{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Errors in Sampling"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "We reuse the setup from the computation of $\\pi$, but run this many times for a number of sample counts."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "metadata": {
        "collapsed": true
      },
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 21,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "sample_counts = np.arange(10, 2000, 20)\n",
        "\n",
        "nexperiments = 1000"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 22,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "avg_errors = []\n",
        "\n",
        "for nsamples in sample_counts:\n",
        "   \n",
        "    errors = []\n",
        "    \n",
        "    for experiment in range(nexperiments):\n",
        "        # draw samples\n",
        "        samples = np.random.rand(2, nsamples)\n",
        "        \n",
        "        # compute p\n",
        "        p = np.zeros(nsamples)\n",
        "        r = samples[0]**2 + samples[1]**2\n",
        "        p[r<=1] = 1\n",
        "        \n",
        "        approx_pi = 4*np.sum(p) / nsamples\n",
        "        errors.append(np.abs(approx_pi - np.pi))\n",
        "        \n",
        "    avg_errors.append(np.sum(errors)/nexperiments)\n",
        "    \n",
        "avg_errors = np.array(avg_errors)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 23,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAY8AAAEWCAYAAACe8xtsAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\ngg==\n",
            "text/plain": [
              "<matplotlib.figure.Figure at 0x7f0b53e695c0>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "plt.loglog(sample_counts, avg_errors)\n",
        "plt.xlabel(\"# Samples\")\n",
        "plt.ylabel(\"Error\")\n",
        "plt.grid()\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Empirically: Average error\n",
        "\n",
        "$$E=O(\\frac1{\\sqrt n})$$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Precise statement: [Central Limit Theorem](https://en.wikipedia.org/wiki/Central_limit_theorem)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": true
      },
      "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": "3.5.2+"
    },
    "widgets": {
      "state": {},
      "version": "1.1.2"
    }
  },
  "nbformat": 4,
  "nbformat_minor": 0
}