{
 "metadata": {
  "name": "12 tables"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Taken from [http://blog.wolfram.com/2013/06/26/is-there-any-point-to-the-12-times-table/](http://blog.wolfram.com/2013/06/26/is-there-any-point-to-the-12-times-table/)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "table = zeros((12, 12), dtype=int16)\n",
      "for i in range(12):\n",
      "    for j in range(12):\n",
      "        table[i, j] = (i + 1) * (j + 1)\n",
      "print table"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[[  1   2   3   4   5   6   7   8   9  10  11  12]\n",
        " [  2   4   6   8  10  12  14  16  18  20  22  24]\n",
        " [  3   6   9  12  15  18  21  24  27  30  33  36]\n",
        " [  4   8  12  16  20  24  28  32  36  40  44  48]\n",
        " [  5  10  15  20  25  30  35  40  45  50  55  60]\n",
        " [  6  12  18  24  30  36  42  48  54  60  66  72]\n",
        " [  7  14  21  28  35  42  49  56  63  70  77  84]\n",
        " [  8  16  24  32  40  48  56  64  72  80  88  96]\n",
        " [  9  18  27  36  45  54  63  72  81  90  99 108]\n",
        " [ 10  20  30  40  50  60  70  80  90 100 110 120]\n",
        " [ 11  22  33  44  55  66  77  88  99 110 121 132]\n",
        " [ 12  24  36  48  60  72  84  96 108 120 132 144]]\n"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The example approximation problem: 7203 * 6892."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print 7203 * 6892\n",
      "print 7000 * 7000\n",
      "print (7000 * 7000 - 7203 * 6892) / float(7203 * 6892) * 100"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "49643076\n",
        "49000000\n",
        "-1.29539918115\n"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "If I knew my 72 times table, I could have made it 7200 * 6900."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print 7203 * 6892\n",
      "print 7200 * 6900\n",
      "print (7200 * 6900 - 7203 * 6892) / float(7203 * 6892) * 100"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "49643076\n",
        "49680000\n",
        "0.074378952666\n"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's build an approximation function that takes into account the lead digit we want to use. This is just the proof of concept."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def approximate(number, lead):\n",
      "    return lead * 10 ** (round(log10(number / lead)))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "approximate(12345, 9)    "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 13,
       "text": [
        "9000.0"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now, let's do the same thing but with a list of lead digits."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def approximate(number, possible_leads):\n",
      "    approx = array([lead * 10 ** (round(log10(number / lead))) for lead in possible_leads])\n",
      "    return approx[abs(number - approx).argmin()]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 30
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "approximate(18345, [1, 2, 3, 4])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 32,
       "text": [
        "20000.0"
       ]
      }
     ],
     "prompt_number": 32
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The approximate product is just the product of approximates."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def approximate_product(a, b, possible_leads):\n",
      "    return approximate(a, possible_leads) * approximate(b, possible_leads)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 33
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "From there follows the relative error."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def relative_error(a, b, possible_leads):\n",
      "    return abs(approximate_product(a, b, possible_leads) / (a * b) - 1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 34
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "relative_error(545, 999, range(1, 11))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 53,
       "text": [
        "0.081650457797246778"
       ]
      }
     ],
     "prompt_number": 53
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "relative_error(549, 999, range(1, 11))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 55,
       "text": [
        "0.088341529142986319"
       ]
      }
     ],
     "prompt_number": 55
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "On to serious things: a random sampling of random numbers and calculating the mean error with 10's table and 12's table."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def mean_error_uniform(max_lead_digit, N=100000):\n",
      "    return mean([relative_error(randint(1, 1e6), randint(1, 1e6), range(1, max_lead_digit + 1)) for i in xrange(N)])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 195
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "mean_error_uniform(10)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 196,
       "text": [
        "0.093972687794082327"
       ]
      }
     ],
     "prompt_number": 196
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "mean_error_uniform(10, 10000)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 197,
       "text": [
        "0.095496434040729439"
       ]
      }
     ],
     "prompt_number": 197
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "mean_error_uniform(12)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 198,
       "text": [
        "0.081577214232046474"
       ]
      }
     ],
     "prompt_number": 198
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "mean_errors = [mean_error_uniform(i, 10000) for i in range(2, 21)]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 199
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(range(2, 21), mean_errors)\n",
      "ylim((0, 0.5))\n",
      "grid(True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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IiLSs1aK+detWUlJSSE5OJj4+nokTJ1JYWNhom/Xr1zNlyhQAhgwZwueff87B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\nSUVORK5CYII=\n"
      }
     ],
     "prompt_number": 200
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "I notice that the routine is quite slow. Maybe I could make it faster with some fancy Numpy trick?\n",
      "\n",
      "- Does approximate support arrays?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "approximate(ones((10, 1)), range(1, 4))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "ename": "TypeError",
       "evalue": "only length-1 arrays can be converted to Python scalars",
       "output_type": "pyerr",
       "traceback": [
        "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m\n\u001b[1;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
        "\u001b[1;32m<ipython-input-75-41cdb1b69303>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m()\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0mapproximate\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mones\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;36m10\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;36m1\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mrange\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;36m1\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;36m4\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
        "\u001b[1;32m<ipython-input-30-95979ec72590>\u001b[0m in \u001b[0;36mapproximate\u001b[1;34m(number, possible_leads)\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[1;32mdef\u001b[0m \u001b[0mapproximate\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mnumber\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mpossible_leads\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 2\u001b[1;33m     \u001b[0mapprox\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0marray\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mlead\u001b[0m \u001b[1;33m*\u001b[0m \u001b[1;36m10\u001b[0m \u001b[1;33m**\u001b[0m \u001b[1;33m(\u001b[0m\u001b[0mround\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mlog10\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mnumber\u001b[0m \u001b[1;33m/\u001b[0m \u001b[0mlead\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;32mfor\u001b[0m \u001b[0mlead\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mpossible_leads\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      3\u001b[0m     \u001b[1;32mreturn\u001b[0m \u001b[0mapprox\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mabs\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mnumber\u001b[0m \u001b[1;33m-\u001b[0m \u001b[0mapprox\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0margmin\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
        "\u001b[1;31mTypeError\u001b[0m: only length-1 arrays can be converted to Python scalars"
       ]
      }
     ],
     "prompt_number": 75
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "No, it doesn't! So let's just use fewer trials in the functions..."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "mean_errors = [mean_error_uniform(i, N=10000) for i in range(2, 51)]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 201
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(range(2, 51), mean_errors)\n",
      "ylim((0, 0.5))\n",
      "grid(True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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dw5QpU7B06VKVRhdc4eHh+OMf/4iDBw9i9+7deOedd3D48GFT5jFo0CBs374d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\nvpWn/sSbOAYAAAAASUVORK5CYII=\n"
      }
     ],
     "prompt_number": 202
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's display the relative improvement for each value."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "mean_errors = array(mean_errors)\n",
      "plot(range(2, 50), (mean_errors[:-1] - mean_errors[1:]) / mean_errors[:-1])\n",
      "ylim((0, 0.6))\n",
      "grid(True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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FmnoTysuBiROBBg+kJCLyK85T9xKHQ31pxr596qvuiIjMwAulXnLFFcBNNwGbN3t/26wX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\n"
      }
     ],
     "prompt_number": 203
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Relative improvement in outcome, per extra fact memorized."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "rng = arange(2, 50)\n",
      "denom = map(lambda n: n ** 2 - (n - 1) ** 2, rng)\n",
      "plot(rng, (mean_errors[:-1] - mean_errors[1:]) / mean_errors[:-1] / (denom))\n",
      "ylim((0, 0.05))\n",
      "grid(True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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3m2MZGRkwdqz5mZFhPhV01N695syn0lJTgAsKzAanKxe41dWZbF5/3WzsKirM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\nAAAAAElFTkSuQmCC\n"
      }
     ],
     "prompt_number": 204
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Simulating with Benford's law: $\\forall d \\in \\[1, 9\\] P(d)=\\log_{10}(1 + \\frac{1}{d})$\n",
      "Steps to follow:\n",
      "\n",
      "- Compute cumulative function of probabilities $G$\n",
      "- for each numbers I want to draw, draw a random number $r$ following $U(0, 1)$, the digit to take is the smallest $a$ such that $G(a) > r$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "probs = log10(1 + 1/arange(1, 10, dtype=float32))\n",
      "G = cumsum(probs)\n",
      "G"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 99,
       "text": [
        "array([ 0.30103001,  0.47712126,  0.60206002,  0.69897002,  0.77815127,\n",
        "        0.84509802,  0.90309   ,  0.95424253,  1.        ], dtype=float32)"
       ]
      }
     ],
     "prompt_number": 99
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "r = rand(1)\n",
      "r"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 120,
       "text": [
        "array([ 0.12225127])"
       ]
      }
     ],
     "prompt_number": 120
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(r < G).argmax()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 121,
       "text": [
        "0"
       ]
      }
     ],
     "prompt_number": 121
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "probs = log10(1 + 1/arange(1, 10, dtype=float32))\n",
      "G = cumsum(probs)\n",
      "def random_benford_number():\n",
      "   return (rand(1) < G).argmax() + 1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 168
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "random_benford_number()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 169,
       "text": [
        "2"
       ]
      }
     ],
     "prompt_number": 169
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "benford_numbers = [random_benford_number() for i in range(1000000)]\n",
      "bar(arange(1, 10), [count_nonzero(benford_numbers == i) for i in arange(1, 10)], label='Empirical')\n",
      "plot(arange(1, 10) , sum(benford_numbers) * log10(2) * log10(1 + 1/arange(1, 10, dtype=float32)), 'or', label=\"Benford's law\")\n",
      "legend()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 174,
       "text": [
        "<matplotlib.legend.Legend at 0xb3558f0>"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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baqPfySIzMxNvvPEGACA7Oxsvv/wytm7delNB3JwcF/ZNROT+4uLiEBcXp7xf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\n8uxZMj+ZQAAAAABJRU5ErkJggg==\n"
      }
     ],
     "prompt_number": 174
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%timeit random_benford_number()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "100000 loops, best of 3: 6.44 us per loop\n"
       ]
      }
     ],
     "prompt_number": 170
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%timeit randint(1, 1e6)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "10000000 loops, best of 3: 114 ns per loop\n"
       ]
      }
     ],
     "prompt_number": 166
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "6.44 / 0.114"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 171,
       "text": [
        "56.49122807017544"
       ]
      }
     ],
     "prompt_number": 171
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def random_integer_benford(n):\n",
      "    r = randint(1, n)\n",
      "    return float(\"\".join([str(random_benford_number()) for digit in str(r)]))\n",
      "        "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 175
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "random_integer_benford(1000)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 189,
       "text": [
        "814.0"
       ]
      }
     ],
     "prompt_number": 189
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Finally, we can rerun the analysis on the data with \"real numbers\"."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def mean_error_benford(max_lead_digit, N=100000):\n",
      "    return mean([relative_error(random_integer_benford(1e6), random_integer_benford(1e6), range(1, max_lead_digit + 1)) for i in xrange(N)])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 205
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "mean_error_benford(10, N=10000)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 206,
       "text": [
        "0.14795717887345317"
       ]
      }
     ],
     "prompt_number": 206
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "mean_errors_benford = [mean_error_benford(i, N=10000) for i in range(2, 21)]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 207
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "mean_errors_benford = array(mean_errors_benford)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 212
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "rng = arange(2, 20)\n",
      "plot(rng + 1, (mean_errors_benford[:-1] - mean_errors_benford[1:]) / mean_errors_benford[:-1])\n",
      "grid(True)\n",
      "ylim((0, 0.5))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 217,
       "text": [
        "(0, 0.5)"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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A1aptSGXXrsDu3cCDD8rP5/bt5p76wLRH6uSc996Td1AqKpI3PFbBuHFy9sFJ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\n"
      }
     ],
     "prompt_number": 217
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "rng = arange(2, 20)\n",
      "denom = map(lambda n: n ** 2 - (n - 1) ** 2, rng)\n",
      "plot(rng, (mean_errors_benford[:-1] - mean_errors_benford[1:]) / mean_errors_benford[:-1] / (denom))\n",
      "ylim((0, 0.05))\n",
      "grid(True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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      }
     ],
     "prompt_number": 218
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The last result is quite different from what is shown on the original link but I'd like to finish here.\n",
      "My conclusions are as follows:\n",
      "\n",
      "- Very interesting blog article\n",
      "- IPython Notebook allows replicating most results\n",
      "- I learned about (and how to simulate) Benford's law\n",
      "- the function I wrote are sloooooooow and it would be lovely to learn (in a more general way) how to make my own functions fast by building more easily on Numpy functions\n",
      "- I often run into confusions because I'm using both Python lists and Numpy arrays. I believe I should try to use only the latter."
     ]
    }
   ],
   "metadata": {}
  }
 ]
}