{
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python [Root]",
   "language": "python",
   "name": "Python [Root]"
  },
  "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"
  },
  "name": "",
  "signature": "sha256:4f36806f83cf7279fc86ca9b836091da47da1fd8d6b5d20a372df321fe52996e"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Practice Problem - Monte-Carlo Error Propagation"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Part 1"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "You have likely encountered the concept of propagation of uncertainty before (see [the usual rules here](http://en.wikipedia.org/wiki/Propagation_of_uncertainty#Example_formulas)). The idea is that given measurements with uncertainties, we can find the uncertainty on the final result of an equation.\n",
      "\n",
      "For example, let us consider the following equation:\n",
      "\n",
      "$$F = \\frac{G~M_1~M_2}{r^2}$$\n",
      "\n",
      "which gives the gravitational force between two masses $M_1$ and $M_2$ separated by a distance $r$.\n",
      "\n",
      "Let us now imagine that we have two masses:\n",
      "\n",
      "$$M_1=40\\times10^4\\pm0.05\\times10^4\\rm{kg}$$\n",
      "\n",
      "and\n",
      "\n",
      "$$M_2=30\\times10^4\\pm0.1\\times10^4\\rm{kg}$$\n",
      "\n",
      "separated by a distance:\n",
      "\n",
      "$$r=3.2\\pm0.01~\\rm{m}$$\n",
      "\n",
      "where the uncertaintes are the standard deviations of Gaussian distributions which could be e.g. measurement errors.\n",
      "\n",
      "We also know:\n",
      "\n",
      "$$G = 6.67384\\times10^{-11}~\\rm{m}^3~\\rm{kg}^{-1}~\\rm{s}^{-2}$$\n",
      "\n",
      "(exact value, no uncertainty)\n",
      "\n",
      "Use the [standard error propagation rules](http://en.wikipedia.org/wiki/Propagation_of_uncertainty#Example_formulas) to determine the resulting force and uncertainty in your script (you can just derive the equation by hand and implement it in a single line in your code).\n",
      "\n",
      "Now, we can try using a **Monte-Carlo** technique instead. The idea behind Monte-Carlo techniques is to generate many possible solutions using random numbers and using these to look at the overall results. In the above case, you can propagate uncertainties with a Monte-Carlo method by doing the following:\n",
      "\n",
      "* randomly sample values of $M_1$, $M_2$, and $r$, 1000000 times, using the means and standard deviations given above\n",
      "\n",
      "* compute the gravitational force for each set of values\n",
      "\n",
      "You should do this with Numpy arrays, and **without any loops**. You should then get an array of 1000000 different values for the forces.\n",
      "\n",
      "Make a plot of the normalized histogram of these values of the force, and then overplot a Gaussian function with the mean and standard deviation derived with the standard error propagation rules. Make sure that you pick the range of x values in the plot wisely, so that the two distributions can be seen. Make sure there are also a sensible number of bins in the histogram so that you can compare the shape of the histogram and the Gaussian function. The two distributions should agree pretty well."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\n",
      "# Your solution here\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": null
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Part 2"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now repeat the experiment above with the following values:\n",
      "\n",
      "$$M_1=40\\times10^4\\pm2\\times10^4\\rm{kg}$$\n",
      "$$M_2=30\\times10^4\\pm10\\times10^4\\rm{kg}$$\n",
      "$$r=3.2\\pm1.0~\\rm{m}$$\n",
      "\n",
      "and as above, produce a plot.\n",
      "\n",
      "In this case, which method do you think is more accurate? Why? What do you think are the advantages of using a Monte-Carlo technique?"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Solution"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\n",
      "# your solution here\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": null
    }
   ],
   "metadata": {}
  }
 ]
}