt9deLV1EroSIiKhSkIcdXO3kKClX4UxyntjlaE0vQo6FhQU6depUbdBw1SDisLCw+163ePFizJ8/H/v27UPnzp1r/RlyuRz29vbVvvSFSi3gZGIOgMpFAImIiPSBRCLRDKGIvGZ46+XoRcgBgOnTp2P16tXYsGED4uPjMWnSJBQVFWHcuHEAgNGjR2PmzJma8z/99FPMmTMHa9euhZ+fH9LT05Geno7CQsPbTOzCrXwUlFbAztIMrb0cxC6HiIhII6wq5BjgooB6M/hj2LBhyMrKwty5c5Geno727dtj3759msHIycnJkEr/zWQrVqyAUqnEkCFDqj3OvHnz8MEHH+iy9IdWlY67NXeCTCoRuRoiIqJ/hd3pYYhNzkNpuQqW5oaz5ZDehBwAmDp1KqZOnVrj/zt8+HC175OSkhq/IB2pSsecVUVERPrGz9kaHvaWSFeUIvpGLnoGGM7YUb3prjJV5So1Tt0ZjxPG8ThERKRnJBKJ5v3J0MblMOSILC41H0VKFRytzRHsoT+DoYmIiKoY6rgchhyRVaXi7s2dIeV4HCIi0kNVLTlnU/JQVFYhcjV1x5AjsqqQw64qIiLSVz5O1mjaxAoVagGnknLELqfOGHJEVFahwukbHI9DRET6zxC7rBhyRHQ2JR+l5Wq42FqgpZut2OUQERHdV9WH8RMGNPiYIUdEmvVx/J0hkXA8DhER6a+qkBOXmg9FabnI1dQNQ46IIq9X7lcVxvVxiIhIz3k6WMHP2RpqAZqlT/QdQ45ISstViLmz2RnH4xARkSEwtPVyGHJEEpOcC2WFGm52cvi72IhdDhER0QN1N7DBxww5Ijlx19RxjschIiJDUDW84mKaAnnFSpGreTCGHJFUpWCOxyEiIkPhZm+JFq42EATgpAGMy2HIEUGJUoXYlDwAHI9DRESGxZDG5TDkiODszTyUqwR42FuimZO12OUQERHVWbfmlSEn+kauyJU8GEOOCGKSK18YHX0dOR6HiIgMSkffJgCA+DQFSpQqkaupHUOOCGJu5AEAOjZrIm4hREREWvJysIS7vRwVagHnbuaJXU6tGHJ0TBAExKZUtuR0aOYobjFERERakkgk6OBT+SH9zJ3xpfqKIUfHUnJKkF2ohLlMgtZeDmKXQ0REpLWOvo4AgBg9H5fDkKNjVeNxWns5wNJcJnI1RERE2qsabhGTnAdBEESu5v4YcnRMM+iY43GIiMhAtfF2gLlMguzCMtzMLRG7nPtiyNGxM3f2q+J4HCIiMlSW5jKE3BlyUfXhXR8x5OhQiVKF+DQFgH+n4BERERmiDj6OAP798K6PGHJ06NzNPFSoBbjby+HlYCl2OURERPVW9WGdLTkEoHKAFlA5HoeLABIRkSHreGfYxcVbCpSW6+eigAw5OnQmmevjEBGRcfB2tIKbXeWigHGp+WKXUyOGHB0RBKFaSw4REZEhk0gk/04l19P1chhydORmbgmyC8tgLpOgjTcXASQiIsOnWRRQT8flMOToSNULIISLABIRkZHooOeLAjLk6EhVU17VlDsiIiJD19bbAWZSCbIK9HNRQIYcHanaxIzr4xARkbGwNJehtZc9AP3crJMhRwdKy1W4eOvOIoCcWUVEREakgx4PPmbI0YFzN/NRoRbgZieHt6OV2OUQERE1mKoeijN6OPiYIUcHYu5aH4eLABIRkTGpGmt6QQ8XBWTI0YFzN/MA/NukR0REZCyaNrGC651FAS/cGZqhLxhydOByegEAINjTXuRKiIiIGpZEIkGQhx0AICGjQORqqmPIaWTKCjWSbhcDAALdbUWuhoiIqOEFut8JOZmFIldSHUNOI0vMLoJKLcBObgYPe+48TkRExqelW+WH+CtsyTEtVb/wAHdbDjomIiKj1LKqJSeDLTkmparpLtDNTuRKiIiIGkfLO8Mx0hWlUJSWi1zNvxhyGlnVIKyWHI9DRERGyt7SXDMkQ59acxhyGllVS05VUx4REZExqvowfzVTf8blMOQ0ImWFGknZRQA4s4qIiIxbyzvDMq6wJcc0JGYXoYIzq4iIyARUfZjXpxlWDDmNKCGTM6uIiMg0/NtdxZYck1DVZFe1fgAREZGxCrjTXZWWrz8zrBhyGlHV4KtADjomIiIj52D17wwrfWnNYchpRJqWHIYcIiIyAVVdVvqyhxVDTiO5e2YVu6uIiMgU6NsMK4acRpJ0u3Jmla3cDJ4OnFlFRETGT9OSw+4q46bZs8qNM6uIiMg0BLK7yjRULWvNRQCJiMhU3D3DqkAPZlgx5DSSBM6sIiIiE+NgZQ53ezkA/eiyYshpJFWDrgI46JiIiExI1Yd7feiyYshpBNX3rGJLDhERmY6qD/f6sBs5Q04j4MwqIiIyVVUf7q+wu8o4JdzVVcWZVUREZEqq1oa7yu4q41Q1fZwzq4iIyNRULQh4Sw9mWOlVyFm+fDn8/PxgaWmJbt26ISoqqtbzf/75ZwQFBcHS0hJt27bFnj17dFRp7apmVlX9oomIiEyFg7U53Oz0Y4aV3oScbdu2Yfr06Zg3bx5iYmIQGhqKiIgIZGZm1nj+8ePHMWLECIwfPx5nzpzB4MGDMXjwYJw/f17Hld8rQbNnFVtyiIjI9FSNy7kq8uBjvQk5S5YswcSJEzFu3DiEhIRg5cqVsLa2xtq1a2s8f9myZXjyySfxzjvvIDg4GPPnz0fHjh3xzTff6Ljy6pQVaiRyZhUREZmwqg/5V0Qel6MXIUepVCI6Ohrh4eGaY1KpFOHh4YiMjKzxmsjIyGrnA0BERMR9zy8rK4NCoaj21RhucGYVERGZuKrhGuyuApCdnQ2VSgV3d/dqx93d3ZGenl7jNenp6Vqdv3DhQjg4OGi+fHx8Gqb4/8gqLIOjtTlnVhERkckKdLeFVFLZuyEmM1F/ug7NnDkT06dP13yvUCgaJej0aOGCM3MeR7FS1eCPTUREZAja+zji4kdPwtJcJmodehFyXFxcIJPJkJGRUe14RkYGPDw8arzGw8NDq/PlcjnkcnnDFPwAEokENnK9uLVEREQ6ZyaTwkzcfANAT7qrLCws0KlTJxw8eFBzTK1W4+DBgwgLC6vxmrCwsGrnA8D+/fvvez4RERGZFr1pbpg+fTrGjBmDzp07o2vXrli6dCmKioowbtw4AMDo0aPh7e2NhQsXAgCmTZuGPn364IsvvsBTTz2FrVu34vTp01i1apWYT4OIiIj0hN6EnGHDhiErKwtz585Feno62rdvj3379mkGFycnJ0Mq/bfhqUePHti8eTNmz56N999/Hy1btsSuXbvQpk0bsZ4CERER6RGJIAiC2EWIQaFQwMHBAfn5+bC3txe7HCIiIqoDbd6/9WJMDhEREVFDY8ghIiIio8SQQ0REREaJIYeIiIiMEkMOERERGSWGHCIiIjJKDDlERERklBhyiIiIyCgx5BAREZFR0pttHXStaqFnhUIhciVERERUV1Xv23XZsMFkQ05BQQEAwMfHR+RKiIiISFsFBQVwcHCo9RyT3btKrVbj1q1bsLOzg0QiadDHVigU8PHxQUpKCvfFakS8z7rB+6wbvM+6w3utG411nwVBQEFBAby8vKpt3F0Tk23JkUqlaNq0aaP+DHt7e/4D0gHeZ93gfdYN3mfd4b3Wjca4zw9qwanCgcdERERklBhyiIiIyCgx5DQCuVyOefPmQS6Xi12KUeN91g3eZ93gfdYd3mvd0If7bLIDj4mIiMi4sSWHiIiIjBJDDhERERklhhwiIiIySgw5REREZJQYcupp+fLl8PPzg6WlJbp164aoqKhaz//5558RFBQES0tLtG3bFnv27NFRpYZNm/u8evVq9O7dG02aNEGTJk0QHh7+wN8LVdL29Vxl69atkEgkGDx4cOMWaCS0vc95eXmYMmUKPD09IZfLERgYyL8ddaDtfV66dClatWoFKysr+Pj44K233kJpaamOqjVMf//9NwYNGgQvLy9IJBLs2rXrgdccPnwYHTt2hFwuR0BAANavX9/odUIgrW3dulWwsLAQ1q5dK1y4cEGYOHGi4OjoKGRkZNR4/rFjxwSZTCYsXrxYuHjxojB79mzB3NxciIuL03HlhkXb+zxy5Ehh+fLlwpkzZ4T4+Hhh7NixgoODg3Dz5k0dV25YtL3PVRITEwVvb2+hd+/ewjPPPKObYg2Ytve5rKxM6Ny5szBgwADh6NGjQmJionD48GEhNjZWx5UbFm3v86ZNmwS5XC5s2rRJSExMFP744w/B09NTeOutt3RcuWHZs2ePMGvWLGHHjh0CAGHnzp21nn/9+nXB2tpamD59unDx4kXh66+/FmQymbBv375GrZMhpx66du0qTJkyRfO9SqUSvLy8hIULF9Z4/gsvvCA89dRT1Y5169ZNePXVVxu1TkOn7X3+r4qKCsHOzk7YsGFDY5VoFOpznysqKoQePXoIa9asEcaMGcOQUwfa3ucVK1YI/v7+glKp1FWJRkHb+zxlyhTh0UcfrXZs+vTpQs+ePRu1TmNSl5AzY8YMoXXr1tWODRs2TIiIiGjEygSB3VVaUiqViI6ORnh4uOaYVCpFeHg4IiMja7wmMjKy2vkAEBERcd/zqX73+b+Ki4tRXl4OJyenxirT4NX3Pn/00Udwc3PD+PHjdVGmwavPff7tt98QFhaGKVOmwN3dHW3atMGCBQugUql0VbbBqc997tGjB6KjozVdWtevX8eePXswYMAAndRsKsR6HzTZDTrrKzs7GyqVCu7u7tWOu7u749KlSzVek56eXuP56enpjVanoavPff6vd999F15eXvf8w6J/1ec+Hz16FN9//z1iY2N1UKFxqM99vn79Ov766y+8+OKL2LNnD65evYrJkyejvLwc8+bN00XZBqc+93nkyJHIzs5Gr169IAgCKioq8Nprr+H999/XRckm437vgwqFAiUlJbCysmqUn8uWHDJKixYtwtatW7Fz505YWlqKXY7RKCgowKhRo7B69Wq4uLiIXY5RU6vVcHNzw6pVq9CpUycMGzYMs2bNwsqVK8UuzagcPnwYCxYswLfffouYmBjs2LEDu3fvxvz588UujRoAW3K05OLiAplMhoyMjGrHMzIy4OHhUeM1Hh4eWp1P9bvPVT7//HMsWrQIBw4cQLt27RqzTIOn7X2+du0akpKSMGjQIM0xtVoNADAzM8Ply5fRokWLxi3aANXn9ezp6Qlzc3PIZDLNseDgYKSnp0OpVMLCwqJRazZE9bnPc+bMwahRozBhwgQAQNu2bVFUVIRXXnkFs2bNglTKtoCGcL/3QXt7+0ZrxQHYkqM1CwsLdOrUCQcPHtQcU6vVOHjwIMLCwmq8JiwsrNr5ALB///77nk/1u88AsHjxYsyfPx/79u1D586ddVGqQdP2PgcFBSEuLg6xsbGar6effhr9+vVDbGwsfHx8dFm+wajP67lnz564evWqJkQCwJUrV+Dp6cmAcx/1uc/FxcX3BJmqYClwa8cGI9r7YKMOazZSW7duFeRyubB+/Xrh4sWLwiuvvCI4OjoK6enpgiAIwqhRo4T33ntPc/6xY8cEMzMz4fPPPxfi4+OFefPmcQp5HWh7nxctWiRYWFgI27dvF9LS0jRfBQUFYj0Fg6Dtff4vzq6qG23vc3JysmBnZydMnTpVuHz5svC///1PcHNzEz7++GOxnoJB0PY+z5s3T7CzsxO2bNkiXL9+Xfjzzz+FFi1aCC+88IJYT8EgFBQUCGfOnBHOnDkjABCWLFkinDlzRrhx44YgCILw3nvvCaNGjdKcXzWF/J133hHi4+OF5cuXcwq5Pvv666+FZs2aCRYWFkLXrl2FEydOaP5fnz59hDFjxlQ7/6effhICAwMFCwsLoXXr1sLu3bt1XLFh0uY++/r6CgDu+Zo3b57uCzcw2r6e78aQU3fa3ufjx48L3bp1E+RyueDv7y988sknQkVFhY6rNjza3Ofy8nLhgw8+EFq0aCFYWloKPj4+wuTJk4Xc3FzdF25ADh06VOPf26p7O2bMGKFPnz73XNO+fXvBwsJC8Pf3F9atW9fodUoEge1xREREZHw4JoeIiIiMEkMOERERGSWGHCIiIjJKDDlERERklBhyiIiIyCgx5BAREZFRYsghIiIio8SQQ0REREaJIYeIiIiMEkMOERmN9PR0SCQSLFu2DB06dIClpSVat26No0ePil0aEYmAIYeIjEZsbCwAYO3atVi6dCliY2PRrFkzvPjii9V28yYi02AmdgFERA3l7NmzMDc3x6+//go/Pz8AwMcff4zOnTsjNTUVPj4+4hZIRDrFlhwiMhqxsbF47rnnNAEHAOzt7cUriIhExZBDREYjNjYW7du3r3YsMjISLi4u8Pb2FqcoIhINQw4RGYWSkhIkJCRApVJpjqnVaixduhRjxoyBVMo/d0Smhv/qicgoxMXFQSKRYOPGjYiMjER8fDyGDRuGvLw8zJ49W+zyiEgEDDlEZBRiY2MRFBSE999/H88//zw6d+4MlUqFI0eOwNHRUezyiEgEEkEQBLGLICJ6WFOmTEFubi42b94sdilEpCfYkkNERiE2Nhbt2rUTuwwi0iMMOURk8ARBQFxcHEMOEVXD7ioiIiIySmzJISIiIqPEkENERERGiSGHiIiIjBJDDhERERklhhwiIiIySgw5REREZJQYcoiIiMgoMeQQERGRUWLIISIiIqPEkENERERGiSGHiIiIjNL/A8RZeKA1i+NFAAAAAElFTkSuQmCC",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# entropy for Bernoulli witt parameter param, with base 2\n",
    "def entropy(param,pmf=Bernoulli,S=SampleSpace):\n",
    "    s=0\n",
    "    for outcome in [S.TAILS,S.HEADS]:\n",
    "        x=X(outcome)\n",
    "        prob=pmf(x,param)\n",
    "        if prob!=0:\n",
    "            s-=prob*np.log2(prob)\n",
    "    return s\n",
    "\n",
    "#displaying the entropy of Bernoulli for p from zero to one\n",
    "ps=np.linspace(0,1,100)\n",
    "hs=[entropy(p) for p in ps]\n",
    "plt.plot(ps,hs)\n",
    "plt.title('Entropy of Bernoulli in terms of $p$')\n",
    "plt.xlabel('$p$')\n",
    "plt.ylabel('$H(X)$')\n",
    "plt.text(.25,.5,'Entropy is maximum for $p=.5$')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "74ae0061",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.11.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}