10_MonteCarlo.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'%.4f'"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "from scipy import stats\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "%matplotlib inline\n",
    "%precision 4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Psedorandom number generators\n",
    "\n",
    "While psuedorandom numbers are generated by a deterministic algorithm, we can mostly treat them as if they were true random numbers and we will drop the “pseudo” prefix. Fundamentally, the algorithm generates random integers which are then normalized to give a floating point number from the standard uniform distribution. Random numbers from other distributions are in turn generated using these uniform random deviates, see later.\n",
    "\n",
    "## Linear congruential generators (LCG)\n",
    "\n",
    "[LCG](https://en.wikipedia.org/wiki/Linear_congruential_generator) is among the simplest and most popular pseudo random number generators. It relies on the recursive and fully deterministic relation:\n",
    "\n",
    "$$\n",
    "z_{i+1}=(a z_i+c)\\mod{m}\n",
    "$$\n",
    "\n",
    "Hull-Dobell Theorem: The LCG will have a period $m$ for all seeds if and only if\n",
    "\n",
    "* $c$ and $m$ are relatively prime,\n",
    "* $a−1$ is divisible by all prime factors of $m$\n",
    "* $a−1$ is a multiple of 4 if $m$ is a multiple of $4$.\n",
    "\n",
    "The number $z_0$ is called the *seed*, and setting it allows us to have a reproducible sequence of (pseudo) random numbers. The LCG is typically coded to return $z/m$, a floating point number in $(0, 1)$. Obviosuly, this can be easily scaled to any other range $(a,b)$.\n",
    "Note that $z \\le m-1$ always holds, the yielded $z/m$ result is thus on purpose strictly smaller than 1. \n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "def lcg(m=2**32, a=1103515245, c=12345):\n",
    "    lcg.current = (a*lcg.current + c) % m\n",
    "    return lcg.current/m\n",
    "\n",
    "# setting the seed\n",
    "lcg.current = 1234"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0.0543,\n",
       " 0.0299,\n",
       " 0.0200,\n",
       " 0.1942,\n",
       " 0.8637,\n",
       " 0.9932,\n",
       " 0.3973,\n",
       " 0.7836,\n",
       " 0.6981,\n",
       " 0.7172]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[lcg() for i in range(10)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.5010449237937108\n",
      "0.288036238746601 0.2886751345948129\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f7a981025e0>]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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vlXrfC0dPI0VNkzBMHr2jtysSycpsx7K6ArqIXCsiIyJyRES2VnjNx0VkWEQOicj/sFrKvKDzuKy7cFFbpN1UqpiBafMzXJ88urm3i56PLOJsBJKV2Y4nNfsuRCQOfBv4PeAE8LKI7DXG/LToNUngr4BrjTHHROTDdouZE/SwxaFjY9pHo5TP7u9bxbqdzwZdjJawPV+wnhb6FcARY8zrxpgJ4HHg5pLX/AGwxxhzLFdI85bdYuYEPWxxPDNZ84yqyZJUsQ4dqtSwZVuf1m7VJtVztKWA40W/n8g/Vux3gE4R+QcROSAiny63IRG5U0QGRWTw1KlTDRd2y/oVgd8sqXY529nh0X/rahL6JXaW7Zw5Gy6/IPBjVrUv20Om64k85Y7H0rA2B7gcuAFYD/wbEfmdGX9kzMPGmB5jTM/ixYsbLizA/DYMlp0dHrs2rmHoa9fQ151ifshGHESJ7YVTHtl/THvpVEXbb1ppdXv1jP87ASwt+v0C4GSZ1/zKGHMWOCsi/xNYDfzMSil5f4RLO06DPzOemRp+1NedYmy8Pce1q/cJejtEBWtzb1cg+dBfBi4SkeUiMhe4Hdhb8pq/A/65iMwRkQ5gLfCazYIGPcKlltGxNNv2HGRgaDTwvv5SbXhREzgN5iqMan7VjTHngC8D+8gF6SeMMYdE5C4RuSv/mteAHwKvAi8B3zHG/MRmQYMe4VKPdCbL3buHOfveObx4+/SctuFFjVKR98j+Y9bT59Y15dIY8wzwTMljD5X83g/02yvadEuSCWfufLdrKgGlVHvp3zditdvFmYtx20lslFIqaJFNn+vH+ntKKRWkhQEMW2wbUcjtEEZRmmulx6hqhO2p/04F9G98chXxKEWHkHA9t0gjJs7pHWhVP9tDnJ0K6H3dKR64dXXQxVBNaOQ0HBOYN8epQ3NKJkpnLzVrkV6xCGDwzdNBF0E1wUDdQzmNgfe0patCoNYRf9XFzc2Yr8S5gP7Yi8drv0i1pf5b6lttStu4qhXm+Nx9m0omeHDjGlJVWuHPHW48p1U1zgV027k2VOt8/QeH2H7TSnZtXKN55VXgzvncPbZl/Qr6ulNVh1zbHrZoby23FrA9q0q11pnxDHfvHg66GEq1xN27h7l79zCpZIIFc+NlF+yI9LBF2+vvRV2HF2urFAVKhdHoWLri6ktnJ85Zbag6FdBdyOfiAi8ubO7tonPBPDJZ7cJSqlGdHXZa1pms3YWinQro7ZbFsF4JL2Z94YTZyGQNj+w/5kxuHKXazRmL48dtNlSdCui2h/i0Sjozyaa1S0nowhdKqRI2G6pOBXTbQ3xa6dH9x9o6n7tSrebFYNfGNUEXI1AJL2418aBTAd3lPnTtqVZqug/M9yKddE8EdmxYFc30ueBuH7pSaiab/dAu8mNKjVMBfcv6FdoPrVRIxEUiP7fE9lBspwJ6X3eKHRtWkdBFMpVyXtYYvvrfDgZdjEBFdoGLgsE3T5PWRTKVCoVKE26iIvLZFh/ZfyzoIiillBW2l9Z0KqDfOxDtyzOl1Eyu9sD6sViPU7tCU+cqpUq52gObnbQ77R8cC+iaOlep9tA+iSzcFun0uTGJ1vqUSrWbXRvXTE2EuXfgYKjvacVFfG9EJi0l+SpwpoU+MDSqwVypgPXvG5kaO/69wRMBl8ZfWWN8vxKxfb5wJqAHnQvdmR2llI9Gx9Lcs3uYS+/7YSTWffW7Dfl22u5sWWfiVNB5XMJ/6CpVHwP8+r1ojx+3JbLj0DWPi1IqTOIxie449C3rV+D5vEq3UlHX4eqgbgdN+nDD1ZlPr687Rf+tq0laXlRVqUaFuVkxnpl0sn4L5rqXtM8Y2PL9V6K7pmhfd4rh+65h18Y12lpXgQn7YCsX6+dqTphM1vD1Hxyytr26ArqIXCsiIyJyRES2VnndPxWRrIjcYq2EZfR1p7hieaefb6GUUi1hMy98zYAuInHg28B1wCXAJhG5pMLr/hzYZ610FQwMjfL80dN+v41SSjmlnhb6FcARY8zrxpgJ4HHg5jKv+yPgSeAti+WbYWBolG17opWkK5nwiIt2MSkVRjbvC9YT0FNAcVasE/nHpohICvgk8FC1DYnInSIyKCKDp041t+Bz/76RyC22PJbOYDSPjVKh48WE7TettLa9egJ6uaZhaXTZBXzFGFM10hpjHjbG9BhjehYvXlxnEaebzQSj0hFZc+PutHp1YpNS4RIT6L91tdVFoutJznUCWFr0+wXAyZLX9ACPS65b4EPA9SJyzhgzYKOQxZYkE4w2GdRL02xOZLXVq9wRjwlZTWgUGpMGq8Ec6muhvwxcJCLLRWQucDuwt/gFxpjlxphlxphlwPeBL/oRzEEXinaJjiy1Q4DNvV08oPMwVA01W+jGmHMi8mVyo1fiwHeNMYdE5K7881X7zf0wb04sVP3oYU0LHMY6BcGQW3rxkf3H6LScblWFS1350I0xzwDPlDxWNpAbYz4z+2KVVxjhEqZgDoUTlPaSq9psjlkOI5e6pfw4OTs1U7SeES4JL+bc1GUN5krNngDnzZvjzBqjN1x6vvVtOlL1nFojXLyYILg5dVkpNTuG3BBfV9pHu186bjWPCzgW0Kul0E0mPJBcciGllGp3mUnD9r328riAYwG93AiXhBdnc28Xv3n3HBkdhqiUcsiY5RWLnFokujBms3/fCCfH0ixJJrjq4sU8eWDU98VclVKq3TkV0CEX1IsH46/b+WzoRr0opaLBdh53p7pcSg0MjTY9axTQhFdKqUB5cbsh2NmAbiPronbTKKWCZLsP3dmAHsWsi0qp8InsEnSQq/y6nc/OqqtFKaXaRf++EWvbcuqmaFin/iuloms2KcFLOdVCj0o3y0UfXhB0EZRSLVJtwmSjnAroNs9k7cqLwZG3zgZdDFWnhCuJQ1Tbuuri5hb7Kcepo9HmmaxdZSY1F41LNLGamq3nDje3HGc5TgX0LetXEG/xqgk6Ul1Vo3MZ1GxFtg998M3TLc11LMAdvV0tez/lHp3L4L+wnzIj24f+2IvHW/p+Bri/b1VL31Mp9b7NvV2hjugJL86W9Susbc+pgN7q1lDCi7F869MtfU+lwijeZFDe/fJx2uUiyHZvb1yEHRtWWV0o2qmA3ur+ynRmUm9QKmVBs5mt2ykltu3e3kljrAZzcCygb1q7tCXvoyurK6X85seoPacC+v19q3J9aj4RmFosQ6kwc+qLH1I2x58XOPe53t+3ipRP49E/duEiXSxDRYKro+edC1hV2Bx/XuBULhfI5XM5+54/Lejnj572ZbtKKTtcPRGV48fMd6dOeIXkXLZzCCulghMXYd2Fi4IuRsst9OFenVMBPSrJuZSKCi8uPHDbah79wpVBF6Xlzk6cs5oLHRwL6FFIzqXan073tyeTNVbzgQetkTVC/ai7UwE92aHDCVXw9Ka5XaNjaest1aCcnWisB8H2Qj1OBXT9HikVTrNdH9hVtq/2nArob+vNUKVCKar3xmxf7TkV0KOQD10pFR2BtNBF5FoRGRGRIyKytczzd4jIq/l/L4jIaqulzPNjZpVSSgWl5S10EYkD3wauAy4BNonIJSUvewP4F8aYS4E/Ax62Wso8P2ZWKaVUUGzPeq+nhX4FcMQY87oxZgJ4HLi5+AXGmBeMMWfyv+4HLrBayjwdtqhUZXObzVGrAmO716GegJ4CileWOJF/rJLPAX9f7gkRuVNEBkVk8NSpxlvb2oeuVGUTbZRqVtXHdq9DPQG93Gm/7JEjIleRC+hfKfe8MeZhY0yPMaZn8eLGz0w2V/ZQKqwE6NQ5G06w3etQT0A/ARQnIr8AOFn6IhG5FPgOcLMx5v/ZKd50fd2phmZiKRVFBnjn3XNs7u0K5eptCS8emhOW7V6HegL6y8BFIrJcROYCtwN7i18gIl3AHuBfGWN+ZrWEJcYbnImlVBRlJg2PvngslCtupTNZzoy7PyfF9nqiUEdAN8acA74M7ANeA54wxhwSkbtE5K78y74GfBD4KxEZFpFBq6UsotP/w82L6YpRtujM6vaTSiaQ/P+21xOFOvOhG2OeAZ4peeyhop8/D3zeasnKGBga5R1dTSi0vLjQf8tq+rpT3DtwkEf2Hwu6SEpZtWX9CutBvJhTC1z07xshY3ulVhWomOQW300lE1MH+8DQKE8eCEeyJtUaheOo3f3xE8MAvgV1MQFdl/X09JjBwcZ6ZpZvfTqUfYLq/S9kwouRzoRpXRqlZurs8LjvxpVNBXYROWCM6Sn3nOZyUW2h0LrSYK6i4Mx4hm17DkZ7gYst61eEchiWUip60plstBe46OtOcUdvV9DFUEopK4KYWNRWej6yCC+m7XSllPuCmFjUVnSki1LKNi+ASCjYT2fiXEC3vQafUmGWTHiBBCvXZCZzI6xaqcOHNCZOfdRhWUhWqVbY3NvFe+cm0YFD9UlnJlsaEM9OZK2PdHEqoH/9B4eCLoJSTkgmPJ47fCqya3U2q9XnPtsjXZwK6GFIyBOEZMKbWhlFbydHw1g6o92TjrA50sWpgK4al/DibL9pJc9vvZpdG9c4EdBTyYT1xXOValc2R7o4FdA1C1/j5hfd6OnfN9LyS8pmjI6lrS+eq1Q7sp1C16mAvv2mlToGvUFnxjPcs3uYewcO6iW4Um0kLmI9ha5T2RYLFb/niWHN9dwAA5qKVqk2kzXGetZFp1rokAvqC+dr14tSyn2RTs5VMJbW0S5KKfdt32t3KLZzAT0Mk4viehtAKYX9xqlTAX1gaJR7dg8HXYxZy2r/v1LKB84E9IGhUbZ87xVdsUipNiPkVuBRjbO935wJ6JplUan2I8AdvV2M6Szuptx340qr23MmoNtOBK+ULVG9JSLAgxvXcH/fKl0eskmRHbbYDgdMKpnQS0s1zebeLpIRPSYK18vrdj6rk9aaFNlhi1vWrwi0sAkvzlUXL+add88FWArVbu7vWxXp7oZte3QG8mxEdk3Rvu4UCwNqCaWSCXZsWMXTr/5S+/HVlEICsai20AFNzztLtruSnZr6H0T63ELLfPveQzqhqY0JtHwEVNYY1nz9R3pcqKbZ7kp2KqDHRVqehS+dyWoeFAcEdd2kwVw1y3amRXCoywUIfUpVP9MDL5gbj+RojLgIC3xYu1GpZhS+g4VuXNujXJxqoScTXqhbRH7W7exE9Po6Bdi0dim7XzoedFGUorPD474bV1oP4sWcaqHbWsQmlUywubcrki3WKDGgN7JV23i3Bat1OxXQbQwPE+Cqixez+6XjmkZglryY0OG19yGk69CqdpHOZK1nVyxV17dRRK4VkREROSIiW8s8LyLyzfzzr4rIZfaLaueOsAGeO3xKW22z1Nnh0X/rajoXzAu6KEo5YyydoftPf+Rb1tiaAV1E4sC3geuAS4BNInJJycuuAy7K/7sT+GvL5QRyk4sS3uxucKWSCU0jMEu7Nq5h6GvX0Ned0n2pVIPOjGfYtuegL0G9nhb6FcARY8zrxpgJ4HHg5pLX3Az8rcnZDyRF5HzLZaWvO8WODatIJRMIjfepe3Fhy/oVbZFGwAaRXBdSq3s9im/qhGVfqtYJU/qMuAipJr4D6UzW+ixRqC+gp4DiYQIn8o81+hpE5E4RGRSRwVOnTjVaViAXTJ7fejVv7LyBB29bU/ffCdB/y2r6ulPWx376yau2GoaBN3bewM//7Q3s2rhmauainxIlZ48t61dUvLmc8GJ641lN6ezwpq7umgmCjYqLIPg7HDhrTNM9B35c3dYT0Mt9J0s7oOt5DcaYh40xPcaYnsWLF9dTvqr6ulN1ne1jkssKV2hZ9nWn2NzbNev3r/h+ZR7zYkJnh4fw/iibwpVGKplg18Y17Nq4ZtrB19nh0X/L6ooHf3HruK87xQO3ra7rwEp4MXZtXDPjtV5Mqp5AYsCODZdOe6yvO8UdZUYMJbw4OzZcyoMb10yrZz2ji4r3D2DtRNXhxaY+g3q/5OXeecHcOLs2rmFzb9dU2QSYW7TvOju8aZ9xMuHN+Pxn231YTuF9K32xY03symb+piDh5fbVL3beMNVNB9UbAuUUPrt6eXHhgdtW88bOGxi+75r88W7/UjaVTEz1HDR64vDj6raecegngKVFv18AnGziNb6478aVbNtzsGJOiUpjP+/vW8Ubp97h+aOny/5dLiCtYvDN0zy6/9iMs1NhdMd4fihSTGDS5D7gwhVA/74RTo6lWZJ/rJ7xp5VeU1rHcrPMCn9beN+FCY9fv5uh+P6vFxN2bLh0xmuXlCn3woSHSG50UbU63N+3ip6PLKpY33J/U7pPC/u73jG6A0OjZffJpy5P8dzhU4yOpadmFqcqlD23jVdJVxhOVry9SvW6v29VXeUtp3ifFe/r+V5sRpm8WK5/LVOy3FW14/vegYM89uJxssYQF2HT2qUzPqerLl48Y3+V7jeYfkycnTg3rRyFhkCl70K5z7SvO1Xxu1VQ6ZgYGBqdUYenXvnl1DyOcvukrztFX3dq6m+L65usse8TXpzLuhbywtHTM47Zwv4p3X6lspX7W5vE1Jh9KSJzgJ8BnwBGgZeBPzDGHCp6zQ3Al4HrgbXAN40xV1Tbbk9PjxkcHJxd6fNKd2K9wROYdtAXlB6Is9m+Lc2WoR3K7le5bNetnfZVubJAc42EVpSt2XIUb6veBoTfKtVvNvW2uc9E5IAxpqfsc7UCen4D1wO7gDjwXWPMN0TkLgBjzEMiIsC3gGuBceCzxpiq0dpmQFdKqaioFtDrmvpvjHkGeKbksYeKfjbAl2ZTSKWUUrPT3tP8lFJK1U0DulJKhYQGdKWUCgkN6EopFRJ1jXLx5Y1FTgFvNvnnHwJ+ZbE4LtA6R4PWORpmU+ePGGPKzswMLKDPhogMVhq2E1Za52jQOkeDX3XWLhellAoJDehKKRUSrgb0h4MuQAC0ztGgdY4GX+rsZB+6UkqpmVxtoSullCqhAV0ppULCuYBea8FqV4jIUhF5TkReE5FDIvKv848vEpH/LiI/z//fWfQ32/L1HhGR9UWPXy4iB/PPfTOf/bJtiUhcRIZE5Kn876Gus4gkReT7InI4/3lfGYE635M/rn8iIo+JyPyw1VlEvisib4nIT4oes1ZHEZknIrvzj78oIstqFsoY48w/cul7jwIfBeYCrwCXBF2uJutyPnBZ/ufzyOWcvwT4C2Br/vGtwJ/nf74kX995wPL8fojnn3sJuJLcwjl/D1wXdP1q1P2Pgf8KPJX/PdR1Bv4z8Pn8z3OBZJjrTG75yTeARP73J4DPhK3OwO8ClwE/KXrMWh2BLwIP5X++Hdhds0xB75QGd+CVwL6i37cB24Iul6W6/R3we8AIcH7+sfOBkXJ1Bfbl98f5wOGixzcBfxN0farU8wLgx8DVvB/QQ1tn4LfywU1KHg9znQtrDC8il6L7KeCaMNYZWFYS0K3VsfCa/M9zyM0slWrlca3Lpa7FqF2Tv5TqBl4EftsY80uA/P8fzr+sUt1T+Z9LH29Xu4A/AYrX+QpznT8KnAL+U76b6TsisoAQ19kYMwr8O+AY8EvgbWPMjwhxnYvYrOPU3xhjzgFvAx+s9uauBfS6FqN2iYh8AHgSuNsY8+tqLy3zmKnyeNsRkd8H3jLGHKj3T8o85lSdybWsLgP+2hjTDZwldyleifN1zvcb30yua2EJsEBENlf7kzKPOVXnOjRTx4br71pAD2wxaj+IiEcumD9qjNmTf/j/isj5+efPB97KP16p7ifyP5c+3o7WATeJyC+Ax4GrReQRwl3nE8AJY8yL+d+/Ty7Ah7nO/xJ4wxhzyhiTAfYAHyPcdS6wWcepv5Hc2s4LgfKr2ue5FtBfBi4SkeUiMpfcjYK9AZepKfk72f8ReM0Y8++LntoL/GH+5z8k17deePz2/J3v5cBFwEv5y7rfiEhvfpufLvqbtmKM2WaMucAYs4zcZ/esMWYz4a7z/wGOi0hhifdPAD8lxHUm19XSKyId+bJ+AniNcNe5wGYdi7d1C7nvS/UrlKBvKjRxE+J6ciNCjgJfDbo8s6jHPyN3+fQqMJz/dz25PrIfAz/P/7+o6G++mq/3CEV3+4Ee4Cf5575FjRsn7fAP+Djv3xQNdZ2BNcBg/rMeADojUOevA4fz5f0v5EZ3hKrOwGPk7hFkyLWmP2ezjsB84HvAEXIjYT5aq0w69V8ppULCtS4XpZRSFWhAV0qpkNCArpRSIaEBXSmlQkIDulJKhYQGdKWUCgkN6EopFRL/H1m0oRjbTdA7AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "rn=[lcg() for i in range(10000)]\n",
    "print (np.mean(rn))\n",
    "print (np.std(rn),1/np.sqrt(12))\n",
    "plt.plot(rn,\"o\")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example: the central limit theorem\n",
    "\n",
    "\n",
    "The sum (and thus the mean if you divide the sum by the $n$) of $n$ i.i.d. variables (regardless their probability distribution) is normally distributed. Check [this video](https://www.youtube.com/watch?v=zeJD6dqJ5lo) for more. \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "# option 1\n",
    "intervals = list(range(6+1)) # uniform distribution\n",
    "# option 2\n",
    "intervals = [0,1.5,2,3,4,4.5,6] # a non-uniform distribution\n",
    "\n",
    "number_of_outcomes = len(intervals)-1 # the spectrum\n",
    "\n",
    "# the number of dice\n",
    "number_of_dice = 10\n",
    "\n",
    "# the number of tosses\n",
    "N=100000"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = []\n",
    "results = []\n",
    "# tossing the dice N times\n",
    "for toss in range(0,N):\n",
    "    # tossing the set of dice\n",
    "    result = np.digitize([lcg()*number_of_outcomes for i in range(number_of_dice)], intervals)\n",
    "    results.extend(result)\n",
    "    data.append(np.sum(result))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 1080x108 with 10 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Draw the dice\n",
    "fig, axs = plt.subplots(1, number_of_dice, figsize=(15, 1.5), sharey=True)\n",
    "faces = {\n",
    "    1:[(0.5, 0.5)],\n",
    "    2:[(0.3, 0.3),(0.7, 0.7)],\n",
    "    3:[(0.3, 0.7),(0.5, 0.5),(0.7, 0.3)],\n",
    "    4:[(0.3, 0.3),(0.7, 0.3), (0.3, 0.7), (0.7, 0.7)],\n",
    "    5:[(0.3, 0.3),(0.7, 0.3), (0.3, 0.7), (0.5, 0.5), (0.7, 0.7)],\n",
    "    6:[(0.3, 0.3),(0.3, 0.5), (0.3, 0.7), (0.7, 0.3), (0.7, 0.5), (0.7, 0.7)]\n",
    "}\n",
    "for i,ax in enumerate(axs): \n",
    "    ax.set_xlim(0, 1)\n",
    "    ax.set_ylim(0, 1)\n",
    "    # Draw the die face\n",
    "    die_face = plt.Rectangle((0.1, 0.1), 0.8, 0.8, facecolor='white', edgecolor='black')\n",
    "    ax.add_patch(die_face)\n",
    "    # Draw the dots on the die face\n",
    "    dot_radius = 0.05\n",
    "    for dot_pos in faces[result[i]]:\n",
    "        dot = plt.Circle(dot_pos, dot_radius, facecolor='black')\n",
    "        ax.add_patch(dot)\n",
    "    ax.axis('off')\n",
    "plt.subplots_adjust(wspace=0.1)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Draw the p.m.f of the dice outcome\n",
    "plt.figure(figsize=(8,5))\n",
    "\n",
    "plt.hist(results, bins=range(1,8), density=True, alpha=0.5, label='simulation')\n",
    "plt.xlabel('Sum of results', fontsize=16)\n",
    "plt.ylabel('Frequency', fontsize=16)\n",
    "plt.tick_params(axis='x', labelsize=14)\n",
    "plt.tick_params(axis='y', labelsize=16)\n",
    "#plt.legend(loc=0, fontsize=20)\n",
    "plt.grid(axis = 'y')\n",
    "plt.grid(axis = 'x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.500179 1.8931962307058925\n"
     ]
    }
   ],
   "source": [
    "# the moments (to be used for normalization)\n",
    "mu = np.mean(results)\n",
    "sigma = np.std(results)\n",
    "print (mu, sigma)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Draw the normalized distribution of the sum of the 10 dice tossing outcomes \n",
    "plt.figure(figsize=(12,8))\n",
    "nbins=6\n",
    "data_rescaled = (data-mu*number_of_dice)/(sigma*np.sqrt(number_of_dice))\n",
    "plt.hist(data_rescaled, \n",
    "         bins=61, range=[-5,5],\n",
    "         #bins = [(i-5*nbins)/nbins for i in range(2*5*(nbins)+1)],\n",
    "         density=True, alpha=0.5, label='simulation',\n",
    "         log=True)\n",
    "plt.xlabel('Sum of results', fontsize=16)\n",
    "plt.ylabel('Frequency', fontsize=16)\n",
    "plt.tick_params(axis='x', labelsize=14)\n",
    "plt.tick_params(axis='y', labelsize=16)\n",
    "#plt.legend(loc=0, fontsize=20)\n",
    "plt.grid(axis = 'y')\n",
    "plt.grid(axis = 'x')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Exercise: fit with a gaussian the distribution you just got.\n",
    "\n",
    "\n",
    "### Beyond LCG\n",
    "\n",
    "LCG though is not sufficiently \"random\" for several complex modern applications. There are nowadays better performing algorithms, like [Mersenne twister](https://en.wikipedia.org/wiki/Mersenne_Twister), a generalized feedback shift-register generator, is used, in particular the numpy random package features it.\n",
    "\n",
    "Numpy uses as default [PCG-64](https://numpy.org/doc/stable/reference/random/bit_generators/index.html) of the [PCG family](https://www.pcg-random.org/), which are considered the ultimate random number generators.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Python libraries for Random Number Generation \n",
    "\n",
    "There are two modules for (pseudo) random numbers that are commonly used. When all you need is to generate random numbers from some distribution, the `numpy.random` moodule is the simplest to use. When you need more information realted to a distribution such as quantiles or the PDF, you can use the `scipy.stats` module."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy.random as npr\n",
    "npr.seed(143) # fix seed for reproducible result"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Random data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "uniform: \n",
      " [[0.7378 0.34   0.4681 0.4239]\n",
      " [0.5683 0.8813 0.0331 0.0088]\n",
      " [0.1992 0.1673 0.969  0.0566]] \n",
      "\n",
      "normal: \n",
      " [[-0.2516 -0.4686 -0.6736  0.1714 -1.8427]\n",
      " [ 0.7976 -0.4144 -0.2424 -0.6527  0.2567]] \n",
      "\n",
      "integers: \n",
      " [[34 19 19  7 22 31]\n",
      " [ 7 17 38 40  8 21]\n",
      " [18 10 27 47 41  4]] \n",
      "\n"
     ]
    }
   ],
   "source": [
    "# random values from a uniform distrubtion in [0.0, 1.0) for a given shape\n",
    "print (\"uniform:\",'\\n', npr.rand(3,4),'\\n') # shape=(3,4)\n",
    " \n",
    "# random values from a standard normal distrubtion  for a given shape\n",
    "print (\"normal:\",'\\n', npr.randn(2,5),'\\n') # shape=(2,5)\n",
    "\n",
    "# random integers between \"low\" and \"high\" edges for a given shape (size)\n",
    "print (\"integers:\",'\\n', npr.randint(1,50,size=(3,6)),'\\n') # shape=(3,6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Operations on arrays"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "reshuffling: \n",
      " [6 0 3 1 8 5 2 9 4 7] \n",
      "\n",
      "choice w/o replace \n",
      " [13 17 10 14 18] \n",
      "\n",
      "choice w/ replace \n",
      " [[16 18 17 13 12 18 14]\n",
      " [13 17 19 19 11 10 11]\n",
      " [11 18 16 19 18 14 16]\n",
      " [13 10 17 16 18 11 12]\n",
      " [11 18 16 12 12 15 13]] \n",
      "\n"
     ]
    }
   ],
   "source": [
    "# randomly shuffling a vector\n",
    "x = np.arange(10)\n",
    "npr.shuffle(x)\n",
    "print (\"reshuffling:\",'\\n', x,'\\n')\n",
    "# npr.permutation works similarly\n",
    "\n",
    "# Generates a random sample from a given 1-D array (without replacement)\n",
    "x = np.arange(10,20)\n",
    "print (\"choice w/o replace\",'\\n',npr.choice(x, 5, replace=False),'\\n')\n",
    "\n",
    "# Generates a random sample of a given dimension from a given 1-D array\n",
    "print (\"choice w/ replace\",'\\n',npr.choice(x, (5, 7), replace=True),'\\n') # this is default"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Probability Density Functions\n",
    "\n",
    "numbers can be drawn from basically any common PDF (normal, binomial, chisquare, etc..)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "exponential: \n",
      " [[17.4746  1.8033]\n",
      " [ 5.3176  7.7107]\n",
      " [ 7.8081 11.7676]] \n",
      "\n"
     ]
    }
   ],
   "source": [
    "# exponential\n",
    "print (\"exponential:\",'\\n', npr.exponential(10, size=(3,2)),'\\n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "frequencies of 6-faces dice tossing (two players): \n",
      " [[3248 3339 3355 3356 3363 3339]\n",
      " [3421 3374 3347 3223 3289 3346]]\n"
     ]
    }
   ],
   "source": [
    "# tossing a dice (the easy way..)\n",
    "roll = 1./6\n",
    "x = npr.multinomial(20000, [roll]*6, size=2)\n",
    "print (\"frequencies of 6-faces dice tossing (two players):\",'\\n', x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's pretend we haven't seen any of the above functionalties available in numpy and scipy and walk through the basics.."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Non-uniform random numbers\n",
    "\n",
    "In several cases the actual random process occur with non-uniform probability, i.e. with a given probability density function (pdf), different from the uniform distribution. Several methods are available, we will see a few of them\n",
    "\n",
    "### Inverst transform method\n",
    "\n",
    "Let'start from a uniform distribution $u(z)$:\n",
    "\n",
    "$$\n",
    "\\left\\{\n",
    "\\begin{array}{ll}\n",
    "      1 & 0\\leq z\\leq 1 \\\\\n",
    "      0 & {\\rm elsewhere}\n",
    "\\end{array} \n",
    "\\right.\n",
    "$$\n",
    "\n",
    "and let's look for a function $x(z)$ such that $x$ is distributed accordingly to a given pdf $p(x)$. The probability to find $x$ between $x$ and $x+dx$ is equal to:\n",
    "\n",
    "$$\n",
    "p(x)dx = dz\n",
    "$$\n",
    "\n",
    "and thus:\n",
    "\n",
    "$$\n",
    "\\int_{-\\infty}^{x(z)} p(x') dx' = \\int_0^z dz'= z\n",
    "$$\n",
    "\n",
    "If (a) we could solve the integral and (b) solve for $x$, then we are done. For most of the pdf at least one of the two is not possible.. The typical solvable analitical example is:\n",
    "\n",
    "$$\n",
    "p(x) = \\mu e^{-\\mu x}\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\int_{0}^{x(z)} p(x') dx' = 1 - e^{-\\mu x} = z\n",
    "$$\n",
    "\n",
    "and thus:\n",
    "\n",
    "$$\n",
    "x(z) = - \\frac{1}{\\mu}\\log{(1-z)}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "def expon_pdf(x, mu=1):\n",
    "    \"\"\"PDF of exponential distribution.\"\"\"\n",
    "    return mu*np.exp(-mu*x)\n",
    "\n",
    "def expon_cdf(x, mu=1):\n",
    "    \"\"\"CDF of exponetial distribution.\"\"\"\n",
    "    return 1 - np.exp(-mu*x)\n",
    "\n",
    "def expon_icdf(z, mu=1):\n",
    "    \"\"\"Inverse CDF of exponential distribution - i.e. quantile function.\"\"\"\n",
    "    return -np.log(1-z)/mu"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "dist = stats.expon()\n",
    "x = np.linspace(0,4,100)\n",
    "y = np.linspace(0,1,100)\n",
    "\n",
    "plt.figure(figsize=(12,4))\n",
    "plt.subplot(121)\n",
    "plt.plot(x, expon_cdf(x))\n",
    "plt.axis([0, 4, 0, 1])\n",
    "for q in [0.5, 0.8]:\n",
    "    plt.arrow(0, q, expon_icdf(q)-0.1, 0, head_width=0.05, head_length=0.1, fc='b', ec='b')\n",
    "    plt.arrow(expon_icdf(q), q, 0, -q+0.1, head_width=0.1, head_length=0.05, fc='b', ec='b')\n",
    "plt.ylabel('1: Generate a (0,1) uniform PRNG')\n",
    "plt.xlabel('2: Find the inverse CDF')\n",
    "plt.title('Inverse transform method');\n",
    "\n",
    "plt.subplot(122)\n",
    "u = np.random.random(10000)\n",
    "v = expon_icdf(u)\n",
    "plt.hist(v, histtype='step', bins=100, density=True, linewidth=2)\n",
    "plt.plot(x, expon_pdf(x), linewidth=2)\n",
    "plt.axis([0,4,0,1])\n",
    "plt.title('Histogram of exponential PRNGs');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Box-Muller for generating normally distributed random numbers\n",
    "\n",
    "The inverse method is not applicable even for the gaussian distribution:\n",
    "\n",
    "$$\n",
    "\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\int_{-\\infty}^{x} \\exp{-\\frac{x'^2}{2\\sigma^2}} dx' = z\n",
    "$$\n",
    "\n",
    "is not solvable. The trick is to consider a two dimensional gaussian function with the same $\\sigma$ on both coordinates:\n",
    "\n",
    "$$\n",
    "p(x)dx \\times p(y)dy = \\frac{1}{\\sqrt{2\\pi\\sigma^2}} \\exp{-\\frac{x^2}{2\\sigma^2}} \\times \\frac{1}{\\sqrt{2\\pi\\sigma^2}} \\exp{-\\frac{y^2}{2\\sigma^2}} = \\frac{1}{2\\pi\\sigma^2} \\exp{-\\frac{(x^2+y^2)}{2\\sigma^2}}dxdy\n",
    "$$\n",
    "\n",
    "which written in radial coordinates:\n",
    "\n",
    "$$\n",
    "x=r\\cos{\\theta};\\,\\,\\,\n",
    "y=r\\sin{\\theta}\n",
    "$$\n",
    "\n",
    "$$\n",
    "p(r,\\theta)dr d\\theta = \\frac{r}{\\sigma^2} \\exp{-\\frac{r^2}{2\\sigma^2}} dr \\times \\frac{d\\theta}{2\\pi} = p(r)dr \\times p(\\theta)d\\theta\n",
    "$$\n",
    "\n",
    "with both $p(r)$ and $p(\\theta)$ normalized to 1. Now, the latter is a simple uniform distribution, whereas the former is solvable:\n",
    "\n",
    "$$\n",
    "\\frac{1}{\\sigma^2} \\int_{0}^{r} \\exp{-\\frac{r^2}{2\\sigma^2}} rdr = z\n",
    "$$\n",
    "\n",
    "which gives:\n",
    "\n",
    "$$\n",
    "r=\\sqrt{-2\\sigma^2\\log{1-z}}\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 10000\n",
    "z = np.random.random(n)\n",
    "theta = 2*np.pi*np.random.random(n)\n",
    "r_squared = -2*np.log(z)\n",
    "r = np.sqrt(r_squared)\n",
    "data = pd.DataFrame({'x':r*np.cos(theta), 'y':r*np.sin(theta)})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<seaborn.axisgrid.JointGrid at 0x7f7ab05e2730>"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x432 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "sns.jointplot(data=data, x='x',y='y', kind=\"hist\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Creating a random number generator for arbitrary distributions\n",
    "\n",
    "Suppose we have some random samples with an unknown distribution. We can still use the inverse transform method to create a random number generator from a random sample, by estimating the inverse CDF function using interpolation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.interpolate import interp1d\n",
    "\n",
    "def extrap1d(interpolator):\n",
    "    \"\"\"From StackOverflow http://bit.ly/1BjyRfk\"\"\"\n",
    "    xs = interpolator.x\n",
    "    ys = interpolator.y\n",
    "\n",
    "    def pointwise(x):\n",
    "        if x < xs[0]:\n",
    "            return ys[0]+(x-xs[0])*(ys[1]-ys[0])/(xs[1]-xs[0])\n",
    "        elif x > xs[-1]:\n",
    "            return ys[-1]+(x-xs[-1])*(ys[-1]-ys[-2])/(xs[-1]-xs[-2])\n",
    "        else:\n",
    "            return interpolator(x)\n",
    "\n",
    "    def ufunclike(xs):\n",
    "        return np.array(list(map(pointwise, np.array(xs))))\n",
    "\n",
    "    return ufunclike"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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us5JeqbqIbfh7SU9GxHWSbtAMnYvt3ZLullSPiA+qNTDl6CRrmLugl/S3kv5CPW7cmmYR8XREXG4/fVat+xKm3fRPg9FHRLwWEc+1P/8ftQJmd7VVDcf2Hkm/L+nBqmvZCttXS/qIpK9IUkRcioj/rrSo4e2Q9Mu2d0h6tyZ8P9FcBb3tOyT9KCK+V3UtI/DHkr5RdREFek2PMXNs1yR9SNJ3Ki5lWH+nVuPm7Yrr2KoPSFqT9E/t7qcHbV9VdVGlIuJHkr6kVi/Ca2rdZ/T0JGtIF/S2v9nuB9v4OCLpC5K+WHWN/Qyof32fL6jVpfBwdZUWK54GY5rZ/hVJ/yLpnoh4s+p6Stn+uKTXI+JM1bVsww5JN0p6ICI+JOn/JM3MtR7bv6rWX7H7JP2GpKts/+Eka0i38EhEfLTb67Z/W61v9PfcmmRoj6TnbB+MiB9PsMS+etW/zvYfSfq4pFtjNm6CKJlCY6rZ/kW1Qv7hiHis6nqGdLOkO2zfLuldkq62/bWImGjQbNOqpNWIWP9L6lHNUNBL+qikVyNiTZJsPybpdyV9bVIFpGvR9xIRL0bEeyOiFhE1tX55bpymkB/E9iFJn5N0R0S8VXU9hWZ6Ggy3WgVfkfRKRPxN1fUMKyI+HxF72r/zRyX924yFvNr/Ri/Y/s32S7dKernCkob1Q0k32X53+/fpVk34YnK6Fn1y/yDplyQ90/6r5NmIOF5tSf1N2TQYW3GzpE9KetH28+3X/jIinqiupLn0p5IebjcWzqs9zcosiIjv2H5U0nNqdbl+VxOeCoEpEAAgubnpugGAeUXQA0ByBD0AJEfQA0ByBD0AJEfQA0ByBD0AJPf/rLb3TCUwg4wAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from statsmodels.distributions.empirical_distribution import ECDF\n",
    "\n",
    "# Make up some random data\n",
    "x = np.concatenate([np.random.normal(0, 1, 10000),\n",
    "                    np.random.normal(4, 1, 10000)])\n",
    "\n",
    "ecdf = ECDF(x)\n",
    "inv_cdf = extrap1d(interp1d(ecdf.y, ecdf.x,\n",
    "                            bounds_error=False, assume_sorted=True))\n",
    "r = np.random.uniform(0, 1, 1000)\n",
    "ys = inv_cdf(r)\n",
    "\n",
    "plt.hist(x, 25, histtype='step', color='red', density=True, linewidth=1)\n",
    "plt.hist(ys, 25, histtype='step', color='blue', density=True, linewidth=1);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Rejection sampling (Accept-reject method)\n",
    "\n",
    "Suppose we want to sample from a (truncated) Cauchy distribution (a Student-t distribution with 1 degree of freedom).\n",
    "We use the uniform as a proposal distibution (highly inefficient)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(-4, 4)\n",
    "\n",
    "dist = stats.cauchy()\n",
    "upper = dist.pdf(0)\n",
    "\n",
    "plt.figure(figsize=(12,4))\n",
    "plt.subplot(121)\n",
    "plt.plot(x, dist.pdf(x))\n",
    "plt.axhline(upper, color='grey')\n",
    "px = 1.0\n",
    "plt.arrow(px,0,0,dist.pdf(1.0)-0.01, linewidth=1,\n",
    "         head_width=0.2, head_length=0.01, fc='g', ec='g')\n",
    "plt.arrow(px,upper,0,-(upper-dist.pdf(px)-0.01), linewidth=1,\n",
    "         head_width=0.3, head_length=0.01, fc='r', ec='r')\n",
    "plt.text(px+.25, 0.2, 'Reject', fontsize=16)\n",
    "plt.text(px+.25, 0.01, 'Accept', fontsize=16)\n",
    "plt.axis([-4,4,0,0.4])\n",
    "plt.title('Rejection sampling concepts', fontsize=20)\n",
    "\n",
    "plt.subplot(122)\n",
    "n = 100000\n",
    "# generate from sampling distribution\n",
    "u = np.random.uniform(-4, 4, n)\n",
    "# accept-reject criterion for each point in sampling distribution\n",
    "r = np.random.uniform(0, upper, n)\n",
    "# accepted points will come from target (Cauchy) distribution\n",
    "v = u[r < dist.pdf(u)]\n",
    "\n",
    "plt.plot(x, dist.pdf(x), linewidth=2)\n",
    "\n",
    "# Plot scaled histogram\n",
    "factor = dist.cdf(4) - dist.cdf(-4)\n",
    "hist, bin_edges = np.histogram(v, bins=100, density=True)\n",
    "bin_centers = (bin_edges[:-1] + bin_edges[1:]) / 2.\n",
    "plt.step(bin_centers, factor*hist, linewidth=2)\n",
    "\n",
    "plt.axis([-4,4,0,0.4])\n",
    "plt.title('Histogram of accepted samples', fontsize=20);\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Mixture representations\n",
    "Sometimee, the target distribution  which we need to generate random numbers from can be expressed as a mixture of “simpler” distributions that we already know how to sample from\n",
    "\n",
    "$$\n",
    "f(x)=\\int g(x|y)p(y)dy\n",
    "$$\n",
    "\n",
    "For example, if $y$ is drawn from the $\\chi^2_\\nu$ distrbution, then ${\\cal N}(0,\\nu/y)$ is a sample from the Student-t distribution with $\\nu$ degrees fo freedom."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "n = 10000\n",
    "df = 5\n",
    "dist = stats.t(df=df)\n",
    "y = stats.chi2(df=df).rvs(size=n)\n",
    "r = stats.norm(0, df/y).rvs(n)\n",
    "\n",
    "\n",
    "plt.plot(x, dist.pdf(x), linewidth=2)\n",
    "\n",
    "# Plot scaled histogram\n",
    "factor = dist.cdf(4) - dist.cdf(-4)\n",
    "hist, bin_edges = np.histogram(v, bins=100, density=True)\n",
    "bin_centers = (bin_edges[:-1] + bin_edges[1:]) / 2.\n",
    "plt.step(bin_centers, factor*hist, linewidth=2)\n",
    "\n",
    "plt.axis([-4,4,0,0.4])\n",
    "plt.title('Histogram of accepted samples', fontsize=20);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Draw from an analytic pdf\n",
    "\n",
    "Obviously scipy stats module features all possible pdf that can come to your mind. You can draw random data from each of them"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import gamma\n",
    "a = 1.99\n",
    "\n",
    "x = np.linspace(gamma.ppf(0.01, a),\n",
    "                gamma.ppf(0.99, a), 100)\n",
    "rv = gamma(a)\n",
    "fig, ax = plt.subplots(1, 1)\n",
    "ax.plot(x, rv.pdf(x), 'k-', lw=2)\n",
    "r = gamma.rvs(a, size=10000)\n",
    "_ = ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Monte Carlo integration\n",
    "\n",
    "Monte Carlo integration is typically less accurate than other integration methods, but very often is the only available tool, e.g. when the integrand has very rapid variations or singular points, or, most importantly, when dealing with high dimensional integrals.\n",
    "\n",
    "The idea is simple, let's the area under the function be $I$ whereas the all possible outcomes lay in a box of area $A$. The probability for a point to fall under the function is $p=I/A$. If we generate $N$ random points, the fraction $k$ which fall under the curve is $k/N$ and approximate $I/A$, thus:\n",
    "\n",
    "$$\n",
    "I\\simeq\\frac{k A}{N}\n",
    "$$\n",
    "\n",
    "Let's try this with the function $f(x) =\\sin^2{\\frac{1}{x(2-x)}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f7a907df640>]"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def f(x):\n",
    "    return (np.sin(1/(x*(2-x))))**2\n",
    "\n",
    "x=np.linspace(0.001,1.999,1000)\n",
    "plt.plot(x,f(x),'r-')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.45158\n"
     ]
    }
   ],
   "source": [
    "# Monte Carlo integration\n",
    "N=100000\n",
    "count=0\n",
    "for i in range(N):\n",
    "    x=2*np.random.random()\n",
    "    y=np.random.random()\n",
    "    if y<f(x): count+=1\n",
    "I=2*count/N\n",
    "print(I)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The mean value method\n",
    "\n",
    "Let's take the integral:\n",
    "\n",
    "$$\n",
    "I=\\int_a^b f(x) dx\n",
    "$$\n",
    "\n",
    "defining $\\langle f \\rangle$ as the mean of $f$:\n",
    "\n",
    "$$\n",
    "\\langle f \\rangle = \\frac{1}{b-a}\\int_a^b f(x) dx\n",
    "$$\n",
    "\n",
    "and estimating $\\langle f \\rangle$ by uniformely probing at random the function domain, such as\n",
    "\n",
    "$$\n",
    "\\langle f \\rangle = \\frac{1}{N} \\sum_{i=1}^{N} f(x_i)\n",
    "$$\n",
    "\n",
    "we get:\n",
    "\n",
    "$$\n",
    "I=\\frac{b-a}{N} \\sum_{i=1}^{N} f(x_i)\n",
    "$$\n",
    "\n",
    "this easily generalize to higher dimensions:\n",
    "\n",
    "$$\n",
    "I=\\frac{V}{N} \\sum_{i=1}^{N} f(\\vec{r}_i)\n",
    "$$\n",
    "\n",
    "where the sampling points $\\vec{r}_i$ are drawn uniformly at random from integration space of volume $V$.\n",
    "\n",
    "It can be proven that the standard deviation of the method scales as $1/\\sqrt{N}$:\n",
    "\n",
    "$$\n",
    "\\sigma = V\\frac{\\sqrt{{\\rm var}\\, f}}{\\sqrt{N}}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Importance sampling\n",
    "\n",
    "There are several general techinques for variance reduction, sometimes known as Monte Carlo swindles since these methods improve the accuracy and convergence rate of Monte Carlo integration without increasing the number of Monte Carlo samples. *Importance sampling* is among the most commonly used.\n",
    "\n",
    "We can define a weighted average of a function $g(x)$:\n",
    "\n",
    "$$\n",
    "\\langle g_w \\rangle = \\frac{\\int_a^b w(x) g(x)dx}{\\int_a^b w(x)dx}\n",
    "$$\n",
    "\n",
    "Consider again the integral of $f(x)$:\n",
    "\n",
    "$$\n",
    "I=\\int_a^b f(x)dx\n",
    "$$\n",
    "\n",
    "Setting $g(x)=f(x)/w(x)$ we have:\n",
    "\n",
    "$$\n",
    "\\left\\langle \\frac{f(x)}{w(x)}\\right\\rangle = \\frac{\\int_a^b w(x)f(x)/w(x) )dx}{\\int_a^b w(x)dx} = \\frac{I}{\\int_a^b w(x)dx}\n",
    "$$\n",
    "\n",
    "and thus:\n",
    "\n",
    "$$\n",
    "I = \\left\\langle \\frac{f(x)}{w(x)}\\right\\rangle \\int_a^b w(x)dx \\simeq \\frac{1}{N}\\sum_{i=1}^N \\frac{f(x_i)}{w(x_i)}\\int_a^b w(x)dx\n",
    "$$\n",
    "\n",
    "which generalizes the mean value method if $w(x)$ is the uniform distribution between $a$ and $b$\n",
    "\n",
    "### Example\n",
    "\n",
    "Suppose we want to estimate the tail probability of ${\\cal N}(0,1)$ for $x>5$. Regular MC integration using samples from ${\\cal N}(0,1)$ is hopeless since nearly all samples will be rejected. However, we can use the exponential density truncated at 5 as the importance function and use importance sampling."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(4, 8, 100)\n",
    "plt.plot(x, stats.expon(5).pdf(x))\n",
    "plt.plot(x, stats.norm().pdf(x))\n",
    "plt.yscale('log')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0000002867"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%precision 10\n",
    "h_true =1 - stats.norm().cdf(5)\n",
    "h_true"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "integral (sum) above 5: 0\n",
      "estimate: 0.0\n",
      "relative error: 1.0\n"
     ]
    }
   ],
   "source": [
    "n = 1000000\n",
    "y = stats.norm().rvs(n)\n",
    "print (\"integral (sum) above 5:\", np.sum(y > 5))\n",
    "h_mc = 1.0/n * np.sum(y > 5)\n",
    "# estimate and relative error\n",
    "print (\"estimate:\", h_mc)\n",
    "print (\"relative error:\", np.abs(h_mc - h_true)/h_true)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "estimate: 2.807731239692683e-07\n",
      "relative error: 0.020507293627662213\n"
     ]
    }
   ],
   "source": [
    "n = 10000\n",
    "y = stats.expon(loc=5).rvs(n)\n",
    "h_is = 1.0/n * np.sum(stats.norm().pdf(y)/stats.expon(loc=5).pdf(y))\n",
    "# estimate and relative error\n",
    "print (\"estimate:\", h_is)\n",
    "print (\"relative error:\", np.abs(h_is - h_true)/h_true)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "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": "3.9.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}