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tonigi/Entropy of a Gaussian in SymPy.ipynb

Last active February 3, 2017 16:05
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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": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAC0AAAA8BAMAAAAAvi6XAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
"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": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAHlBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAACGjDitAAAACXRSTlMAVO8Qq5l2zWYZcMvdAAAACXBIWXMAAA7EAAAOxAGV\nKw4bAAAAHUlEQVQIHWNgAANGZQYGk5DJQDYbqQSr03QPsBkAJYgIYEZbtZEAAAAASUVORK5CYII=\n",
"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": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAMcAAAAqBAMAAAD18VKpAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrdd/if0hOn//KvIHcKH5RdqFq4AAAAAASUVORK5CYII=\n",
"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\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWYiuzKJ\nRN0MreaOAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACVklEQVRIDe2VP2gTURzHv8mlSS+XxCDoZimK\nm2goBSk4BEVUKM0hKRTtkEWhWHApZLDSLOqaRdFF4yKoSxQdXGwGGwcD3uYgmENcxIJKBAf/nL/f\nXe7l7r0LN7j2B3fv+/u83zfv3Xv3LsB/xO6Sak4cUJlMtJMV1TlbHch1Ufll1Yn0jjOwVDsrFFiM\nKFk9staQ+cTxn6dl5ua3h3SyGdktoHGqI7Qr9Jafn/HFmHamHO7IiKGOhjv2hlMgKzmviYJcTUgW\nsc7PolybFpJFnFO3R+VXR3KMU+uuWsjPL11sAWmTLoeC2q1Y58smHqBfKjRq9NgNoPdk5fodct2I\ndW4Dm+0FpGyqLBRhFHPNS2yaAg/ufCP5ul7/WK+vMd3lQhLZsvGd0s5Zz5m0mOEE12zyTUTECiX+\nkLN1rz1TpCrX+Qw2G2Kd7pjmre4brqbZAktwv24fGIiIGBMHgb61zyvhFdIGoGnEr1ALT9tYx7v3\nL7g6Z9Ku2DjEeplvIuQxjcqPpr78ykLGcWhs5G13Z+bZcAW4ee4xKw7h1BcftT3k3e9jbqpGcvT2\n0euklfHWGlbNDVvMIuc+zDBPtIBJuhB641MlJKcJhoJ2bH8AaF9olryu2aZP6ZQlbaR++bnfHgYq\nweke666e577RyX4IZAYRzo1S2On/IqSvSSHqj+trSZSPF/2O2pf/rTKVfFIRCuUIKKMJUyaUL0Yw\nBV1QiHf6I3AYJUzwGQ/HHuhi68I9gew5sBJIXWmYSMc69e1e1Zadd3tbCzJTcjoGjuLccJy/SqUE\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": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAEYAAAA0BAMAAADF6lN9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n",
"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
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAKsAAAAPBAMAAABpSyLSAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nq1hZM5F7gvlP7R+nkw9HSGtrSgAAAABJRU5ErkJggg==\n",
"text/latex": [
"$$0.342198280312217$$"
],
"text/plain": [
"0.342198280312217"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"N(r[0])"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": []
}
],
"metadata": {
"anaconda-cloud": {},
"kernelspec": {
"display_name": "Python [default]",
"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": "3.5.2"
}
},
"nbformat": 4,
"nbformat_minor": 1
}
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