Entropy of a Gaussian in SymPy.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "from sympy import *\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "IPython console for SymPy 1.0 (Python 3.5.2-64-bit) (ground types: python)\n",
      "\n",
      "These commands were executed:\n",
      ">>> from __future__ import division\n",
      ">>> from sympy import *\n",
      ">>> x, y, z, t = symbols('x y z t')\n",
      ">>> k, m, n = symbols('k m n', integer=True)\n",
      ">>> f, g, h = symbols('f g h', cls=Function)\n",
      ">>> init_printing()\n",
      "\n",
      "Documentation can be found at http://docs.sympy.org/1.0/\n"
     ]
    }
   ],
   "source": [
    "init_session()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "a=Symbol('a',real=True,positive=True)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\frac{e^{- \\frac{x^{2}}{a^{2}}}}{\\sqrt{\\pi} a}$$"
      ],
      "text/plain": [
       "   2 \n",
       " -x  \n",
       " ────\n",
       "   2 \n",
       "  a  \n",
       "ℯ    \n",
       "─────\n",
       " √π⋅a"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f=exp(-(x**2)/a**2)/sqrt(pi)/a\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(f,(x,-oo,oo) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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      "text/latex": [
       "$$\\log{\\left (a \\right )} + \\frac{1}{2} + \\frac{1}{2} \\log{\\left (\\pi \\right )}$$"
      ],
      "text/plain": [
       "         1   log(π)\n",
       "log(a) + ─ + ──────\n",
       "         2     2   "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h=integrate(-f*log(f),(x,-oo,oo) )\n",
    "h"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7fbf2722a908>"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(h,(a,0.001,4))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHIAAAAqBAMAAACdLhLUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n/gFrR5zXPkHhMgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{1}{2} + \\frac{1}{2} \\log{\\left (\\pi \\right )}$$"
      ],
      "text/plain": [
       "1   log(π)\n",
       "─ + ──────\n",
       "2     2   "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h.subs(a,1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\left [ \\frac{1}{\\sqrt{\\pi} e^{\\frac{1}{2}}}\\right ]$$"
      ],
      "text/plain": [
       "⎡ -1/2⎤\n",
       "⎢ℯ    ⎥\n",
       "⎢─────⎥\n",
       "⎣  √π ⎦"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r=solve(h,a)\n",
    "r"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
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   "outputs": [
    {
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      "text/latex": [
       "$$0.342198280312217$$"
      ],
      "text/plain": [
       "0.342198280312217"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N(r[0])"
   ]
  },
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   "execution_count": null,
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