Inverse_of_a_Distribution_Function_Example_Python.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### MIT License (c) 2018 by Andrew Lyasoff"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Jupyter notebook written in Python 3. It illustrates the use of SymPy to compute the distribution function of the Gaussian law and its inverse, which is then used to transform a uniform Monte Carlo sample into a Gaussian Monte Carlo sample."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, compute the distribution function and its inverse in symbolic form:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from numpy import *\n",
    "from sympy import *\n",
    "x,y,n,u=symbols('x y n u')\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAHlBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACGjDitAAAACXRSTlMAVO8Qq5l2zWYZcMvdAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAHUlEQVQIHWNgAANGZQYGk5DJQDYbqQSr03QPsBkAJYgIYEZbtZEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate((1/sqrt(2*pi))*exp(-u**2/2),(u,-oo,+oo))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{1}{2} \\operatorname{erf}{\\left (\\frac{\\sqrt{2} x}{2} \\right )} + \\frac{1}{2}$$"
      ],
      "text/plain": [
       "   ⎛√2⋅x⎞    \n",
       "erf⎜────⎟    \n",
       "   ⎝ 2  ⎠   1\n",
       "───────── + ─\n",
       "    2       2"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AA=integrate(sqrt(1/(2*pi))*exp(-u**2/2),(u,-oo,x))\n",
    "AA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- y + \\frac{1}{2} \\operatorname{erf}{\\left (\\frac{\\sqrt{2} x}{2} \\right )} + \\frac{1}{2}$$"
      ],
      "text/plain": [
       "        ⎛√2⋅x⎞    \n",
       "     erf⎜────⎟    \n",
       "        ⎝ 2  ⎠   1\n",
       "-y + ───────── + ─\n",
       "         2       2"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(1+erf(sqrt(2)*x/2))/2-y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\sqrt{2} \\operatorname{erfinv}{\\left (2 y - 1 \\right )}$$"
      ],
      "text/plain": [
       "√2⋅erfinv(2⋅y - 1)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BB=solve((1+erf(sqrt(2)*x/2))/2-y,x)[0]\n",
    "BB"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABsAAAAVBAMAAABF8IgWAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAInarRM2ZVBDdiWbvuzJCz3LGAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAtElEQVQYGWNggADG/yDwAcpjYHaAscC0KAqPIR2FyzEBhctWgMLtZGAQy70HF1rNwGHAsN4BygdqZW5g4DkA5TJtYOABivwDcl2AmJOBgfUHmMt0FMhVAqvi/MHAIKHvwMAAsdV/AVC0/wIDN4hmYJgMItg+MnA2gBkJIJLxG8MuEM2gBiYZ7BtmgBjsCQyCIPr98gQQ9YyBQRFEsx5tAJKMs1Ylgy1g/wYWBIYNxL5AEBcCALUeKW//iOF2AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\sqrt{2}$$"
      ],
      "text/plain": [
       "√2"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sqrt(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$1.4142135623731$$"
      ],
      "text/plain": [
       "1.41421356237310"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sqrt(2.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "def F(xxx):\n",
    "    return erf(sqrt(2)*xxx/2)/2 + 1/2\n",
    "\n",
    "def Finv(yyy):\n",
    "    return sqrt(2)*erfinv(2*yyy - 1)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABoAAAAOBAMAAADDIxFwAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJZjLNVN0i77urRHZ72Yd1AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAp0lEQVQIHWNgEDIxZQABuce6QDKdQWICiKfAwFPAwLKAgTkByGF3YOBMYOA+wMD+Echj3sDA2MDAf4CB9zuQx/tbgNmBQV6BgfcfkMdw/osJA8N8AwbWzyAe8/9EIE8Byqu58esCXCXLAYb3C0GmsINM4ZrAwPqdgXsDAwvIBnkgPgKynSkBIsewk4GhnUEqgPEbA28iA8sDBgbhd0cZGHoYGGKUdRkAVcEoloZwKkkAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$0.8$$"
      ],
      "text/plain": [
       "0.800000000000000"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F(Finv(0.8))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAOBAMAAAAPuiubAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAzXYQMplU74mrIma7RN0SDTw+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAATklEQVQIHWNgVGYJY2Bg//+/gYGBsVOLgYGBE4jh5G5VBwYGHgOuv2DBc2AyfgHDRAaG/gsMhQwM8QYMAQwMDxkYljJwVzEwcEVMNWAAADWrDnXF0pOGAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$5$$"
      ],
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Finv(F(5))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABkAAAAOBAMAAAAoFKpzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAzXYQMplU74mrIma7RN0SDTw+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAoklEQVQIHWNgVGYJYwAC5ogdDgwM7P//N4B4jAnMr4BkpxaIw7CdgaGRgYETzGZgOMnAkG8A5/1kYLifwMC5W9UBKM38FchTYOAx4PoL5HH9YGDwbwAyGM6BeEA5CC9+AVzlRAaG/gtASaAp+QkMhQwM8QZA3hkGhv0GDAEMDA+BHJDtgQwMSxm4qxjsBRj4EphLgIZFTDVg4JvAwNyk6cAAAKBqJMZnpa9LAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$5.0$$"
      ],
      "text/plain": [
       "5.00000000000000"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Finv(F(5.))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now plot the distribution function for the standard normal ${\\cal N}(0,1)$ distribution law. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f26d8d254e0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = arange(-3, 3, 0.01)\n",
    "s=[F(u) for u in t]\n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(t, s)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f26d04e0940>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = arange(.001, 1, 0.001)\n",
    "s=[Finv(u) for u in t]\n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(t, s)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot the inverse:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute the \"exact\" probability for falling outside the interval $[-3\\sigma,3\\sigma]$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALIAAAAmBAMAAACSU9wOAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMiKZZomrRO92VLsN82QsAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADVklEQVRIDZ1WTWgTQRR+m2Tz2ySLRdG2kLX+YFEh/iCKFHoqFA8NiL0oJKIgPViDlwoVDCJo8WAQaVGKBr3bCqKChQbBPyq0XkRQaAUvgkoraouo8b3Zmc3mZ2ZT57Dvzfd98830vdlNARQjkFWQ6xScK/VEpQibKlbNhe8p+ZdKVkm2dilpj6pWypVwXUIfKdFY0BclvCvs+yGR3OC4bGfJsjLsyZdzZ6abfNZrOOEV5OlUfbF/luOeYn2BK3pMomgWeGBCZCuMkxL9SYHrv0TWaAyPzaNU/+7UbzO6eRG0JEDfwEciHzkVznzNxQvOqZ0faKHDBBYY8M5gYaq0zPlgFrQ8XCng9CqHyoGrt0NzqgyWswGW+vFsoI3cspy3jAkpljlkQGQC2Rmx5r2VCDU2IGgKzhmte+qdZ9iM5Zy3Bc8BIkkIUa3mxG7cGfdi6lgeonVfI8s5nqlxDhcQSgJ4Fi3n3lkmAahyjuch/IdTFPQTw6nbk2839Txm1vEi4/iZR88eB5g2AZos2EtnSmSZpMY5kYHwb05RuGPgx+3B6yW4yUC+jDt/gHRBb8POeq0V4zmMfPMa53QOfKLhJMc7NAkbMLGc09aBuDNALAnaX4AzJAXYR494jp44qqqRzpDzql00NgL4fg4OjkIH6ixnXkTb2Y8fqFMp2MGs/BkKVivOt7fvb2+nhaKDVdUIsJpvRpo7O88cMiGIpUhnAjkygGfsyZtcc2bsYNTRQR+redm5ohoRE/zYs9Ci1yDPaAYOYUjkaIKjqhqxeQg4b91OgDYoOyeKbJFVDT+2zsSKLb9hYDfAU0z45jXO+Kb4UW2P3iwchT04/cog3h501r6BnoGhFMJb1xOn773Un8RY9z6jGjqhtYuEfGgjL4y7pS/QU/pMSCRPz/6pjixdhL7hozSdy9DTgz9W5DxeoBkOUQ2hXvvqmkXUfcbMOrCHDm6PcwZPhbPNKJPghJIm8rJQrBZJQ7H+N6ViKXtdKpCGJtqSm8xdIXH4JMFtOCr7t8FWSJIhCW7DFXfWRhtIErMuokjWRSCjm0wZw/FpF15O35dTjHnowsvp03KKGNe/Sb7cU5RzyBxOKWkVqe1Wsp0q1oU7qDpV7H9vBm2qdym2blFwRP0D7C3Ef1++1l0AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$- \\operatorname{erf}{\\left (1.5 \\sqrt{2} \\right )} + 1.0$$"
      ],
      "text/plain": [
       "-erf(1.5⋅√2) + 1.0"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2*(1-F(3.0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\operatorname{erf}{\\left (1.5 \\sqrt{2} \\right )}$$"
      ],
      "text/plain": [
       "erf(1.5⋅√2)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1-(_)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now try to compute the above probability by using Monte Carlo simulation and sampling from the uniform distribution in $[0,1]$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- \\operatorname{erf}{\\left (1.5 \\sqrt{2} \\right )} + 0.997359$$"
      ],
      "text/plain": [
       "-erf(1.5⋅√2) + 0.997359"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2*(1-F(3.0))-0.002641"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHaJmUSrVGYizbvd7zLJaKAlAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAWElEQVQIHWNgEFJ2ZWBgcGBgK2Bgb2DgXMDAvIGB8QAD628B5gYGhvwfFkBZ5v8LgWRZzN8ABo4EBv2LDLwTGPi+M8gDBTNAbIadDKwLGTgUGBieObsyAADQ0RHsm3pGXQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$8$$"
      ],
      "text/plain": [
       "8"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Finv(F(8))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAOBAMAAAAPuiubAAAALVBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAIpm7MhCriUTv3c12VLgeopIAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABOSURBVAgdY2BUMnZgYAhjYH/BwJDKwDCTgWEWA0OeA8O+ABAJBOsCgATHcxCTKwFEKoEIHgUQeYmBUYCBRYGBR4BBqrwoi4Fh37t3rxgAK5QOlzv7snYAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$2$$"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1+1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.6.4"
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 },
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