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   "source": [
    "# Practice Problem - Monte-Carlo Error Propagation"
   ]
  },
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   "metadata": {},
   "source": [
    "## Part 1"
   ]
  },
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   "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",
    "$$M_1=(40\\pm0.05)\\times10^4\\,\\mathrm{kg}$$\n",
    "\n",
    "and\n",
    "\n",
    "$$M_2=(30\\pm0.1)\\times10^4\\,\\mathrm{kg}$$\n",
    "\n",
    "separated by a distance:\n",
    "\n",
    "$$r=3.2\\pm0.01\\,\\mathrm{m}$$\n",
    "\n",
    "where the uncertaintes are normally distributed (Gaussian measurement uncertainties).\n",
    "\n",
    "We also know:\n",
    "\n",
    "$$G = 6.67384\\times10^{-11}\\,\\mathrm{m}^3\\,\\mathrm{kg}^{-1}\\,\\mathrm{s}^{-2}$$\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 1000000 times values of $M_1$, $M_2$, and $r$ 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."
   ]
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   "source": [
    "\n",
    "# Your solution here\n"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Part 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now repeat the experiment above with the following values:\n",
    "\n",
    "$$M_1=(40\\pm2)\\times10^4\\,\\mathrm{kg}$$\n",
    "$$M_2=(30\\pm10)\\times10^4\\,\\mathrm{kg}$$\n",
    "$$r=(3.2\\pm1.0)\\,\\mathrm{m}$$\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": "markdown",
   "metadata": {},
   "source": [
    "## Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "# your solution here\n"
   ]
  }
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