03. Statistics.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Statistics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Statistics\n",
    "using StatsBase\n",
    "using RDatasets\n",
    "using Plots\n",
    "using StatsPlots\n",
    "using KernelDensity\n",
    "using Distributions\n",
    "using LinearAlgebra\n",
    "using HypothesisTests\n",
    "using PyCall\n",
    "using MLBase"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Getting the data\n",
    "\n",
    "The *RDatasets* package contains the R datasets."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "names(D) = [\"Eruptions\", \"Waiting\"]\n",
      "typeof(D) = DataFrame\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<table class=\"data-frame\"><thead><tr><th></th><th>Eruptions</th><th>Waiting</th></tr><tr><th></th><th>Float64</th><th>Int64</th></tr></thead><tbody><p>272 rows × 2 columns</p><tr><th>1</th><td>3.6</td><td>79</td></tr><tr><th>2</th><td>1.8</td><td>54</td></tr><tr><th>3</th><td>3.333</td><td>74</td></tr><tr><th>4</th><td>2.283</td><td>62</td></tr><tr><th>5</th><td>4.533</td><td>85</td></tr><tr><th>6</th><td>2.883</td><td>55</td></tr><tr><th>7</th><td>4.7</td><td>88</td></tr><tr><th>8</th><td>3.6</td><td>85</td></tr><tr><th>9</th><td>1.95</td><td>51</td></tr><tr><th>10</th><td>4.35</td><td>85</td></tr><tr><th>11</th><td>1.833</td><td>54</td></tr><tr><th>12</th><td>3.917</td><td>84</td></tr><tr><th>13</th><td>4.2</td><td>78</td></tr><tr><th>14</th><td>1.75</td><td>47</td></tr><tr><th>15</th><td>4.7</td><td>83</td></tr><tr><th>16</th><td>2.167</td><td>52</td></tr><tr><th>17</th><td>1.75</td><td>62</td></tr><tr><th>18</th><td>4.8</td><td>84</td></tr><tr><th>19</th><td>1.6</td><td>52</td></tr><tr><th>20</th><td>4.25</td><td>79</td></tr><tr><th>21</th><td>1.8</td><td>51</td></tr><tr><th>22</th><td>1.75</td><td>47</td></tr><tr><th>23</th><td>3.45</td><td>78</td></tr><tr><th>24</th><td>3.067</td><td>69</td></tr><tr><th>25</th><td>4.533</td><td>74</td></tr><tr><th>26</th><td>3.6</td><td>83</td></tr><tr><th>27</th><td>1.967</td><td>55</td></tr><tr><th>28</th><td>4.083</td><td>76</td></tr><tr><th>29</th><td>3.85</td><td>78</td></tr><tr><th>30</th><td>4.433</td><td>79</td></tr><tr><th>&vellip;</th><td>&vellip;</td><td>&vellip;</td></tr></tbody></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{r|cc}\n",
       "\t& Eruptions & Waiting\\\\\n",
       "\t\\hline\n",
       "\t& Float64 & Int64\\\\\n",
       "\t\\hline\n",
       "\t1 & 3.6 & 79 \\\\\n",
       "\t2 & 1.8 & 54 \\\\\n",
       "\t3 & 3.333 & 74 \\\\\n",
       "\t4 & 2.283 & 62 \\\\\n",
       "\t5 & 4.533 & 85 \\\\\n",
       "\t6 & 2.883 & 55 \\\\\n",
       "\t7 & 4.7 & 88 \\\\\n",
       "\t8 & 3.6 & 85 \\\\\n",
       "\t9 & 1.95 & 51 \\\\\n",
       "\t10 & 4.35 & 85 \\\\\n",
       "\t11 & 1.833 & 54 \\\\\n",
       "\t12 & 3.917 & 84 \\\\\n",
       "\t13 & 4.2 & 78 \\\\\n",
       "\t14 & 1.75 & 47 \\\\\n",
       "\t15 & 4.7 & 83 \\\\\n",
       "\t16 & 2.167 & 52 \\\\\n",
       "\t17 & 1.75 & 62 \\\\\n",
       "\t18 & 4.8 & 84 \\\\\n",
       "\t19 & 1.6 & 52 \\\\\n",
       "\t20 & 4.25 & 79 \\\\\n",
       "\t21 & 1.8 & 51 \\\\\n",
       "\t22 & 1.75 & 47 \\\\\n",
       "\t23 & 3.45 & 78 \\\\\n",
       "\t24 & 3.067 & 69 \\\\\n",
       "\t25 & 4.533 & 74 \\\\\n",
       "\t26 & 3.6 & 83 \\\\\n",
       "\t27 & 1.967 & 55 \\\\\n",
       "\t28 & 4.083 & 76 \\\\\n",
       "\t29 & 3.85 & 78 \\\\\n",
       "\t30 & 4.433 & 79 \\\\\n",
       "\t$\\dots$ & $\\dots$ & $\\dots$ \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "272×2 DataFrame\n",
       "│ Row │ Eruptions │ Waiting │\n",
       "│     │ \u001b[90mFloat64\u001b[39m   │ \u001b[90mInt64\u001b[39m   │\n",
       "├─────┼───────────┼─────────┤\n",
       "│ 1   │ 3.6       │ 79      │\n",
       "│ 2   │ 1.8       │ 54      │\n",
       "│ 3   │ 3.333     │ 74      │\n",
       "│ 4   │ 2.283     │ 62      │\n",
       "│ 5   │ 4.533     │ 85      │\n",
       "│ 6   │ 2.883     │ 55      │\n",
       "│ 7   │ 4.7       │ 88      │\n",
       "│ 8   │ 3.6       │ 85      │\n",
       "│ 9   │ 1.95      │ 51      │\n",
       "│ 10  │ 4.35      │ 85      │\n",
       "⋮\n",
       "│ 262 │ 4.533     │ 84      │\n",
       "│ 263 │ 1.85      │ 58      │\n",
       "│ 264 │ 4.25      │ 83      │\n",
       "│ 265 │ 1.983     │ 43      │\n",
       "│ 266 │ 2.25      │ 60      │\n",
       "│ 267 │ 4.75      │ 75      │\n",
       "│ 268 │ 4.117     │ 81      │\n",
       "│ 269 │ 2.15      │ 46      │\n",
       "│ 270 │ 4.417     │ 90      │\n",
       "│ 271 │ 1.817     │ 46      │\n",
       "│ 272 │ 4.467     │ 74      │"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D = RDatasets.dataset(\"datasets\", \"faithful\")\n",
    "@show names(D)\n",
    "@show typeof(D)\n",
    "D"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"data-frame\"><thead><tr><th></th><th>variable</th><th>mean</th><th>min</th><th>median</th><th>max</th><th>nunique</th><th>nmissing</th><th>eltype</th></tr><tr><th></th><th>Symbol</th><th>Float64</th><th>Real</th><th>Float64</th><th>Real</th><th>Nothing</th><th>Nothing</th><th>DataType</th></tr></thead><tbody><p>2 rows × 8 columns</p><tr><th>1</th><td>Eruptions</td><td>3.48778</td><td>1.6</td><td>4.0</td><td>5.1</td><td></td><td></td><td>Float64</td></tr><tr><th>2</th><td>Waiting</td><td>70.8971</td><td>43</td><td>76.0</td><td>96</td><td></td><td></td><td>Int64</td></tr></tbody></table>"
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       "\\begin{tabular}{r|cccccccc}\n",
       "\t& variable & mean & min & median & max & nunique & nmissing & eltype\\\\\n",
       "\t\\hline\n",
       "\t& Symbol & Float64 & Real & Float64 & Real & Nothing & Nothing & DataType\\\\\n",
       "\t\\hline\n",
       "\t1 & Eruptions & 3.48778 & 1.6 & 4.0 & 5.1 &  &  & Float64 \\\\\n",
       "\t2 & Waiting & 70.8971 & 43 & 76.0 & 96 &  &  & Int64 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "2×8 DataFrame. Omitted printing of 1 columns\n",
       "│ Row │ variable  │ mean    │ min  │ median  │ max  │ nunique │ nmissing │\n",
       "│     │ \u001b[90mSymbol\u001b[39m    │ \u001b[90mFloat64\u001b[39m │ \u001b[90mReal\u001b[39m │ \u001b[90mFloat64\u001b[39m │ \u001b[90mReal\u001b[39m │ \u001b[90mNothing\u001b[39m │ \u001b[90mNothing\u001b[39m  │\n",
       "├─────┼───────────┼─────────┼──────┼─────────┼──────┼─────────┼──────────┤\n",
       "│ 1   │ Eruptions │ 3.48778 │ 1.6  │ 4.0     │ 5.1  │         │          │\n",
       "│ 2   │ Waiting   │ 70.8971 │ 43   │ 76.0    │ 96   │         │          │"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "describe(D)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
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      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "wait_times = D.Waiting\n",
    "eruptions = D.Eruptions\n",
    "scatter(eruptions, label=\"eruptions\", color=\"orange\")\n",
    "#scatter!(wait_times, label=\"wait time\", color=\"blue\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Statistics plots"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Boxplot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{verbatim}\n",
       "boxplot(x, y)\n",
       "boxplot!(x, y)\n",
       "\\end{verbatim}\n",
       "Make a box and whisker plot.\n",
       "\n",
       "\\section{Keyword arguments}\n",
       "\\begin{itemize}\n",
       "\\item \\texttt{notch}: Bool. Notch the box plot? (false)\n",
       "\n",
       "\n",
       "\\item \\texttt{range}: Real. Values more than range*IQR below the first quartile          or above the third quartile are shown as outliers (1.5)\n",
       "\n",
       "\n",
       "\\item \\texttt{outliers}: Bool. Show outliers? (true)\n",
       "\n",
       "\n",
       "\\item \\texttt{whisker\\_width}: Real or Symbol. Length of whiskers (:match)\n",
       "\n",
       "\\end{itemize}\n",
       "\\section{Example}\n",
       "\\begin{verbatim}\n",
       "julia> using StatsPlots\n",
       "julia> boxplot(repeat([1,2,3],outer=100),randn(300))\n",
       "\\end{verbatim}\n"
      ],
      "text/markdown": [
       "```\n",
       "boxplot(x, y)\n",
       "boxplot!(x, y)\n",
       "```\n",
       "\n",
       "Make a box and whisker plot.\n",
       "\n",
       "# Keyword arguments\n",
       "\n",
       "  * `notch`: Bool. Notch the box plot? (false)\n",
       "  * `range`: Real. Values more than range*IQR below the first quartile          or above the third quartile are shown as outliers (1.5)\n",
       "  * `outliers`: Bool. Show outliers? (true)\n",
       "  * `whisker_width`: Real or Symbol. Length of whiskers (:match)\n",
       "\n",
       "# Example\n",
       "\n",
       "```julia-repl\n",
       "julia> using StatsPlots\n",
       "julia> boxplot(repeat([1,2,3],outer=100),randn(300))\n",
       "```\n"
      ],
      "text/plain": [
       "\u001b[36m  boxplot(x, y)\u001b[39m\n",
       "\u001b[36m  boxplot!(x, y)\u001b[39m\n",
       "\n",
       "  Make a box and whisker plot.\n",
       "\n",
       "\u001b[1m  Keyword arguments\u001b[22m\n",
       "\u001b[1m  ≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡\u001b[22m\n",
       "\n",
       "    •    \u001b[36mnotch\u001b[39m: Bool. Notch the box plot? (false)\n",
       "\n",
       "    •    \u001b[36mrange\u001b[39m: Real. Values more than range*IQR below the first quartile\n",
       "        or above the third quartile are shown as outliers (1.5)\n",
       "\n",
       "    •    \u001b[36moutliers\u001b[39m: Bool. Show outliers? (true)\n",
       "\n",
       "    •    \u001b[36mwhisker_width\u001b[39m: Real or Symbol. Length of whiskers (:match)\n",
       "\n",
       "\u001b[1m  Example\u001b[22m\n",
       "\u001b[1m  ≡≡≡≡≡≡≡≡≡\u001b[22m\n",
       "\n",
       "\u001b[36m  julia> using StatsPlots\u001b[39m\n",
       "\u001b[36m  julia> boxplot(repeat([1,2,3],outer=100),randn(300))\u001b[39m"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "?StatsPlots.boxplot "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
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       "</svg>\n"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "boxplot([\"eruption length\"], eruptions, legend=false, size=(200,400), whisker_width=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Histogram"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{verbatim}\n",
       "histogram(x)\n",
       "histogram!(x)\n",
       "\\end{verbatim}\n",
       "Plot a histogram.\n",
       "\n",
       "\\section{Arguments}\n",
       "\\begin{itemize}\n",
       "\\item \\texttt{x}: AbstractVector of values to be binned\n",
       "\n",
       "\n",
       "\\item \\texttt{bins}: Integer, NTuple\\{2,Integer\\}, AbstractVector or          Symbol. Default is :auto (the Freedman-Diaconis rule). For          histogram-types, defines the approximate number of bins to aim for,          or the auto-binning algorithm to use (:sturges,          :sqrt, :rice, :scott or :fd). For fine-grained control          pass a Vector of break values, e.g. \\texttt{range(minimum(x),          stop = maximum(x), length = 25)}\n",
       "\n",
       "\n",
       "\\item \\texttt{weights}: Vector of weights for the values in \\texttt{x}, for weighted bin counts\n",
       "\n",
       "\n",
       "\\item \\texttt{normalize}: Bool or Symbol. Histogram normalization mode.               Possible values are: false/:none (no normalization,               default), true/:pdf (normalize to a discrete               Probability Density Function, where the total area of the               bins is 1), :probability (bin heights sum to 1)               and :density (the area of each bin, rather than               the height, is equal to the counts - useful for               uneven bin sizes).\n",
       "\n",
       "\n",
       "\\item \\texttt{bar\\_position}: Symbol.  Choose from \\texttt{:overlay} (default),                  \\texttt{:stack}.  (warning: May not be implemented fully)\n",
       "\n",
       "\n",
       "\\item \\texttt{bar\\_width}: nothing or Number. Width of bars in data               coordinates. When nothing, chooses based on x (or y when               \\texttt{orientation = :h}).\n",
       "\n",
       "\n",
       "\\item \\texttt{bar\\_edges}: Bool.  Align bars to edges (true), or centers               (the default)?\n",
       "\n",
       "\n",
       "\\item \\texttt{orientation}: Symbol.  Horizontal or vertical orientation                 for bar types.  Values \\texttt{:h}, \\texttt{:hor},                 \\texttt{:horizontal} correspond to horizontal (sideways,                 anchored to y-axis), and \\texttt{:v}, \\texttt{:vert}, and                 \\texttt{:vertical} correspond to vertical (the default).\n",
       "\n",
       "\\end{itemize}\n",
       "\\section{Example}\n",
       "\\begin{verbatim}\n",
       "julia> histogram([1,2,1,1,4,3,8],bins=0:8)\n",
       "\\end{verbatim}\n"
      ],
      "text/markdown": [
       "```\n",
       "histogram(x)\n",
       "histogram!(x)\n",
       "```\n",
       "\n",
       "Plot a histogram.\n",
       "\n",
       "# Arguments\n",
       "\n",
       "  * `x`: AbstractVector of values to be binned\n",
       "  * `bins`: Integer, NTuple{2,Integer}, AbstractVector or          Symbol. Default is :auto (the Freedman-Diaconis rule). For          histogram-types, defines the approximate number of bins to aim for,          or the auto-binning algorithm to use (:sturges,          :sqrt, :rice, :scott or :fd). For fine-grained control          pass a Vector of break values, e.g. `range(minimum(x),          stop = maximum(x), length = 25)`\n",
       "  * `weights`: Vector of weights for the values in `x`, for weighted bin counts\n",
       "  * `normalize`: Bool or Symbol. Histogram normalization mode.               Possible values are: false/:none (no normalization,               default), true/:pdf (normalize to a discrete               Probability Density Function, where the total area of the               bins is 1), :probability (bin heights sum to 1)               and :density (the area of each bin, rather than               the height, is equal to the counts - useful for               uneven bin sizes).\n",
       "  * `bar_position`: Symbol.  Choose from `:overlay` (default),                  `:stack`.  (warning: May not be implemented fully)\n",
       "  * `bar_width`: nothing or Number. Width of bars in data               coordinates. When nothing, chooses based on x (or y when               `orientation = :h`).\n",
       "  * `bar_edges`: Bool.  Align bars to edges (true), or centers               (the default)?\n",
       "  * `orientation`: Symbol.  Horizontal or vertical orientation                 for bar types.  Values `:h`, `:hor`,                 `:horizontal` correspond to horizontal (sideways,                 anchored to y-axis), and `:v`, `:vert`, and                 `:vertical` correspond to vertical (the default).\n",
       "\n",
       "# Example\n",
       "\n",
       "```julia-repl\n",
       "julia> histogram([1,2,1,1,4,3,8],bins=0:8)\n",
       "```\n"
      ],
      "text/plain": [
       "\u001b[36m  histogram(x)\u001b[39m\n",
       "\u001b[36m  histogram!(x)\u001b[39m\n",
       "\n",
       "  Plot a histogram.\n",
       "\n",
       "\u001b[1m  Arguments\u001b[22m\n",
       "\u001b[1m  ≡≡≡≡≡≡≡≡≡≡≡\u001b[22m\n",
       "\n",
       "    •    \u001b[36mx\u001b[39m: AbstractVector of values to be binned\n",
       "\n",
       "    •    \u001b[36mbins\u001b[39m: Integer, NTuple{2,Integer}, AbstractVector or Symbol.\n",
       "        Default is :auto (the Freedman-Diaconis rule). For\n",
       "        histogram-types, defines the approximate number of bins to aim\n",
       "        for, or the auto-binning algorithm to use (:sturges, :sqrt, :rice,\n",
       "        :scott or :fd). For fine-grained control pass a Vector of break\n",
       "        values, e.g. \u001b[36mrange(minimum(x),          stop = maximum(x), length\n",
       "        = 25)\u001b[39m\n",
       "\n",
       "    •    \u001b[36mweights\u001b[39m: Vector of weights for the values in \u001b[36mx\u001b[39m, for weighted bin\n",
       "        counts\n",
       "\n",
       "    •    \u001b[36mnormalize\u001b[39m: Bool or Symbol. Histogram normalization mode. Possible\n",
       "        values are: false/:none (no normalization, default), true/:pdf\n",
       "        (normalize to a discrete Probability Density Function, where the\n",
       "        total area of the bins is 1), :probability (bin heights sum to 1)\n",
       "        and :density (the area of each bin, rather than the height, is\n",
       "        equal to the counts - useful for uneven bin sizes).\n",
       "\n",
       "    •    \u001b[36mbar_position\u001b[39m: Symbol. Choose from \u001b[36m:overlay\u001b[39m (default), \u001b[36m:stack\u001b[39m.\n",
       "        (warning: May not be implemented fully)\n",
       "\n",
       "    •    \u001b[36mbar_width\u001b[39m: nothing or Number. Width of bars in data coordinates.\n",
       "        When nothing, chooses based on x (or y when \u001b[36morientation = :h\u001b[39m).\n",
       "\n",
       "    •    \u001b[36mbar_edges\u001b[39m: Bool. Align bars to edges (true), or centers (the\n",
       "        default)?\n",
       "\n",
       "    •    \u001b[36morientation\u001b[39m: Symbol. Horizontal or vertical orientation for bar\n",
       "        types. Values \u001b[36m:h\u001b[39m, \u001b[36m:hor\u001b[39m, \u001b[36m:horizontal\u001b[39m correspond to horizontal\n",
       "        (sideways, anchored to y-axis), and \u001b[36m:v\u001b[39m, \u001b[36m:vert\u001b[39m, and \u001b[36m:vertical\u001b[39m\n",
       "        correspond to vertical (the default).\n",
       "\n",
       "\u001b[1m  Example\u001b[22m\n",
       "\u001b[1m  ≡≡≡≡≡≡≡≡≡\u001b[22m\n",
       "\n",
       "\u001b[36m  julia> histogram([1,2,1,1,4,3,8],bins=0:8)\u001b[39m"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "?StatsPlots.histogram"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
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     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "histogram(eruptions, bins=:sqrt, orientation=:v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Correlation plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{verbatim}\n",
       "corrplot\n",
       "\\end{verbatim}\n",
       "This plot type shows the correlation among input variables.   A correlation plot may be produced by a matrix.\n",
       "\n",
       "A correlation matrix can also be created from the columns of a \\texttt{DataFrame} using the \\href{@ref}{\\texttt{@df}} macro like so:\n",
       "\n",
       "\\begin{verbatim}\n",
       "@df iris corrplot([:SepalLength :SepalWidth :PetalLength :PetalWidth])\n",
       "\\end{verbatim}\n",
       "The marker color in scatter plots reveals the degree of correlation.  Pass the desired colorgradient to \\texttt{markercolor}.\n",
       "\n",
       "With the default gradient positive correlations are blue, neutral are yellow  and negative are red. In the 2d-histograms, the color gradient shows the frequency  of points in that bin (as usual, controlled by \\texttt{seriescolor}).\n",
       "\n"
      ],
      "text/markdown": [
       "```\n",
       "corrplot\n",
       "```\n",
       "\n",
       "This plot type shows the correlation among input variables.   A correlation plot may be produced by a matrix.\n",
       "\n",
       "A correlation matrix can also be created from the columns of a `DataFrame` using the [`@df`](@ref) macro like so:\n",
       "\n",
       "```julia\n",
       "@df iris corrplot([:SepalLength :SepalWidth :PetalLength :PetalWidth])\n",
       "```\n",
       "\n",
       "The marker color in scatter plots reveals the degree of correlation.  Pass the desired colorgradient to `markercolor`.\n",
       "\n",
       "With the default gradient positive correlations are blue, neutral are yellow  and negative are red. In the 2d-histograms, the color gradient shows the frequency  of points in that bin (as usual, controlled by `seriescolor`).\n"
      ],
      "text/plain": [
       "\u001b[36m  corrplot\u001b[39m\n",
       "\n",
       "  This plot type shows the correlation among input variables. A correlation\n",
       "  plot may be produced by a matrix.\n",
       "\n",
       "  A correlation matrix can also be created from the columns of a \u001b[36mDataFrame\u001b[39m\n",
       "  using the \u001b[36m@df\u001b[39m macro like so:\n",
       "\n",
       "\u001b[36m  @df iris corrplot([:SepalLength :SepalWidth :PetalLength :PetalWidth])\u001b[39m\n",
       "\n",
       "  The marker color in scatter plots reveals the degree of correlation. Pass\n",
       "  the desired colorgradient to \u001b[36mmarkercolor\u001b[39m.\n",
       "\n",
       "  With the default gradient positive correlations are blue, neutral are yellow\n",
       "  and negative are red. In the 2d-histograms, the color gradient shows the\n",
       "  frequency of points in that bin (as usual, controlled by \u001b[36mseriescolor\u001b[39m)."
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "?StatsPlots.corrplot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot = @df D corrplot([:Eruptions :Waiting]);\n",
    "plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{verbatim}\n",
       "cor(x::AbstractVector)\n",
       "\\end{verbatim}\n",
       "Return the number one.\n",
       "\n",
       "\\rule{\\textwidth}{1pt}\n",
       "\\begin{verbatim}\n",
       "cor(X::AbstractMatrix; dims::Int=1)\n",
       "\\end{verbatim}\n",
       "Compute the Pearson correlation matrix of the matrix \\texttt{X} along the dimension \\texttt{dims}.\n",
       "\n",
       "\\rule{\\textwidth}{1pt}\n",
       "\\begin{verbatim}\n",
       "cor(x::AbstractVector, y::AbstractVector)\n",
       "\\end{verbatim}\n",
       "Compute the Pearson correlation between the vectors \\texttt{x} and \\texttt{y}.\n",
       "\n",
       "\\rule{\\textwidth}{1pt}\n",
       "\\begin{verbatim}\n",
       "cor(X::AbstractVecOrMat, Y::AbstractVecOrMat; dims=1)\n",
       "\\end{verbatim}\n",
       "Compute the Pearson correlation between the vectors or matrices \\texttt{X} and \\texttt{Y} along the dimension \\texttt{dims}.\n",
       "\n",
       "\\rule{\\textwidth}{1pt}\n",
       "\\begin{verbatim}\n",
       "cor(X, w::AbstractWeights, dims=1)\n",
       "\\end{verbatim}\n",
       "Compute the Pearson correlation matrix of \\texttt{X} along the dimension \\texttt{dims} with a weighting \\texttt{w} .\n",
       "\n",
       "\\rule{\\textwidth}{1pt}\n",
       "\\begin{verbatim}\n",
       "cor(d::MultivariateDistribution)\n",
       "\\end{verbatim}\n",
       "Computes the correlation matrix for distribution \\texttt{d}.\n",
       "\n"
      ],
      "text/markdown": [
       "```\n",
       "cor(x::AbstractVector)\n",
       "```\n",
       "\n",
       "Return the number one.\n",
       "\n",
       "---\n",
       "\n",
       "```\n",
       "cor(X::AbstractMatrix; dims::Int=1)\n",
       "```\n",
       "\n",
       "Compute the Pearson correlation matrix of the matrix `X` along the dimension `dims`.\n",
       "\n",
       "---\n",
       "\n",
       "```\n",
       "cor(x::AbstractVector, y::AbstractVector)\n",
       "```\n",
       "\n",
       "Compute the Pearson correlation between the vectors `x` and `y`.\n",
       "\n",
       "---\n",
       "\n",
       "```\n",
       "cor(X::AbstractVecOrMat, Y::AbstractVecOrMat; dims=1)\n",
       "```\n",
       "\n",
       "Compute the Pearson correlation between the vectors or matrices `X` and `Y` along the dimension `dims`.\n",
       "\n",
       "---\n",
       "\n",
       "```\n",
       "cor(X, w::AbstractWeights, dims=1)\n",
       "```\n",
       "\n",
       "Compute the Pearson correlation matrix of `X` along the dimension `dims` with a weighting `w` .\n",
       "\n",
       "---\n",
       "\n",
       "```\n",
       "cor(d::MultivariateDistribution)\n",
       "```\n",
       "\n",
       "Computes the correlation matrix for distribution `d`.\n"
      ],
      "text/plain": [
       "\u001b[36m  cor(x::AbstractVector)\u001b[39m\n",
       "\n",
       "  Return the number one.\n",
       "\n",
       "  ────────────────────────────────────────────────────────────────────────────\n",
       "\n",
       "\u001b[36m  cor(X::AbstractMatrix; dims::Int=1)\u001b[39m\n",
       "\n",
       "  Compute the Pearson correlation matrix of the matrix \u001b[36mX\u001b[39m along the dimension\n",
       "  \u001b[36mdims\u001b[39m.\n",
       "\n",
       "  ────────────────────────────────────────────────────────────────────────────\n",
       "\n",
       "\u001b[36m  cor(x::AbstractVector, y::AbstractVector)\u001b[39m\n",
       "\n",
       "  Compute the Pearson correlation between the vectors \u001b[36mx\u001b[39m and \u001b[36my\u001b[39m.\n",
       "\n",
       "  ────────────────────────────────────────────────────────────────────────────\n",
       "\n",
       "\u001b[36m  cor(X::AbstractVecOrMat, Y::AbstractVecOrMat; dims=1)\u001b[39m\n",
       "\n",
       "  Compute the Pearson correlation between the vectors or matrices \u001b[36mX\u001b[39m and \u001b[36mY\u001b[39m\n",
       "  along the dimension \u001b[36mdims\u001b[39m.\n",
       "\n",
       "  ────────────────────────────────────────────────────────────────────────────\n",
       "\n",
       "\u001b[36m  cor(X, w::AbstractWeights, dims=1)\u001b[39m\n",
       "\n",
       "  Compute the Pearson correlation matrix of \u001b[36mX\u001b[39m along the dimension \u001b[36mdims\u001b[39m with a\n",
       "  weighting \u001b[36mw\u001b[39m .\n",
       "\n",
       "  ────────────────────────────────────────────────────────────────────────────\n",
       "\n",
       "\u001b[36m  cor(d::MultivariateDistribution)\u001b[39m\n",
       "\n",
       "  Computes the correlation matrix for distribution \u001b[36md\u001b[39m."
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "?Statistics.cor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9008111683218128"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cor(eruptions, wait_times)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. Kernel density estimates"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can fit a kernel density estimation function to a dataset using the KernelDensity.jl package."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "UnivariateKDE{StepRangeLen{Float64,Base.TwicePrecision{Float64},Base.TwicePrecision{Float64}}}(0.2608918621442271:0.0030181808870110138:6.439108137855772, [1.1161139442217305e-5, 1.1244427155304715e-5, 1.1344179473224414e-5, 1.146054328619861e-5, 1.1593684794627968e-5, 1.1743789689266926e-5, 1.1911063348919726e-5, 1.209573105311057e-5, 1.2298038211913664e-5, 1.2518250611666393e-5  …  1.1215022627797389e-5, 1.113759304488017e-5, 1.1076192604054722e-5, 1.1030781609218734e-5, 1.1001338720893414e-5, 1.0987860983070058e-5, 1.0990363864676889e-5, 1.1008881316295835e-5, 1.1043465842321826e-5, 1.1094188587988663e-5])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = kde(eruptions)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we want the histogram and the kernel density graph to be aligned we need to remember that the \"density contribution\" of every point added to one of these histograms is 1/(nb of elements)*bin width."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
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       "  334.694,1444.82 335.704,1444.79 336.713,1444.76 337.723,1444.73 338.733,1444.7 339.743,1444.67 340.753,1444.63 341.763,1444.6 342.772,1444.56 343.782,1444.53 \n",
       "  344.792,1444.49 345.802,1444.45 346.812,1444.41 347.822,1444.37 348.832,1444.33 349.841,1444.29 350.851,1444.24 351.861,1444.2 352.871,1444.15 353.881,1444.1 \n",
       "  354.891,1444.06 355.9,1444 356.91,1443.95 357.92,1443.9 358.93,1443.84 359.94,1443.79 360.95,1443.73 361.959,1443.67 362.969,1443.61 363.979,1443.55 \n",
       "  364.989,1443.48 365.999,1443.41 367.009,1443.35 368.018,1443.28 369.028,1443.2 370.038,1443.13 371.048,1443.05 372.058,1442.98 373.068,1442.9 374.077,1442.81 \n",
       "  375.087,1442.73 376.097,1442.64 377.107,1442.55 378.117,1442.46 379.127,1442.37 380.136,1442.27 381.146,1442.17 382.156,1442.07 383.166,1441.97 384.176,1441.86 \n",
       "  385.186,1441.75 386.195,1441.64 387.205,1441.53 388.215,1441.41 389.225,1441.29 390.235,1441.17 391.245,1441.04 392.254,1440.91 393.264,1440.78 394.274,1440.64 \n",
       "  395.284,1440.5 396.294,1440.36 397.304,1440.21 398.313,1440.06 399.323,1439.91 400.333,1439.75 401.343,1439.59 402.353,1439.42 403.363,1439.25 404.372,1439.08 \n",
       "  405.382,1438.9 406.392,1438.72 407.402,1438.53 408.412,1438.34 409.422,1438.14 410.431,1437.94 411.441,1437.74 412.451,1437.53 413.461,1437.32 414.471,1437.1 \n",
       "  415.481,1436.87 416.49,1436.64 417.5,1436.41 418.51,1436.17 419.52,1435.92 420.53,1435.67 421.54,1435.41 422.549,1435.15 423.559,1434.88 424.569,1434.61 \n",
       "  425.579,1434.32 426.589,1434.04 427.599,1433.74 428.608,1433.44 429.618,1433.14 430.628,1432.82 431.638,1432.5 432.648,1432.18 433.658,1431.84 434.667,1431.5 \n",
       "  435.677,1431.15 436.687,1430.8 437.697,1430.43 438.707,1430.06 439.717,1429.68 440.726,1429.29 441.736,1428.9 442.746,1428.49 443.756,1428.08 444.766,1427.66 \n",
       "  445.776,1427.23 446.785,1426.79 447.795,1426.34 448.805,1425.89 449.815,1425.42 450.825,1424.95 451.835,1424.46 452.844,1423.97 453.854,1423.46 454.864,1422.95 \n",
       "  455.874,1422.42 456.884,1421.89 457.894,1421.34 458.903,1420.79 459.913,1420.22 460.923,1419.64 461.933,1419.05 462.943,1418.45 463.953,1417.84 464.962,1417.22 \n",
       "  465.972,1416.58 466.982,1415.93 467.992,1415.27 469.002,1414.6 470.012,1413.92 471.021,1413.22 472.031,1412.51 473.041,1411.79 474.051,1411.05 475.061,1410.3 \n",
       "  476.071,1409.54 477.08,1408.76 478.09,1407.97 479.1,1407.16 480.11,1406.34 481.12,1405.51 482.13,1404.66 483.139,1403.8 484.149,1402.92 485.159,1402.02 \n",
       "  486.169,1401.12 487.179,1400.19 488.189,1399.25 489.198,1398.29 490.208,1397.32 491.218,1396.33 492.228,1395.32 493.238,1394.3 494.248,1393.26 495.257,1392.2 \n",
       "  496.267,1391.13 497.277,1390.04 498.287,1388.93 499.297,1387.8 500.307,1386.66 501.316,1385.49 502.326,1384.31 503.336,1383.11 504.346,1381.89 505.356,1380.65 \n",
       "  506.366,1379.39 507.375,1378.11 508.385,1376.82 509.395,1375.5 510.405,1374.16 511.415,1372.81 512.425,1371.43 513.434,1370.03 514.444,1368.61 515.454,1367.17 \n",
       "  516.464,1365.71 517.474,1364.23 518.484,1362.72 519.493,1361.2 520.503,1359.65 521.513,1358.08 522.523,1356.49 523.533,1354.87 524.543,1353.24 525.552,1351.58 \n",
       "  526.562,1349.9 527.572,1348.19 528.582,1346.46 529.592,1344.71 530.602,1342.93 531.611,1341.13 532.621,1339.31 533.631,1337.46 534.641,1335.59 535.651,1333.7 \n",
       "  536.661,1331.78 537.67,1329.83 538.68,1327.86 539.69,1325.87 540.7,1323.85 541.71,1321.8 542.72,1319.73 543.729,1317.64 544.739,1315.52 545.749,1313.37 \n",
       "  546.759,1311.2 547.769,1309 548.779,1306.77 549.788,1304.52 550.798,1302.25 551.808,1299.94 552.818,1297.61 553.828,1295.26 554.838,1292.87 555.847,1290.46 \n",
       "  556.857,1288.03 557.867,1285.57 558.877,1283.08 559.887,1280.56 560.897,1278.01 561.906,1275.44 562.916,1272.84 563.926,1270.22 564.936,1267.57 565.946,1264.89 \n",
       "  566.956,1262.18 567.965,1259.44 568.975,1256.68 569.985,1253.89 570.995,1251.07 572.005,1248.23 573.015,1245.36 574.024,1242.46 575.034,1239.53 576.044,1236.58 \n",
       "  577.054,1233.6 578.064,1230.59 579.074,1227.55 580.083,1224.49 581.093,1221.4 582.103,1218.28 583.113,1215.13 584.123,1211.96 585.133,1208.76 586.142,1205.54 \n",
       "  587.152,1202.28 588.162,1199 589.172,1195.7 590.182,1192.36 591.192,1189 592.201,1185.62 593.211,1182.21 594.221,1178.77 595.231,1175.3 596.241,1171.81 \n",
       "  597.251,1168.3 598.26,1164.76 599.27,1161.19 600.28,1157.6 601.29,1153.98 602.3,1150.34 603.31,1146.67 604.319,1142.98 605.329,1139.26 606.339,1135.53 \n",
       "  607.349,1131.76 608.359,1127.98 609.369,1124.16 610.378,1120.33 611.388,1116.48 612.398,1112.6 613.408,1108.69 614.418,1104.77 615.428,1100.83 616.437,1096.86 \n",
       "  617.447,1092.87 618.457,1088.86 619.467,1084.83 620.477,1080.78 621.487,1076.71 622.496,1072.62 623.506,1068.51 624.516,1064.38 625.526,1060.23 626.536,1056.07 \n",
       "  627.546,1051.88 628.555,1047.68 629.565,1043.46 630.575,1039.22 631.585,1034.97 632.595,1030.7 633.605,1026.42 634.614,1022.12 635.624,1017.8 636.634,1013.47 \n",
       "  637.644,1009.13 638.654,1004.77 639.664,1000.4 640.673,996.018 641.683,991.622 642.693,987.214 643.703,982.795 644.713,978.364 645.723,973.924 646.732,969.473 \n",
       "  647.742,965.012 648.752,960.543 649.762,956.065 650.772,951.579 651.782,947.085 652.791,942.584 653.801,938.077 654.811,933.564 655.821,929.045 656.831,924.521 \n",
       "  657.841,919.993 658.85,915.461 659.86,910.925 660.87,906.387 661.88,901.847 662.89,897.305 663.9,892.761 664.909,888.217 665.919,883.674 666.929,879.13 \n",
       "  667.939,874.588 668.949,870.048 669.959,865.51 670.969,860.974 671.978,856.443 672.988,851.915 673.998,847.392 675.008,842.874 676.018,838.363 677.028,833.857 \n",
       "  678.037,829.359 679.047,824.868 680.057,820.386 681.067,815.913 682.077,811.449 683.087,806.995 684.096,802.552 685.106,798.12 686.116,793.7 687.126,789.293 \n",
       "  688.136,784.899 689.146,780.518 690.155,776.153 691.165,771.802 692.175,767.467 693.185,763.148 694.195,758.846 695.205,754.561 696.214,750.295 697.224,746.047 \n",
       "  698.234,741.819 699.244,737.61 700.254,733.422 701.264,729.256 702.273,725.111 703.283,720.988 704.293,716.889 705.303,712.813 706.313,708.761 707.323,704.734 \n",
       "  708.332,700.732 709.342,696.756 710.352,692.807 711.362,688.885 712.372,684.991 713.382,681.125 714.391,677.287 715.401,673.48 716.411,669.702 717.421,665.955 \n",
       "  718.431,662.238 719.441,658.554 720.45,654.902 721.46,651.282 722.47,647.696 723.48,644.144 724.49,640.626 725.5,637.143 726.509,633.695 727.519,630.283 \n",
       "  728.529,626.908 729.539,623.57 730.549,620.269 731.559,617.006 732.568,613.781 733.578,610.595 734.588,607.449 735.598,604.342 736.608,601.275 737.618,598.25 \n",
       "  738.627,595.265 739.637,592.322 740.647,589.421 741.657,586.563 742.667,583.747 743.677,580.974 744.686,578.246 745.696,575.561 746.706,572.921 747.716,570.325 \n",
       "  748.726,567.775 749.736,565.27 750.745,562.811 751.755,560.398 752.765,558.032 753.775,555.713 754.785,553.441 755.795,551.216 756.804,549.04 757.814,546.911 \n",
       "  758.824,544.831 759.834,542.799 760.844,540.816 761.854,538.883 762.863,536.999 763.873,535.164 764.883,533.38 765.893,531.645 766.903,529.961 767.913,528.327 \n",
       "  768.922,526.744 769.932,525.212 770.942,523.731 771.952,522.302 772.962,520.923 773.972,519.596 774.981,518.321 775.991,517.098 777.001,515.926 778.011,514.806 \n",
       "  779.021,513.739 780.031,512.724 781.04,511.761 782.05,510.85 783.06,509.992 784.07,509.186 785.08,508.433 786.09,507.732 787.099,507.083 788.109,506.488 \n",
       "  789.119,505.944 790.129,505.454 791.139,505.015 792.149,504.63 793.158,504.297 794.168,504.016 795.178,503.787 796.188,503.611 797.198,503.487 798.208,503.416 \n",
       "  799.217,503.396 800.227,503.428 801.237,503.512 802.247,503.648 803.257,503.835 804.267,504.074 805.276,504.365 806.286,504.706 807.296,505.099 808.306,505.542 \n",
       "  809.316,506.036 810.326,506.581 811.335,507.175 812.345,507.82 813.355,508.515 814.365,509.26 815.375,510.054 816.385,510.897 817.394,511.789 818.404,512.73 \n",
       "  819.414,513.719 820.424,514.757 821.434,515.842 822.444,516.976 823.453,518.156 824.463,519.384 825.473,520.658 826.483,521.979 827.493,523.346 828.503,524.759 \n",
       "  829.512,526.218 830.522,527.722 831.532,529.27 832.542,530.863 833.552,532.501 834.562,534.182 835.571,535.907 836.581,537.674 837.591,539.485 838.601,541.337 \n",
       "  839.611,543.232 840.621,545.168 841.63,547.146 842.64,549.164 843.65,551.223 844.66,553.321 845.67,555.459 846.68,557.637 847.689,559.853 848.699,562.107 \n",
       "  849.709,564.399 850.719,566.729 851.729,569.095 852.739,571.498 853.748,573.938 854.758,576.412 855.768,578.922 856.778,581.467 857.788,584.046 858.798,586.659 \n",
       "  859.807,589.305 860.817,591.985 861.827,594.696 862.837,597.44 863.847,600.215 864.857,603.02 865.866,605.857 866.876,608.723 867.886,611.619 868.896,614.545 \n",
       "  869.906,617.498 870.916,620.48 871.925,623.489 872.935,626.526 873.945,629.589 874.955,632.678 875.965,635.793 876.975,638.933 877.984,642.097 878.994,645.286 \n",
       "  880.004,648.498 881.014,651.734 882.024,654.992 883.034,658.272 884.043,661.573 885.053,664.896 886.063,668.24 887.073,671.604 888.083,674.987 889.093,678.389 \n",
       "  890.102,681.81 891.112,685.249 892.122,688.706 893.132,692.18 894.142,695.671 895.152,699.178 896.161,702.7 897.171,706.237 898.181,709.79 899.191,713.356 \n",
       "  900.201,716.936 901.211,720.529 902.22,724.135 903.23,727.753 904.24,731.383 905.25,735.024 906.26,738.676 907.27,742.339 908.279,746.011 909.289,749.692 \n",
       "  910.299,753.382 911.309,757.081 912.319,760.788 913.329,764.502 914.338,768.223 915.348,771.951 916.358,775.685 917.368,779.424 918.378,783.169 919.388,786.918 \n",
       "  920.397,790.672 921.407,794.429 922.417,798.19 923.427,801.954 924.437,805.721 925.447,809.489 926.456,813.259 927.466,817.031 928.476,820.803 929.486,824.576 \n",
       "  930.496,828.349 931.506,832.121 932.515,835.892 933.525,839.663 934.535,843.431 935.545,847.198 936.555,850.962 937.565,854.723 938.574,858.481 939.584,862.236 \n",
       "  940.594,865.987 941.604,869.733 942.614,873.474 943.624,877.211 944.633,880.942 945.643,884.667 946.653,888.386 947.663,892.098 948.673,895.804 949.683,899.502 \n",
       "  950.692,903.193 951.702,906.876 952.712,910.55 953.722,914.216 954.732,917.873 955.742,921.521 956.751,925.16 957.761,928.788 958.771,932.406 959.781,936.014 \n",
       "  960.791,939.611 961.801,943.197 962.81,946.772 963.82,950.335 964.83,953.886 965.84,957.425 966.85,960.951 967.86,964.465 968.869,967.965 969.879,971.452 \n",
       "  970.889,974.926 971.899,978.386 972.909,981.831 973.919,985.263 974.928,988.679 975.938,992.081 976.948,995.468 977.958,998.839 978.968,1002.2 979.978,1005.54 \n",
       "  980.987,1008.86 981.997,1012.17 983.007,1015.46 984.017,1018.73 985.027,1021.99 986.037,1025.23 987.047,1028.46 988.056,1031.66 989.066,1034.85 990.076,1038.02 \n",
       "  991.086,1041.18 992.096,1044.31 993.106,1047.43 994.115,1050.53 995.125,1053.61 996.135,1056.67 997.145,1059.71 998.155,1062.73 999.165,1065.73 1000.17,1068.72 \n",
       "  1001.18,1071.68 1002.19,1074.62 1003.2,1077.55 1004.21,1080.45 1005.22,1083.34 1006.23,1086.2 1007.24,1089.04 1008.25,1091.87 1009.26,1094.67 1010.27,1097.45 \n",
       "  1011.28,1100.21 1012.29,1102.95 1013.3,1105.67 1014.31,1108.37 1015.32,1111.04 1016.33,1113.7 1017.34,1116.33 1018.35,1118.94 1019.36,1121.53 1020.37,1124.1 \n",
       "  1021.38,1126.64 1022.39,1129.17 1023.4,1131.67 1024.41,1134.15 1025.42,1136.61 1026.43,1139.04 1027.44,1141.46 1028.45,1143.85 1029.46,1146.22 1030.47,1148.56 \n",
       "  1031.48,1150.89 1032.49,1153.19 1033.5,1155.46 1034.51,1157.72 1035.52,1159.95 1036.53,1162.16 1037.54,1164.35 1038.55,1166.52 1039.56,1168.66 1040.57,1170.78 \n",
       "  1041.58,1172.87 1042.59,1174.95 1043.6,1177 1044.61,1179.02 1045.62,1181.03 1046.63,1183.01 1047.64,1184.97 1048.65,1186.9 1049.66,1188.81 1050.67,1190.7 \n",
       "  1051.68,1192.57 1052.69,1194.41 1053.7,1196.23 1054.71,1198.03 1055.72,1199.8 1056.73,1201.55 1057.73,1203.28 1058.74,1204.98 1059.75,1206.66 1060.76,1208.32 \n",
       "  1061.77,1209.95 1062.78,1211.57 1063.79,1213.15 1064.8,1214.72 1065.81,1216.26 1066.82,1217.78 1067.83,1219.28 1068.84,1220.75 1069.85,1222.2 1070.86,1223.63 \n",
       "  1071.87,1225.03 1072.88,1226.41 1073.89,1227.77 1074.9,1229.11 1075.91,1230.42 1076.92,1231.71 1077.93,1232.98 1078.94,1234.22 1079.95,1235.44 1080.96,1236.64 \n",
       "  1081.97,1237.82 1082.98,1238.97 1083.99,1240.1 1085,1241.21 1086.01,1242.3 1087.02,1243.36 1088.03,1244.4 1089.04,1245.42 1090.05,1246.42 1091.06,1247.39 \n",
       "  1092.07,1248.34 1093.08,1249.27 1094.09,1250.18 1095.1,1251.06 1096.11,1251.93 1097.12,1252.77 1098.13,1253.58 1099.14,1254.38 1100.15,1255.15 1101.16,1255.91 \n",
       "  1102.17,1256.64 1103.18,1257.35 1104.19,1258.03 1105.2,1258.7 1106.21,1259.34 1107.22,1259.96 1108.23,1260.56 1109.24,1261.14 1110.25,1261.7 1111.26,1262.23 \n",
       "  1112.27,1262.75 1113.28,1263.24 1114.29,1263.71 1115.3,1264.16 1116.31,1264.59 1117.32,1264.99 1118.32,1265.38 1119.33,1265.75 1120.34,1266.09 1121.35,1266.41 \n",
       "  1122.36,1266.71 1123.37,1266.99 1124.38,1267.25 1125.39,1267.49 1126.4,1267.71 1127.41,1267.91 1128.42,1268.09 1129.43,1268.24 1130.44,1268.38 1131.45,1268.49 \n",
       "  1132.46,1268.59 1133.47,1268.66 1134.48,1268.71 1135.49,1268.75 1136.5,1268.76 1137.51,1268.75 1138.52,1268.73 1139.53,1268.68 1140.54,1268.61 1141.55,1268.52 \n",
       "  1142.56,1268.41 1143.57,1268.28 1144.58,1268.14 1145.59,1267.97 1146.6,1267.78 1147.61,1267.57 1148.62,1267.35 1149.63,1267.1 1150.64,1266.83 1151.65,1266.54 \n",
       "  1152.66,1266.24 1153.67,1265.91 1154.68,1265.57 1155.69,1265.2 1156.7,1264.82 1157.71,1264.41 1158.72,1263.99 1159.73,1263.55 1160.74,1263.09 1161.75,1262.61 \n",
       "  1162.76,1262.11 1163.77,1261.59 1164.78,1261.05 1165.79,1260.49 1166.8,1259.91 1167.81,1259.32 1168.82,1258.7 1169.83,1258.07 1170.84,1257.42 1171.85,1256.75 \n",
       "  1172.86,1256.06 1173.87,1255.35 1174.88,1254.62 1175.89,1253.87 1176.9,1253.11 1177.91,1252.33 1178.91,1251.52 1179.92,1250.7 1180.93,1249.86 1181.94,1249.01 \n",
       "  1182.95,1248.13 1183.96,1247.23 1184.97,1246.32 1185.98,1245.39 1186.99,1244.44 1188,1243.47 1189.01,1242.49 1190.02,1241.48 1191.03,1240.46 1192.04,1239.42 \n",
       "  1193.05,1238.36 1194.06,1237.28 1195.07,1236.19 1196.08,1235.07 1197.09,1233.94 1198.1,1232.79 1199.11,1231.63 1200.12,1230.44 1201.13,1229.24 1202.14,1228.02 \n",
       "  1203.15,1226.78 1204.16,1225.52 1205.17,1224.25 1206.18,1222.96 1207.19,1221.65 1208.2,1220.32 1209.21,1218.97 1210.22,1217.61 1211.23,1216.23 1212.24,1214.83 \n",
       "  1213.25,1213.42 1214.26,1211.98 1215.27,1210.53 1216.28,1209.06 1217.29,1207.58 1218.3,1206.08 1219.31,1204.56 1220.32,1203.02 1221.33,1201.46 1222.34,1199.89 \n",
       "  1223.35,1198.3 1224.36,1196.69 1225.37,1195.07 1226.38,1193.43 1227.39,1191.77 1228.4,1190.09 1229.41,1188.4 1230.42,1186.69 1231.43,1184.96 1232.44,1183.21 \n",
       "  1233.45,1181.45 1234.46,1179.67 1235.47,1177.88 1236.48,1176.06 1237.49,1174.23 1238.5,1172.38 1239.5,1170.52 1240.51,1168.64 1241.52,1166.74 1242.53,1164.83 \n",
       "  1243.54,1162.89 1244.55,1160.94 1245.56,1158.98 1246.57,1157 1247.58,1155 1248.59,1152.98 1249.6,1150.95 1250.61,1148.9 1251.62,1146.83 1252.63,1144.75 \n",
       "  1253.64,1142.65 1254.65,1140.53 1255.66,1138.4 1256.67,1136.25 1257.68,1134.08 1258.69,1131.9 1259.7,1129.7 1260.71,1127.48 1261.72,1125.25 1262.73,1123 \n",
       "  1263.74,1120.73 1264.75,1118.45 1265.76,1116.15 1266.77,1113.84 1267.78,1111.51 1268.79,1109.16 1269.8,1106.79 1270.81,1104.41 1271.82,1102.02 1272.83,1099.6 \n",
       "  1273.84,1097.17 1274.85,1094.73 1275.86,1092.26 1276.87,1089.79 1277.88,1087.29 1278.89,1084.78 1279.9,1082.25 1280.91,1079.71 1281.92,1077.15 1282.93,1074.58 \n",
       "  1283.94,1071.99 1284.95,1069.38 1285.96,1066.76 1286.97,1064.12 1287.98,1061.46 1288.99,1058.79 1290,1056.1 1291.01,1053.4 1292.02,1050.68 1293.03,1047.95 \n",
       "  1294.04,1045.2 1295.05,1042.43 1296.06,1039.65 1297.07,1036.85 1298.08,1034.04 1299.09,1031.21 1300.09,1028.37 1301.1,1025.51 1302.11,1022.64 1303.12,1019.75 \n",
       "  1304.13,1016.84 1305.14,1013.92 1306.15,1010.98 1307.16,1008.03 1308.17,1005.07 1309.18,1002.08 1310.19,999.087 1311.2,996.075 1312.21,993.048 1313.22,990.006 \n",
       "  1314.23,986.949 1315.24,983.877 1316.25,980.79 1317.26,977.688 1318.27,974.572 1319.28,971.441 1320.29,968.295 1321.3,965.135 1322.31,961.96 1323.32,958.77 \n",
       "  1324.33,955.566 1325.34,952.348 1326.35,949.116 1327.36,945.869 1328.37,942.608 1329.38,939.333 1330.39,936.043 1331.4,932.74 1332.41,929.423 1333.42,926.092 \n",
       "  1334.43,922.747 1335.44,919.389 1336.45,916.017 1337.46,912.631 1338.47,909.232 1339.48,905.819 1340.49,902.393 1341.5,898.954 1342.51,895.502 1343.52,892.036 \n",
       "  1344.53,888.558 1345.54,885.067 1346.55,881.562 1347.56,878.046 1348.57,874.516 1349.58,870.974 1350.59,867.419 1351.6,863.852 1352.61,860.273 1353.62,856.682 \n",
       "  1354.63,853.079 1355.64,849.463 1356.65,845.836 1357.66,842.197 1358.67,838.546 1359.68,834.884 1360.69,831.211 1361.69,827.526 1362.7,823.83 1363.71,820.123 \n",
       "  1364.72,816.405 1365.73,812.676 1366.74,808.936 1367.75,805.186 1368.76,801.426 1369.77,797.655 1370.78,793.874 1371.79,790.083 1372.8,786.282 1373.81,782.471 \n",
       "  1374.82,778.65 1375.83,774.82 1376.84,770.981 1377.85,767.132 1378.86,763.275 1379.87,759.408 1380.88,755.533 1381.89,751.649 1382.9,747.756 1383.91,743.855 \n",
       "  1384.92,739.946 1385.93,736.03 1386.94,732.105 1387.95,728.172 1388.96,724.232 1389.97,720.285 1390.98,716.331 1391.99,712.369 1393,708.401 1394.01,704.426 \n",
       "  1395.02,700.445 1396.03,696.457 1397.04,692.464 1398.05,688.464 1399.06,684.459 1400.07,680.448 1401.08,676.432 1402.09,672.41 1403.1,668.384 1404.11,664.353 \n",
       "  1405.12,660.317 1406.13,656.277 1407.14,652.233 1408.15,648.185 1409.16,644.133 1410.17,640.077 1411.18,636.018 1412.19,631.956 1413.2,627.891 1414.21,623.824 \n",
       "  1415.22,619.754 1416.23,615.681 1417.24,611.607 1418.25,607.531 1419.26,603.453 1420.27,599.374 1421.28,595.294 1422.28,591.213 1423.29,587.131 1424.3,583.048 \n",
       "  1425.31,578.966 1426.32,574.883 1427.33,570.801 1428.34,566.72 1429.35,562.639 1430.36,558.559 1431.37,554.48 1432.38,550.403 1433.39,546.327 1434.4,542.254 \n",
       "  1435.41,538.183 1436.42,534.114 1437.43,530.048 1438.44,525.985 1439.45,521.925 1440.46,517.869 1441.47,513.817 1442.48,509.769 1443.49,505.725 1444.5,501.685 \n",
       "  1445.51,497.651 1446.52,493.621 1447.53,489.597 1448.54,485.579 1449.55,481.566 1450.56,477.56 1451.57,473.56 1452.58,469.567 1453.59,465.581 1454.6,461.603 \n",
       "  1455.61,457.632 1456.62,453.669 1457.63,449.714 1458.64,445.767 1459.65,441.829 1460.66,437.9 1461.67,433.981 1462.68,430.071 1463.69,426.171 1464.7,422.281 \n",
       "  1465.71,418.401 1466.72,414.533 1467.73,410.675 1468.74,406.829 1469.75,402.994 1470.76,399.171 1471.77,395.36 1472.78,391.562 1473.79,387.777 1474.8,384.005 \n",
       "  1475.81,380.246 1476.82,376.501 1477.83,372.769 1478.84,369.053 1479.85,365.35 1480.86,361.663 1481.87,357.991 1482.87,354.334 1483.88,350.693 1484.89,347.068 \n",
       "  1485.9,343.459 1486.91,339.868 1487.92,336.293 1488.93,332.735 1489.94,329.195 1490.95,325.673 1491.96,322.17 1492.97,318.684 1493.98,315.218 1494.99,311.77 \n",
       "  1496,308.342 1497.01,304.934 1498.02,301.546 1499.03,298.178 1500.04,294.83 1501.05,291.504 1502.06,288.199 1503.07,284.915 1504.08,281.653 1505.09,278.413 \n",
       "  1506.1,275.196 1507.11,272.002 1508.12,268.83 1509.13,265.682 1510.14,262.557 1511.15,259.456 1512.16,256.38 1513.17,253.328 1514.18,250.301 1515.19,247.299 \n",
       "  1516.2,244.322 1517.21,241.371 1518.22,238.446 1519.23,235.547 1520.24,232.675 1521.25,229.829 1522.26,227.011 1523.27,224.22 1524.28,221.457 1525.29,218.722 \n",
       "  1526.3,216.016 1527.31,213.337 1528.32,210.688 1529.33,208.068 1530.34,205.477 1531.35,202.916 1532.36,200.385 1533.37,197.884 1534.38,195.413 1535.39,192.974 \n",
       "  1536.4,190.565 1537.41,188.188 1538.42,185.842 1539.43,183.528 1540.44,181.247 1541.45,178.997 1542.46,176.781 1543.46,174.597 1544.47,172.447 1545.48,170.33 \n",
       "  1546.49,168.246 1547.5,166.197 1548.51,164.181 1549.52,162.201 1550.53,160.254 1551.54,158.343 1552.55,156.467 1553.56,154.626 1554.57,152.821 1555.58,151.052 \n",
       "  1556.59,149.319 1557.6,147.622 1558.61,145.962 1559.62,144.339 1560.63,142.752 1561.64,141.203 1562.65,139.692 1563.66,138.218 1564.67,136.781 1565.68,135.383 \n",
       "  1566.69,134.024 1567.7,132.702 1568.71,131.42 1569.72,130.176 1570.73,128.972 1571.74,127.807 1572.75,126.681 1573.76,125.595 1574.77,124.549 1575.78,123.542 \n",
       "  1576.79,122.577 1577.8,121.651 1578.81,120.766 1579.82,119.922 1580.83,119.118 1581.84,118.356 1582.85,117.635 1583.86,116.955 1584.87,116.317 1585.88,115.721 \n",
       "  1586.89,115.166 1587.9,114.654 1588.91,114.183 1589.92,113.755 1590.93,113.369 1591.94,113.025 1592.95,112.724 1593.96,112.466 1594.97,112.251 1595.98,112.079 \n",
       "  1596.99,111.95 1598,111.864 1599.01,111.821 1600.02,111.822 1601.03,111.866 1602.04,111.954 1603.05,112.085 1604.05,112.26 1605.06,112.479 1606.07,112.742 \n",
       "  1607.08,113.049 1608.09,113.399 1609.1,113.794 1610.11,114.233 1611.12,114.716 1612.13,115.244 1613.14,115.815 1614.15,116.431 1615.16,117.092 1616.17,117.796 \n",
       "  1617.18,118.546 1618.19,119.339 1619.2,120.177 1620.21,121.06 1621.22,121.986 1622.23,122.958 1623.24,123.974 1624.25,125.034 1625.26,126.139 1626.27,127.288 \n",
       "  1627.28,128.481 1628.29,129.719 1629.3,131.002 1630.31,132.328 1631.32,133.699 1632.33,135.114 1633.34,136.574 1634.35,138.077 1635.36,139.625 1636.37,141.217 \n",
       "  1637.38,142.852 1638.39,144.532 1639.4,146.255 1640.41,148.022 1641.42,149.833 1642.43,151.687 1643.44,153.585 1644.45,155.526 1645.46,157.51 1646.47,159.537 \n",
       "  1647.48,161.608 1648.49,163.721 1649.5,165.878 1650.51,168.076 1651.52,170.318 1652.53,172.602 1653.54,174.928 1654.55,177.296 1655.56,179.706 1656.57,182.157 \n",
       "  1657.58,184.651 1658.59,187.186 1659.6,189.762 1660.61,192.379 1661.62,195.038 1662.63,197.736 1663.64,200.476 1664.65,203.256 1665.65,206.076 1666.66,208.936 \n",
       "  1667.67,211.835 1668.68,214.775 1669.69,217.753 1670.7,220.771 1671.71,223.827 1672.72,226.922 1673.73,230.056 1674.74,233.228 1675.75,236.437 1676.76,239.684 \n",
       "  1677.77,242.969 1678.78,246.29 1679.79,249.649 1680.8,253.044 1681.81,256.476 1682.82,259.943 1683.83,263.447 1684.84,266.986 1685.85,270.56 1686.86,274.169 \n",
       "  1687.87,277.812 1688.88,281.49 1689.89,285.202 1690.9,288.948 1691.91,292.727 1692.92,296.539 1693.93,300.384 1694.94,304.261 1695.95,308.171 1696.96,312.112 \n",
       "  1697.97,316.085 1698.98,320.088 1699.99,324.123 1701,328.188 1702.01,332.283 1703.02,336.408 1704.03,340.562 1705.04,344.745 1706.05,348.957 1707.06,353.197 \n",
       "  1708.07,357.465 1709.08,361.76 1710.09,366.083 1711.1,370.432 1712.11,374.808 1713.12,379.21 1714.13,383.638 1715.14,388.091 1716.15,392.568 1717.16,397.071 \n",
       "  1718.17,401.597 1719.18,406.147 1720.19,410.72 1721.2,415.316 1722.21,419.935 1723.22,424.576 1724.23,429.238 1725.24,433.922 1726.24,438.626 1727.25,443.351 \n",
       "  1728.26,448.096 1729.27,452.861 1730.28,457.645 1731.29,462.447 1732.3,467.268 1733.31,472.107 1734.32,476.963 1735.33,481.836 1736.34,486.726 1737.35,491.632 \n",
       "  1738.36,496.554 1739.37,501.492 1740.38,506.444 1741.39,511.411 1742.4,516.391 1743.41,521.386 1744.42,526.393 1745.43,531.414 1746.44,536.446 1747.45,541.491 \n",
       "  1748.46,546.547 1749.47,551.614 1750.48,556.691 1751.49,561.779 1752.5,566.876 1753.51,571.983 1754.52,577.098 1755.53,582.222 1756.54,587.353 1757.55,592.493 \n",
       "  1758.56,597.639 1759.57,602.792 1760.58,607.951 1761.59,613.116 1762.6,618.286 1763.61,623.461 1764.62,628.64 1765.63,633.824 1766.64,639.011 1767.65,644.201 \n",
       "  1768.66,649.394 1769.67,654.59 1770.68,659.787 1771.69,664.986 1772.7,670.185 1773.71,675.386 1774.72,680.586 1775.73,685.787 1776.74,690.986 1777.75,696.185 \n",
       "  1778.76,701.382 1779.77,706.577 1780.78,711.771 1781.79,716.961 1782.8,722.148 1783.81,727.332 1784.82,732.511 1785.83,737.687 1786.83,742.858 1787.84,748.023 \n",
       "  1788.85,753.184 1789.86,758.338 1790.87,763.486 1791.88,768.627 1792.89,773.762 1793.9,778.888 1794.91,784.008 1795.92,789.118 1796.93,794.221 1797.94,799.314 \n",
       "  1798.95,804.399 1799.96,809.473 1800.97,814.538 1801.98,819.592 1802.99,824.635 1804,829.668 1805.01,834.689 1806.02,839.698 1807.03,844.696 1808.04,849.68 \n",
       "  1809.05,854.652 1810.06,859.611 1811.07,864.557 1812.08,869.489 1813.09,874.407 1814.1,879.31 1815.11,884.199 1816.12,889.073 1817.13,893.931 1818.14,898.774 \n",
       "  1819.15,903.601 1820.16,908.412 1821.17,913.206 1822.18,917.984 1823.19,922.744 1824.2,927.488 1825.21,932.213 1826.22,936.921 1827.23,941.611 1828.24,946.282 \n",
       "  1829.25,950.935 1830.26,955.569 1831.27,960.184 1832.28,964.779 1833.29,969.355 1834.3,973.911 1835.31,978.447 1836.32,982.962 1837.33,987.457 1838.34,991.931 \n",
       "  1839.35,996.384 1840.36,1000.82 1841.37,1005.23 1842.38,1009.62 1843.39,1013.98 1844.4,1018.33 1845.41,1022.65 1846.42,1026.95 1847.42,1031.23 1848.43,1035.49 \n",
       "  1849.44,1039.72 1850.45,1043.93 1851.46,1048.12 1852.47,1052.28 1853.48,1056.42 1854.49,1060.53 1855.5,1064.63 1856.51,1068.69 1857.52,1072.74 1858.53,1076.76 \n",
       "  1859.54,1080.75 1860.55,1084.72 1861.56,1088.67 1862.57,1092.59 1863.58,1096.49 1864.59,1100.36 1865.6,1104.21 1866.61,1108.03 1867.62,1111.83 1868.63,1115.6 \n",
       "  1869.64,1119.34 1870.65,1123.06 1871.66,1126.76 1872.67,1130.43 1873.68,1134.07 1874.69,1137.69 1875.7,1141.28 1876.71,1144.85 1877.72,1148.39 1878.73,1151.9 \n",
       "  1879.74,1155.39 1880.75,1158.85 1881.76,1162.29 1882.77,1165.7 1883.78,1169.08 1884.79,1172.44 1885.8,1175.77 1886.81,1179.07 1887.82,1182.35 1888.83,1185.6 \n",
       "  1889.84,1188.83 1890.85,1192.03 1891.86,1195.2 1892.87,1198.35 1893.88,1201.47 1894.89,1204.57 1895.9,1207.63 1896.91,1210.68 1897.92,1213.69 1898.93,1216.68 \n",
       "  1899.94,1219.65 1900.95,1222.58 1901.96,1225.5 1902.97,1228.38 1903.98,1231.24 1904.99,1234.07 1906,1236.88 1907.01,1239.66 1908.01,1242.42 1909.02,1245.15 \n",
       "  1910.03,1247.85 1911.04,1250.53 1912.05,1253.19 1913.06,1255.81 1914.07,1258.42 1915.08,1260.99 1916.09,1263.55 1917.1,1266.07 1918.11,1268.57 1919.12,1271.05 \n",
       "  1920.13,1273.5 1921.14,1275.93 1922.15,1278.33 1923.16,1280.71 1924.17,1283.06 1925.18,1285.39 1926.19,1287.69 1927.2,1289.97 1928.21,1292.22 1929.22,1294.46 \n",
       "  1930.23,1296.66 1931.24,1298.85 1932.25,1301.01 1933.26,1303.14 1934.27,1305.25 1935.28,1307.34 1936.29,1309.41 1937.3,1311.45 1938.31,1313.47 1939.32,1315.47 \n",
       "  1940.33,1317.44 1941.34,1319.39 1942.35,1321.32 1943.36,1323.23 1944.37,1325.11 1945.38,1326.97 1946.39,1328.81 1947.4,1330.63 1948.41,1332.43 1949.42,1334.2 \n",
       "  1950.43,1335.95 1951.44,1337.69 1952.45,1339.4 1953.46,1341.09 1954.47,1342.76 1955.48,1344.4 1956.49,1346.03 1957.5,1347.64 1958.51,1349.23 1959.52,1350.79 \n",
       "  1960.53,1352.34 1961.54,1353.87 1962.55,1355.38 1963.56,1356.86 1964.57,1358.33 1965.58,1359.78 1966.59,1361.21 1967.6,1362.62 1968.61,1364.02 1969.61,1365.39 \n",
       "  1970.62,1366.75 1971.63,1368.08 1972.64,1369.4 1973.65,1370.7 1974.66,1371.99 1975.67,1373.25 1976.68,1374.5 1977.69,1375.73 1978.7,1376.95 1979.71,1378.14 \n",
       "  1980.72,1379.32 1981.73,1380.49 1982.74,1381.63 1983.75,1382.76 1984.76,1383.88 1985.77,1384.98 1986.78,1386.06 1987.79,1387.13 1988.8,1388.18 1989.81,1389.21 \n",
       "  1990.82,1390.23 1991.83,1391.24 1992.84,1392.23 1993.85,1393.2 1994.86,1394.16 1995.87,1395.11 1996.88,1396.04 1997.89,1396.96 1998.9,1397.86 1999.91,1398.75 \n",
       "  2000.92,1399.63 2001.93,1400.49 2002.94,1401.34 2003.95,1402.18 2004.96,1403 2005.97,1403.81 2006.98,1404.61 2007.99,1405.39 2009,1406.16 2010.01,1406.92 \n",
       "  2011.02,1407.67 2012.03,1408.4 2013.04,1409.13 2014.05,1409.84 2015.06,1410.54 2016.07,1411.23 2017.08,1411.9 2018.09,1412.57 2019.1,1413.22 2020.11,1413.87 \n",
       "  2021.12,1414.5 2022.13,1415.12 2023.14,1415.73 2024.15,1416.33 2025.16,1416.92 2026.17,1417.5 2027.18,1418.08 2028.19,1418.64 2029.2,1419.19 2030.2,1419.73 \n",
       "  2031.21,1420.26 2032.22,1420.78 2033.23,1421.3 2034.24,1421.8 2035.25,1422.3 2036.26,1422.78 2037.27,1423.26 2038.28,1423.73 2039.29,1424.19 2040.3,1424.65 \n",
       "  2041.31,1425.09 2042.32,1425.53 2043.33,1425.96 2044.34,1426.38 2045.35,1426.79 2046.36,1427.19 2047.37,1427.59 2048.38,1427.98 2049.39,1428.37 2050.4,1428.74 \n",
       "  2051.41,1429.11 2052.42,1429.47 2053.43,1429.83 2054.44,1430.17 2055.45,1430.52 2056.46,1430.85 2057.47,1431.18 2058.48,1431.5 2059.49,1431.82 2060.5,1432.13 \n",
       "  2061.51,1432.43 2062.52,1432.73 2063.53,1433.02 2064.54,1433.31 2065.55,1433.59 2066.56,1433.86 2067.57,1434.13 2068.58,1434.4 2069.59,1434.65 2070.6,1434.91 \n",
       "  2071.61,1435.16 2072.62,1435.4 2073.63,1435.64 2074.64,1435.87 2075.65,1436.1 2076.66,1436.32 2077.67,1436.54 2078.68,1436.76 2079.69,1436.97 2080.7,1437.18 \n",
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     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "histogram(eruptions, label=\"eruptions\")\n",
    "plot!(p.x, p.density .* length(eruptions), linewidth=3, color=2, label=\"kdef fit\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
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       "  905.75,995.583 906.77,998.165 907.79,1000.75 908.81,1003.34 909.83,1005.92 910.851,1008.51 911.871,1011.1 912.891,1013.7 913.911,1016.29 914.931,1018.88 \n",
       "  915.951,1021.47 916.972,1024.07 917.992,1026.66 919.012,1029.25 920.032,1031.84 921.052,1034.43 922.072,1037.01 923.093,1039.6 924.113,1042.18 925.133,1044.76 \n",
       "  926.153,1047.34 927.173,1049.91 928.193,1052.48 929.214,1055.05 930.234,1057.61 931.254,1060.17 932.274,1062.73 933.294,1065.28 934.314,1067.83 935.334,1070.37 \n",
       "  936.355,1072.91 937.375,1075.44 938.395,1077.96 939.415,1080.48 940.435,1082.99 941.455,1085.5 942.476,1088 943.496,1090.49 944.516,1092.98 945.536,1095.46 \n",
       "  946.556,1097.93 947.576,1100.4 948.597,1102.85 949.617,1105.3 950.637,1107.74 951.657,1110.17 952.677,1112.6 953.697,1115.01 954.718,1117.42 955.738,1119.81 \n",
       "  956.758,1122.2 957.778,1124.58 958.798,1126.94 959.818,1129.3 960.839,1131.65 961.859,1133.99 962.879,1136.31 963.899,1138.63 964.919,1140.94 965.939,1143.23 \n",
       "  966.959,1145.52 967.98,1147.79 969,1150.05 970.02,1152.3 971.04,1154.54 972.06,1156.77 973.08,1158.98 974.101,1161.19 975.121,1163.38 976.141,1165.56 \n",
       "  977.161,1167.73 978.181,1169.88 979.201,1172.02 980.222,1174.15 981.242,1176.27 982.262,1178.37 983.282,1180.46 984.302,1182.54 985.322,1184.6 986.343,1186.65 \n",
       "  987.363,1188.69 988.383,1190.71 989.403,1192.72 990.423,1194.71 991.443,1196.7 992.463,1198.66 993.484,1200.62 994.504,1202.56 995.524,1204.48 996.544,1206.4 \n",
       "  997.564,1208.29 998.584,1210.17 999.605,1212.04 1000.62,1213.9 1001.64,1215.73 1002.67,1217.56 1003.69,1219.37 1004.71,1221.16 1005.73,1222.94 1006.75,1224.71 \n",
       "  1007.77,1226.46 1008.79,1228.19 1009.81,1229.91 1010.83,1231.61 1011.85,1233.3 1012.87,1234.98 1013.89,1236.63 1014.91,1238.28 1015.93,1239.91 1016.95,1241.52 \n",
       "  1017.97,1243.11 1018.99,1244.7 1020.01,1246.26 1021.03,1247.81 1022.05,1249.35 1023.07,1250.86 1024.09,1252.37 1025.11,1253.86 1026.13,1255.33 1027.15,1256.78 \n",
       "  1028.17,1258.22 1029.19,1259.65 1030.21,1261.06 1031.23,1262.45 1032.25,1263.83 1033.27,1265.19 1034.29,1266.53 1035.31,1267.86 1036.33,1269.18 1037.35,1270.48 \n",
       "  1038.37,1271.76 1039.39,1273.02 1040.41,1274.27 1041.43,1275.51 1042.45,1276.73 1043.47,1277.93 1044.49,1279.12 1045.51,1280.29 1046.53,1281.44 1047.55,1282.58 \n",
       "  1048.57,1283.71 1049.59,1284.81 1050.61,1285.9 1051.63,1286.98 1052.65,1288.04 1053.67,1289.08 1054.69,1290.11 1055.71,1291.12 1056.73,1292.12 1057.75,1293.1 \n",
       "  1058.77,1294.07 1059.79,1295.02 1060.81,1295.95 1061.83,1296.87 1062.85,1297.77 1063.87,1298.66 1064.89,1299.53 1065.92,1300.38 1066.94,1301.22 1067.96,1302.05 \n",
       "  1068.98,1302.85 1070,1303.65 1071.02,1304.42 1072.04,1305.19 1073.06,1305.93 1074.08,1306.66 1075.1,1307.38 1076.12,1308.08 1077.14,1308.76 1078.16,1309.43 \n",
       "  1079.18,1310.09 1080.2,1310.72 1081.22,1311.35 1082.24,1311.95 1083.26,1312.55 1084.28,1313.12 1085.3,1313.69 1086.32,1314.23 1087.34,1314.77 1088.36,1315.28 \n",
       "  1089.38,1315.78 1090.4,1316.27 1091.42,1316.74 1092.44,1317.2 1093.46,1317.64 1094.48,1318.07 1095.5,1318.48 1096.52,1318.88 1097.54,1319.26 1098.56,1319.63 \n",
       "  1099.58,1319.98 1100.6,1320.32 1101.62,1320.64 1102.64,1320.95 1103.66,1321.25 1104.68,1321.53 1105.7,1321.79 1106.72,1322.04 1107.74,1322.28 1108.76,1322.5 \n",
       "  1109.78,1322.71 1110.8,1322.9 1111.82,1323.08 1112.84,1323.24 1113.86,1323.39 1114.88,1323.53 1115.9,1323.65 1116.92,1323.76 1117.94,1323.85 1118.96,1323.93 \n",
       "  1119.98,1324 1121,1324.05 1122.02,1324.08 1123.04,1324.11 1124.06,1324.12 1125.08,1324.11 1126.1,1324.09 1127.12,1324.06 1128.14,1324.01 1129.17,1323.95 \n",
       "  1130.19,1323.88 1131.21,1323.79 1132.23,1323.69 1133.25,1323.57 1134.27,1323.44 1135.29,1323.3 1136.31,1323.14 1137.33,1322.97 1138.35,1322.79 1139.37,1322.59 \n",
       "  1140.39,1322.38 1141.41,1322.16 1142.43,1321.92 1143.45,1321.67 1144.47,1321.41 1145.49,1321.13 1146.51,1320.84 1147.53,1320.53 1148.55,1320.22 1149.57,1319.89 \n",
       "  1150.59,1319.54 1151.61,1319.19 1152.63,1318.82 1153.65,1318.43 1154.67,1318.04 1155.69,1317.63 1156.71,1317.2 1157.73,1316.77 1158.75,1316.32 1159.77,1315.86 \n",
       "  1160.79,1315.39 1161.81,1314.9 1162.83,1314.4 1163.85,1313.89 1164.87,1313.36 1165.89,1312.82 1166.91,1312.27 1167.93,1311.71 1168.95,1311.13 1169.97,1310.54 \n",
       "  1170.99,1309.94 1172.01,1309.32 1173.03,1308.7 1174.05,1308.06 1175.07,1307.4 1176.09,1306.74 1177.11,1306.06 1178.13,1305.37 1179.15,1304.67 1180.17,1303.95 \n",
       "  1181.19,1303.22 1182.21,1302.48 1183.23,1301.73 1184.25,1300.97 1185.27,1300.19 1186.29,1299.4 1187.31,1298.6 1188.33,1297.78 1189.35,1296.96 1190.37,1296.12 \n",
       "  1191.39,1295.27 1192.41,1294.4 1193.44,1293.53 1194.46,1292.64 1195.48,1291.74 1196.5,1290.83 1197.52,1289.9 1198.54,1288.97 1199.56,1288.02 1200.58,1287.06 \n",
       "  1201.6,1286.08 1202.62,1285.1 1203.64,1284.1 1204.66,1283.09 1205.68,1282.07 1206.7,1281.04 1207.72,1280 1208.74,1278.94 1209.76,1277.87 1210.78,1276.79 \n",
       "  1211.8,1275.7 1212.82,1274.59 1213.84,1273.48 1214.86,1272.35 1215.88,1271.21 1216.9,1270.06 1217.92,1268.89 1218.94,1267.72 1219.96,1266.53 1220.98,1265.33 \n",
       "  1222,1264.12 1223.02,1262.9 1224.04,1261.66 1225.06,1260.42 1226.08,1259.16 1227.1,1257.89 1228.12,1256.61 1229.14,1255.31 1230.16,1254.01 1231.18,1252.69 \n",
       "  1232.2,1251.37 1233.22,1250.03 1234.24,1248.68 1235.26,1247.31 1236.28,1245.94 1237.3,1244.55 1238.32,1243.16 1239.34,1241.75 1240.36,1240.33 1241.38,1238.9 \n",
       "  1242.4,1237.45 1243.42,1236 1244.44,1234.53 1245.46,1233.06 1246.48,1231.57 1247.5,1230.07 1248.52,1228.55 1249.54,1227.03 1250.56,1225.5 1251.58,1223.95 \n",
       "  1252.6,1222.39 1253.62,1220.83 1254.64,1219.25 1255.66,1217.66 1256.69,1216.05 1257.71,1214.44 1258.73,1212.82 1259.75,1211.18 1260.77,1209.53 1261.79,1207.87 \n",
       "  1262.81,1206.2 1263.83,1204.52 1264.85,1202.83 1265.87,1201.13 1266.89,1199.41 1267.91,1197.69 1268.93,1195.95 1269.95,1194.2 1270.97,1192.45 1271.99,1190.68 \n",
       "  1273.01,1188.9 1274.03,1187.1 1275.05,1185.3 1276.07,1183.49 1277.09,1181.66 1278.11,1179.83 1279.13,1177.98 1280.15,1176.12 1281.17,1174.26 1282.19,1172.38 \n",
       "  1283.21,1170.49 1284.23,1168.59 1285.25,1166.68 1286.27,1164.75 1287.29,1162.82 1288.31,1160.88 1289.33,1158.92 1290.35,1156.96 1291.37,1154.98 1292.39,1153 \n",
       "  1293.41,1151 1294.43,1148.99 1295.45,1146.98 1296.47,1144.95 1297.49,1142.91 1298.51,1140.86 1299.53,1138.8 1300.55,1136.73 1301.57,1134.65 1302.59,1132.56 \n",
       "  1303.61,1130.46 1304.63,1128.35 1305.65,1126.23 1306.67,1124.1 1307.69,1121.96 1308.71,1119.8 1309.73,1117.64 1310.75,1115.47 1311.77,1113.29 1312.79,1111.1 \n",
       "  1313.81,1108.9 1314.83,1106.68 1315.85,1104.46 1316.87,1102.23 1317.89,1099.99 1318.91,1097.74 1319.94,1095.48 1320.96,1093.21 1321.98,1090.93 1323,1088.64 \n",
       "  1324.02,1086.34 1325.04,1084.03 1326.06,1081.72 1327.08,1079.39 1328.1,1077.06 1329.12,1074.71 1330.14,1072.36 1331.16,1069.99 1332.18,1067.62 1333.2,1065.24 \n",
       "  1334.22,1062.85 1335.24,1060.45 1336.26,1058.04 1337.28,1055.62 1338.3,1053.2 1339.32,1050.76 1340.34,1048.32 1341.36,1045.87 1342.38,1043.41 1343.4,1040.94 \n",
       "  1344.42,1038.47 1345.44,1035.98 1346.46,1033.49 1347.48,1030.99 1348.5,1028.48 1349.52,1025.96 1350.54,1023.44 1351.56,1020.91 1352.58,1018.37 1353.6,1015.82 \n",
       "  1354.62,1013.27 1355.64,1010.7 1356.66,1008.13 1357.68,1005.56 1358.7,1002.97 1359.72,1000.38 1360.74,997.783 1361.76,995.178 1362.78,992.566 1363.8,989.947 \n",
       "  1364.82,987.322 1365.84,984.69 1366.86,982.052 1367.88,979.407 1368.9,976.756 1369.92,974.099 1370.94,971.436 1371.96,968.767 1372.98,966.092 1374,963.412 \n",
       "  1375.02,960.726 1376.04,958.034 1377.06,955.337 1378.08,952.635 1379.1,949.927 1380.12,947.215 1381.14,944.497 1382.16,941.775 1383.19,939.048 1384.21,936.317 \n",
       "  1385.23,933.581 1386.25,930.841 1387.27,928.096 1388.29,925.348 1389.31,922.596 1390.33,919.839 1391.35,917.079 1392.37,914.316 1393.39,911.549 1394.41,908.779 \n",
       "  1395.43,906.006 1396.45,903.23 1397.47,900.45 1398.49,897.669 1399.51,894.884 1400.53,892.097 1401.55,889.308 1402.57,886.517 1403.59,883.724 1404.61,880.928 \n",
       "  1405.63,878.132 1406.65,875.333 1407.67,872.533 1408.69,869.732 1409.71,866.93 1410.73,864.127 1411.75,861.323 1412.77,858.519 1413.79,855.714 1414.81,852.908 \n",
       "  1415.83,850.103 1416.85,847.298 1417.87,844.492 1418.89,841.688 1419.91,838.883 1420.93,836.08 1421.95,833.277 1422.97,830.475 1423.99,827.675 1425.01,824.875 \n",
       "  1426.03,822.078 1427.05,819.282 1428.07,816.488 1429.09,813.696 1430.11,810.906 1431.13,808.119 1432.15,805.334 1433.17,802.552 1434.19,799.773 1435.21,796.997 \n",
       "  1436.23,794.225 1437.25,791.456 1438.27,788.691 1439.29,785.929 1440.31,783.172 1441.33,780.419 1442.35,777.671 1443.37,774.927 1444.39,772.188 1445.41,769.454 \n",
       "  1446.43,766.725 1447.46,764.001 1448.48,761.283 1449.5,758.571 1450.52,755.865 1451.54,753.166 1452.56,750.472 1453.58,747.785 1454.6,745.105 1455.62,742.432 \n",
       "  1456.64,739.766 1457.66,737.108 1458.68,734.457 1459.7,731.814 1460.72,729.179 1461.74,726.552 1462.76,723.933 1463.78,721.323 1464.8,718.722 1465.82,716.13 \n",
       "  1466.84,713.547 1467.86,710.973 1468.88,708.409 1469.9,705.855 1470.92,703.311 1471.94,700.777 1472.96,698.253 1473.98,695.74 1475,693.238 1476.02,690.747 \n",
       "  1477.04,688.268 1478.06,685.8 1479.08,683.343 1480.1,680.898 1481.12,678.466 1482.14,676.046 1483.16,673.638 1484.18,671.243 1485.2,668.861 1486.22,666.492 \n",
       "  1487.24,664.136 1488.26,661.794 1489.28,659.466 1490.3,657.151 1491.32,654.851 1492.34,652.565 1493.36,650.294 1494.38,648.037 1495.4,645.796 1496.42,643.57 \n",
       "  1497.44,641.359 1498.46,639.163 1499.48,636.984 1500.5,634.821 1501.52,632.673 1502.54,630.543 1503.56,628.428 1504.58,626.331 1505.6,624.251 1506.62,622.188 \n",
       "  1507.64,620.142 1508.66,618.115 1509.68,616.105 1510.71,614.113 1511.73,612.139 1512.75,610.184 1513.77,608.247 1514.79,606.329 1515.81,604.43 1516.83,602.551 \n",
       "  1517.85,600.691 1518.87,598.851 1519.89,597.03 1520.91,595.229 1521.93,593.449 1522.95,591.689 1523.97,589.95 1524.99,588.231 1526.01,586.533 1527.03,584.857 \n",
       "  1528.05,583.202 1529.07,581.568 1530.09,579.956 1531.11,578.366 1532.13,576.799 1533.15,575.253 1534.17,573.73 1535.19,572.229 1536.21,570.751 1537.23,569.297 \n",
       "  1538.25,567.865 1539.27,566.456 1540.29,565.072 1541.31,563.71 1542.33,562.373 1543.35,561.06 1544.37,559.77 1545.39,558.506 1546.41,557.265 1547.43,556.049 \n",
       "  1548.45,554.858 1549.47,553.693 1550.49,552.552 1551.51,551.436 1552.53,550.346 1553.55,549.281 1554.57,548.243 1555.59,547.23 1556.61,546.243 1557.63,545.282 \n",
       "  1558.65,544.348 1559.67,543.44 1560.69,542.559 1561.71,541.704 1562.73,540.876 1563.75,540.076 1564.77,539.302 1565.79,538.556 1566.81,537.837 1567.83,537.145 \n",
       "  1568.85,536.482 1569.87,535.846 1570.89,535.237 1571.91,534.657 1572.93,534.105 1573.96,533.581 1574.98,533.086 1576,532.619 1577.02,532.18 1578.04,531.771 \n",
       "  1579.06,531.389 1580.08,531.037 1581.1,530.714 1582.12,530.419 1583.14,530.154 1584.16,529.918 1585.18,529.711 1586.2,529.534 1587.22,529.386 1588.24,529.268 \n",
       "  1589.26,529.179 1590.28,529.12 1591.3,529.091 1592.32,529.091 1593.34,529.122 1594.36,529.182 1595.38,529.272 1596.4,529.392 1597.42,529.543 1598.44,529.723 \n",
       "  1599.46,529.934 1600.48,530.175 1601.5,530.447 1602.52,530.748 1603.54,531.08 1604.56,531.443 1605.58,531.836 1606.6,532.259 1607.62,532.713 1608.64,533.197 \n",
       "  1609.66,533.712 1610.68,534.257 1611.7,534.833 1612.72,535.439 1613.74,536.076 1614.76,536.744 1615.78,537.442 1616.8,538.17 1617.82,538.929 1618.84,539.719 \n",
       "  1619.86,540.539 1620.88,541.39 1621.9,542.271 1622.92,543.183 1623.94,544.125 1624.96,545.097 1625.98,546.1 1627,547.133 1628.02,548.197 1629.04,549.291 \n",
       "  1630.06,550.415 1631.08,551.569 1632.1,552.753 1633.12,553.967 1634.14,555.211 1635.16,556.486 1636.18,557.79 1637.21,559.124 1638.23,560.487 1639.25,561.88 \n",
       "  1640.27,563.303 1641.29,564.755 1642.31,566.237 1643.33,567.748 1644.35,569.288 1645.37,570.858 1646.39,572.456 1647.41,574.083 1648.43,575.74 1649.45,577.424 \n",
       "  1650.47,579.138 1651.49,580.88 1652.51,582.65 1653.53,584.449 1654.55,586.275 1655.57,588.13 1656.59,590.012 1657.61,591.923 1658.63,593.861 1659.65,595.826 \n",
       "  1660.67,597.819 1661.69,599.838 1662.71,601.885 1663.73,603.959 1664.75,606.059 1665.77,608.186 1666.79,610.339 1667.81,612.519 1668.83,614.724 1669.85,616.956 \n",
       "  1670.87,619.213 1671.89,621.495 1672.91,623.803 1673.93,626.136 1674.95,628.494 1675.97,630.877 1676.99,633.285 1678.01,635.717 1679.03,638.173 1680.05,640.653 \n",
       "  1681.07,643.156 1682.09,645.684 1683.11,648.235 1684.13,650.809 1685.15,653.405 1686.17,656.025 1687.19,658.667 1688.21,661.332 1689.23,664.018 1690.25,666.726 \n",
       "  1691.27,669.456 1692.29,672.208 1693.31,674.98 1694.33,677.773 1695.35,680.587 1696.37,683.422 1697.39,686.277 1698.41,689.151 1699.43,692.045 1700.45,694.959 \n",
       "  1701.48,697.892 1702.5,700.844 1703.52,703.814 1704.54,706.803 1705.56,709.81 1706.58,712.835 1707.6,715.877 1708.62,718.937 1709.64,722.014 1710.66,725.108 \n",
       "  1711.68,728.219 1712.7,731.345 1713.72,734.488 1714.74,737.646 1715.76,740.82 1716.78,744.009 1717.8,747.213 1718.82,750.432 1719.84,753.664 1720.86,756.911 \n",
       "  1721.88,760.172 1722.9,763.446 1723.92,766.733 1724.94,770.034 1725.96,773.346 1726.98,776.672 1728,780.009 1729.02,783.358 1730.04,786.718 1731.06,790.089 \n",
       "  1732.08,793.472 1733.1,796.864 1734.12,800.267 1735.14,803.68 1736.16,807.103 1737.18,810.535 1738.2,813.976 1739.22,817.426 1740.24,820.884 1741.26,824.351 \n",
       "  1742.28,827.825 1743.3,831.307 1744.32,834.796 1745.34,838.292 1746.36,841.795 1747.38,845.304 1748.4,848.819 1749.42,852.34 1750.44,855.867 1751.46,859.398 \n",
       "  1752.48,862.935 1753.5,866.476 1754.52,870.021 1755.54,873.57 1756.56,877.123 1757.58,880.679 1758.6,884.238 1759.62,887.8 1760.64,891.365 1761.66,894.931 \n",
       "  1762.68,898.5 1763.7,902.07 1764.73,905.642 1765.75,909.214 1766.77,912.787 1767.79,916.361 1768.81,919.934 1769.83,923.508 1770.85,927.081 1771.87,930.654 \n",
       "  1772.89,934.225 1773.91,937.795 1774.93,941.364 1775.95,944.93 1776.97,948.495 1777.99,952.057 1779.01,955.616 1780.03,959.173 1781.05,962.726 1782.07,966.276 \n",
       "  1783.09,969.822 1784.11,973.364 1785.13,976.901 1786.15,980.434 1787.17,983.963 1788.19,987.486 1789.21,991.003 1790.23,994.516 1791.25,998.022 1792.27,1001.52 \n",
       "  1793.29,1005.02 1794.31,1008.5 1795.33,1011.98 1796.35,1015.46 1797.37,1018.92 1798.39,1022.38 1799.41,1025.83 1800.43,1029.27 1801.45,1032.71 1802.47,1036.13 \n",
       "  1803.49,1039.55 1804.51,1042.96 1805.53,1046.36 1806.55,1049.74 1807.57,1053.12 1808.59,1056.49 1809.61,1059.85 1810.63,1063.2 1811.65,1066.54 1812.67,1069.87 \n",
       "  1813.69,1073.19 1814.71,1076.49 1815.73,1079.79 1816.75,1083.07 1817.77,1086.34 1818.79,1089.6 1819.81,1092.85 1820.83,1096.08 1821.85,1099.31 1822.87,1102.52 \n",
       "  1823.89,1105.71 1824.91,1108.9 1825.93,1112.07 1826.95,1115.23 1827.98,1118.37 1829,1121.5 1830.02,1124.62 1831.04,1127.72 1832.06,1130.81 1833.08,1133.88 \n",
       "  1834.1,1136.94 1835.12,1139.99 1836.14,1143.02 1837.16,1146.04 1838.18,1149.04 1839.2,1152.02 1840.22,1155 1841.24,1157.95 1842.26,1160.89 1843.28,1163.81 \n",
       "  1844.3,1166.72 1845.32,1169.62 1846.34,1172.49 1847.36,1175.35 1848.38,1178.2 1849.4,1181.03 1850.42,1183.84 1851.44,1186.63 1852.46,1189.41 1853.48,1192.18 \n",
       "  1854.5,1194.92 1855.52,1197.65 1856.54,1200.36 1857.56,1203.06 1858.58,1205.73 1859.6,1208.4 1860.62,1211.04 1861.64,1213.67 1862.66,1216.27 1863.68,1218.87 \n",
       "  1864.7,1221.44 1865.72,1224 1866.74,1226.54 1867.76,1229.06 1868.78,1231.56 1869.8,1234.05 1870.82,1236.51 1871.84,1238.96 1872.86,1241.4 1873.88,1243.81 \n",
       "  1874.9,1246.21 1875.92,1248.59 1876.94,1250.95 1877.96,1253.29 1878.98,1255.62 1880,1257.92 1881.02,1260.21 1882.04,1262.48 1883.06,1264.74 1884.08,1266.97 \n",
       "  1885.1,1269.19 1886.12,1271.39 1887.14,1273.57 1888.16,1275.73 1889.18,1277.88 1890.2,1280 1891.23,1282.11 1892.25,1284.2 1893.27,1286.27 1894.29,1288.33 \n",
       "  1895.31,1290.37 1896.33,1292.38 1897.35,1294.39 1898.37,1296.37 1899.39,1298.33 1900.41,1300.28 1901.43,1302.21 1902.45,1304.12 1903.47,1306.01 1904.49,1307.89 \n",
       "  1905.51,1309.75 1906.53,1311.59 1907.55,1313.41 1908.57,1315.22 1909.59,1317.01 1910.61,1318.78 1911.63,1320.53 1912.65,1322.27 1913.67,1323.99 1914.69,1325.69 \n",
       "  1915.71,1327.37 1916.73,1329.04 1917.75,1330.69 1918.77,1332.32 1919.79,1333.94 1920.81,1335.54 1921.83,1337.12 1922.85,1338.69 1923.87,1340.24 1924.89,1341.77 \n",
       "  1925.91,1343.29 1926.93,1344.79 1927.95,1346.27 1928.97,1347.74 1929.99,1349.19 1931.01,1350.63 1932.03,1352.05 1933.05,1353.45 1934.07,1354.84 1935.09,1356.21 \n",
       "  1936.11,1357.57 1937.13,1358.91 1938.15,1360.23 1939.17,1361.54 1940.19,1362.84 1941.21,1364.12 1942.23,1365.38 1943.25,1366.63 1944.27,1367.87 1945.29,1369.08 \n",
       "  1946.31,1370.29 1947.33,1371.48 1948.35,1372.66 1949.37,1373.82 1950.39,1374.96 1951.41,1376.1 1952.43,1377.22 1953.45,1378.32 1954.47,1379.41 1955.5,1380.49 \n",
       "  1956.52,1381.55 1957.54,1382.6 1958.56,1383.64 1959.58,1384.66 1960.6,1385.67 1961.62,1386.66 1962.64,1387.65 1963.66,1388.62 1964.68,1389.57 1965.7,1390.52 \n",
       "  1966.72,1391.45 1967.74,1392.37 1968.76,1393.27 1969.78,1394.17 1970.8,1395.05 1971.82,1395.92 1972.84,1396.78 1973.86,1397.63 1974.88,1398.46 1975.9,1399.28 \n",
       "  1976.92,1400.09 1977.94,1400.89 1978.96,1401.68 1979.98,1402.46 1981,1403.22 1982.02,1403.98 1983.04,1404.72 1984.06,1405.45 1985.08,1406.18 1986.1,1406.89 \n",
       "  1987.12,1407.59 1988.14,1408.28 1989.16,1408.96 1990.18,1409.63 1991.2,1410.29 1992.22,1410.94 1993.24,1411.58 1994.26,1412.21 1995.28,1412.83 1996.3,1413.44 \n",
       "  1997.32,1414.05 1998.34,1414.64 1999.36,1415.22 2000.38,1415.8 2001.4,1416.36 2002.42,1416.92 2003.44,1417.47 2004.46,1418 2005.48,1418.54 2006.5,1419.06 \n",
       "  2007.52,1419.57 2008.54,1420.08 2009.56,1420.57 2010.58,1421.06 2011.6,1421.54 2012.62,1422.01 2013.64,1422.48 2014.66,1422.94 2015.68,1423.39 2016.7,1423.83 \n",
       "  2017.72,1424.26 2018.75,1424.69 2019.77,1425.11 2020.79,1425.52 2021.81,1425.93 2022.83,1426.33 2023.85,1426.72 2024.87,1427.11 2025.89,1427.49 2026.91,1427.86 \n",
       "  2027.93,1428.22 2028.95,1428.58 2029.97,1428.94 2030.99,1429.28 2032.01,1429.62 2033.03,1429.96 2034.05,1430.29 2035.07,1430.61 2036.09,1430.93 2037.11,1431.24 \n",
       "  2038.13,1431.54 2039.15,1431.84 2040.17,1432.14 2041.19,1432.43 2042.21,1432.71 2043.23,1432.99 2044.25,1433.26 2045.27,1433.53 2046.29,1433.79 2047.31,1434.05 \n",
       "  2048.33,1434.3 2049.35,1434.55 2050.37,1434.8 2051.39,1435.04 2052.41,1435.27 2053.43,1435.5 2054.45,1435.73 2055.47,1435.95 2056.49,1436.17 2057.51,1436.38 \n",
       "  2058.53,1436.59 2059.55,1436.79 2060.57,1436.99 2061.59,1437.19 2062.61,1437.38 2063.63,1437.57 2064.65,1437.76 2065.67,1437.94 2066.69,1438.11 2067.71,1438.29 \n",
       "  2068.73,1438.46 2069.75,1438.63 2070.77,1438.79 2071.79,1438.95 2072.81,1439.11 2073.83,1439.26 2074.85,1439.41 2075.87,1439.56 2076.89,1439.71 2077.91,1439.85 \n",
       "  2078.93,1439.99 2079.95,1440.12 2080.97,1440.25 2082,1440.38 2083.02,1440.51 2084.04,1440.64 2085.06,1440.76 2086.08,1440.88 2087.1,1440.99 2088.12,1441.11 \n",
       "  2089.14,1441.22 2090.16,1441.33 2091.18,1441.44 2092.2,1441.54 2093.22,1441.64 2094.24,1441.74 2095.26,1441.84 2096.28,1441.94 2097.3,1442.03 2098.32,1442.12 \n",
       "  2099.34,1442.21 2100.36,1442.3 2101.38,1442.39 2102.4,1442.47 2103.42,1442.55 2104.44,1442.63 2105.46,1442.71 2106.48,1442.78 2107.5,1442.86 2108.52,1442.93 \n",
       "  2109.54,1443 2110.56,1443.07 2111.58,1443.14 2112.6,1443.21 2113.62,1443.27 2114.64,1443.34 2115.66,1443.4 2116.68,1443.46 2117.7,1443.52 2118.72,1443.57 \n",
       "  2119.74,1443.63 2120.76,1443.68 2121.78,1443.74 2122.8,1443.79 2123.82,1443.84 2124.84,1443.89 2125.86,1443.94 2126.88,1443.99 2127.9,1444.03 2128.92,1444.08 \n",
       "  2129.94,1444.12 2130.96,1444.17 2131.98,1444.21 2133,1444.25 2134.02,1444.29 2135.04,1444.33 2136.06,1444.36 2137.08,1444.4 2138.1,1444.44 2139.12,1444.47 \n",
       "  2140.14,1444.51 2141.16,1444.54 2142.18,1444.57 2143.2,1444.6 2144.22,1444.64 2145.25,1444.67 2146.27,1444.69 2147.29,1444.72 2148.31,1444.75 2149.33,1444.78 \n",
       "  2150.35,1444.8 2151.37,1444.83 2152.39,1444.86 2153.41,1444.88 2154.43,1444.9 2155.45,1444.93 2156.47,1444.95 2157.49,1444.97 2158.51,1444.99 2159.53,1445.01 \n",
       "  2160.55,1445.03 2161.57,1445.05 2162.59,1445.07 2163.61,1445.09 2164.63,1445.11 2165.65,1445.13 2166.67,1445.14 2167.69,1445.16 2168.71,1445.18 2169.73,1445.19 \n",
       "  2170.75,1445.21 2171.77,1445.22 2172.79,1445.24 2173.81,1445.25 2174.83,1445.27 2175.85,1445.28 2176.87,1445.29 2177.89,1445.31 2178.91,1445.32 2179.93,1445.33 \n",
       "  2180.95,1445.34 2181.97,1445.35 2182.99,1445.36 2184.01,1445.37 2185.03,1445.39 2186.05,1445.4 2187.07,1445.41 2188.09,1445.41 2189.11,1445.42 2190.13,1445.43 \n",
       "  2191.15,1445.44 2192.17,1445.45 2193.19,1445.46 2194.21,1445.47 2195.23,1445.47 2196.25,1445.48 2197.27,1445.49 2198.29,1445.5 2199.31,1445.5 2200.33,1445.51 \n",
       "  2201.35,1445.52 2202.37,1445.52 2203.39,1445.53 2204.41,1445.53 2205.43,1445.54 2206.45,1445.55 2207.47,1445.55 2208.49,1445.56 2209.52,1445.56 2210.54,1445.57 \n",
       "  2211.56,1445.57 2212.58,1445.58 2213.6,1445.58 2214.62,1445.58 2215.64,1445.59 2216.66,1445.59 2217.68,1445.6 2218.7,1445.6 2219.72,1445.6 2220.74,1445.61 \n",
       "  2221.76,1445.61 2222.78,1445.62 2223.8,1445.62 2224.82,1445.62 2225.84,1445.62 2226.86,1445.63 2227.88,1445.63 2228.9,1445.63 2229.92,1445.64 2230.94,1445.64 \n",
       "  2231.96,1445.64 2232.98,1445.64 2234,1445.65 2235.02,1445.65 2236.04,1445.65 2237.06,1445.65 2238.08,1445.65 2239.1,1445.66 2240.12,1445.66 2241.14,1445.66 \n",
       "  2242.16,1445.66 2243.18,1445.66 2244.2,1445.67 2245.22,1445.67 2246.24,1445.67 2247.26,1445.67 2248.28,1445.67 2249.3,1445.67 2250.32,1445.67 2251.34,1445.68 \n",
       "  2252.36,1445.68 2253.38,1445.68 2254.4,1445.68 2255.42,1445.68 2256.44,1445.68 2257.46,1445.68 2258.48,1445.68 2259.5,1445.68 2260.52,1445.68 2261.54,1445.69 \n",
       "  2262.56,1445.69 2263.58,1445.69 2264.6,1445.69 2265.62,1445.69 2266.64,1445.69 2267.66,1445.69 2268.68,1445.69 2269.7,1445.69 2270.72,1445.69 2271.74,1445.69 \n",
       "  2272.77,1445.69 2273.79,1445.69 2274.81,1445.69 2275.83,1445.69 2276.85,1445.69 2277.87,1445.69 2278.89,1445.69 2279.91,1445.69 2280.93,1445.69 2281.95,1445.69 \n",
       "  2282.97,1445.69 2283.99,1445.69 2285.01,1445.69 2286.03,1445.69 2287.05,1445.69 2288.07,1445.69 2289.09,1445.69 2290.11,1445.69 \n",
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     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "histogram(eruptions,bins=:sqrt,label=\"eruptions\")\n",
    "plot!(p.x,p.density .* length(eruptions) .*0.2, linewidth=3,color=2,label=\"kde fit\") # nb of elements*bin width"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "Next, we will take a look at one probablity distribution, namely the normal distribution and verify that it generates a bell curve."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
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       "  463.342,1445 464.337,1444.99 465.333,1444.97 466.328,1444.95 467.323,1444.93 468.319,1444.9 469.314,1444.88 470.309,1444.86 471.304,1444.83 472.3,1444.8 \n",
       "  473.295,1444.78 474.29,1444.75 475.285,1444.72 476.281,1444.69 477.276,1444.65 478.271,1444.62 479.266,1444.58 480.262,1444.55 481.257,1444.51 482.252,1444.47 \n",
       "  483.247,1444.43 484.243,1444.39 485.238,1444.35 486.233,1444.31 487.228,1444.27 488.224,1444.22 489.219,1444.18 490.214,1444.14 491.21,1444.09 492.205,1444.05 \n",
       "  493.2,1444.01 494.195,1443.96 495.191,1443.92 496.186,1443.88 497.181,1443.84 498.176,1443.8 499.172,1443.76 500.167,1443.72 501.162,1443.68 502.157,1443.64 \n",
       "  503.153,1443.6 504.148,1443.57 505.143,1443.53 506.138,1443.5 507.134,1443.47 508.129,1443.44 509.124,1443.41 510.12,1443.38 511.115,1443.35 512.11,1443.32 \n",
       "  513.105,1443.3 514.101,1443.27 515.096,1443.24 516.091,1443.22 517.086,1443.19 518.082,1443.16 519.077,1443.14 520.072,1443.11 521.067,1443.08 522.063,1443.06 \n",
       "  523.058,1443.03 524.053,1443 525.048,1442.96 526.044,1442.93 527.039,1442.9 528.034,1442.86 529.029,1442.82 530.025,1442.78 531.02,1442.74 532.015,1442.7 \n",
       "  533.011,1442.65 534.006,1442.6 535.001,1442.55 535.996,1442.5 536.992,1442.45 537.987,1442.39 538.982,1442.33 539.977,1442.27 540.973,1442.21 541.968,1442.15 \n",
       "  542.963,1442.08 543.958,1442.01 544.954,1441.95 545.949,1441.88 546.944,1441.81 547.939,1441.74 548.935,1441.67 549.93,1441.6 550.925,1441.52 551.92,1441.45 \n",
       "  552.916,1441.38 553.911,1441.31 554.906,1441.24 555.902,1441.16 556.897,1441.09 557.892,1441.02 558.887,1440.95 559.883,1440.88 560.878,1440.81 561.873,1440.74 \n",
       "  562.868,1440.68 563.864,1440.61 564.859,1440.54 565.854,1440.48 566.849,1440.41 567.845,1440.35 568.84,1440.29 569.835,1440.22 570.83,1440.16 571.826,1440.1 \n",
       "  572.821,1440.04 573.816,1439.98 574.811,1439.92 575.807,1439.86 576.802,1439.8 577.797,1439.74 578.793,1439.68 579.788,1439.62 580.783,1439.56 581.778,1439.5 \n",
       "  582.774,1439.44 583.769,1439.38 584.764,1439.32 585.759,1439.26 586.755,1439.19 587.75,1439.12 588.745,1439.06 589.74,1438.99 590.736,1438.92 591.731,1438.84 \n",
       "  592.726,1438.76 593.721,1438.68 594.717,1438.6 595.712,1438.52 596.707,1438.43 597.702,1438.33 598.698,1438.23 599.693,1438.13 600.688,1438.02 601.684,1437.91 \n",
       "  602.679,1437.8 603.674,1437.67 604.669,1437.55 605.665,1437.41 606.66,1437.28 607.655,1437.13 608.65,1436.98 609.646,1436.83 610.641,1436.67 611.636,1436.5 \n",
       "  612.631,1436.33 613.627,1436.15 614.622,1435.97 615.617,1435.78 616.612,1435.59 617.608,1435.39 618.603,1435.19 619.598,1434.98 620.593,1434.77 621.589,1434.56 \n",
       "  622.584,1434.34 623.579,1434.12 624.575,1433.9 625.57,1433.68 626.565,1433.46 627.56,1433.23 628.556,1433 629.551,1432.78 630.546,1432.55 631.541,1432.32 \n",
       "  632.537,1432.1 633.532,1431.87 634.527,1431.64 635.522,1431.42 636.518,1431.2 637.513,1430.97 638.508,1430.75 639.503,1430.53 640.499,1430.31 641.494,1430.09 \n",
       "  642.489,1429.87 643.484,1429.66 644.48,1429.44 645.475,1429.22 646.47,1429 647.466,1428.78 648.461,1428.56 649.456,1428.34 650.451,1428.12 651.447,1427.89 \n",
       "  652.442,1427.66 653.437,1427.43 654.432,1427.2 655.428,1426.97 656.423,1426.73 657.418,1426.48 658.413,1426.24 659.409,1425.99 660.404,1425.73 661.399,1425.47 \n",
       "  662.394,1425.21 663.39,1424.94 664.385,1424.67 665.38,1424.4 666.376,1424.12 667.371,1423.83 668.366,1423.55 669.361,1423.25 670.357,1422.96 671.352,1422.66 \n",
       "  672.347,1422.36 673.342,1422.05 674.338,1421.74 675.333,1421.43 676.328,1421.11 677.323,1420.79 678.319,1420.47 679.314,1420.15 680.309,1419.82 681.304,1419.49 \n",
       "  682.3,1419.17 683.295,1418.83 684.29,1418.5 685.285,1418.17 686.281,1417.83 687.276,1417.49 688.271,1417.15 689.267,1416.81 690.262,1416.47 691.257,1416.12 \n",
       "  692.252,1415.77 693.248,1415.42 694.243,1415.07 695.238,1414.72 696.233,1414.36 697.229,1414 698.224,1413.63 699.219,1413.27 700.214,1412.89 701.21,1412.52 \n",
       "  702.205,1412.13 703.2,1411.75 704.195,1411.35 705.191,1410.95 706.186,1410.55 707.181,1410.14 708.176,1409.72 709.172,1409.29 710.167,1408.86 711.162,1408.41 \n",
       "  712.158,1407.96 713.153,1407.5 714.148,1407.03 715.143,1406.55 716.139,1406.06 717.134,1405.56 718.129,1405.06 719.124,1404.54 720.12,1404.01 721.115,1403.47 \n",
       "  722.11,1402.92 723.105,1402.37 724.101,1401.8 725.096,1401.23 726.091,1400.64 727.086,1400.05 728.082,1399.44 729.077,1398.83 730.072,1398.21 731.067,1397.59 \n",
       "  732.063,1396.95 733.058,1396.31 734.053,1395.66 735.049,1395.01 736.044,1394.35 737.039,1393.68 738.034,1393.01 739.03,1392.33 740.025,1391.65 741.02,1390.96 \n",
       "  742.015,1390.27 743.011,1389.57 744.006,1388.87 745.001,1388.17 745.996,1387.46 746.992,1386.75 747.987,1386.03 748.982,1385.3 749.977,1384.58 750.973,1383.84 \n",
       "  751.968,1383.11 752.963,1382.36 753.958,1381.61 754.954,1380.86 755.949,1380.1 756.944,1379.33 757.94,1378.55 758.935,1377.77 759.93,1376.98 760.925,1376.18 \n",
       "  761.921,1375.37 762.916,1374.55 763.911,1373.72 764.906,1372.88 765.902,1372.03 766.897,1371.17 767.892,1370.3 768.887,1369.42 769.883,1368.52 770.878,1367.62 \n",
       "  771.873,1366.7 772.868,1365.77 773.864,1364.83 774.859,1363.87 775.854,1362.91 776.849,1361.93 777.845,1360.94 778.84,1359.94 779.835,1358.93 780.831,1357.9 \n",
       "  781.826,1356.87 782.821,1355.83 783.816,1354.78 784.812,1353.72 785.807,1352.66 786.802,1351.58 787.797,1350.51 788.793,1349.42 789.788,1348.33 790.783,1347.23 \n",
       "  791.778,1346.13 792.774,1345.03 793.769,1343.93 794.764,1342.82 795.759,1341.71 796.755,1340.6 797.75,1339.48 798.745,1338.37 799.74,1337.25 800.736,1336.14 \n",
       "  801.731,1335.02 802.726,1333.9 803.722,1332.78 804.717,1331.66 805.712,1330.53 806.707,1329.4 807.703,1328.27 808.698,1327.14 809.693,1325.99 810.688,1324.85 \n",
       "  811.684,1323.69 812.679,1322.53 813.674,1321.35 814.669,1320.17 815.665,1318.97 816.66,1317.76 817.655,1316.54 818.65,1315.3 819.646,1314.05 820.641,1312.77 \n",
       "  821.636,1311.48 822.631,1310.17 823.627,1308.84 824.622,1307.49 825.617,1306.11 826.613,1304.72 827.608,1303.3 828.603,1301.85 829.598,1300.38 830.594,1298.89 \n",
       "  831.589,1297.38 832.584,1295.84 833.579,1294.28 834.575,1292.7 835.57,1291.09 836.565,1289.47 837.56,1287.82 838.556,1286.15 839.551,1284.47 840.546,1282.77 \n",
       "  841.541,1281.05 842.537,1279.32 843.532,1277.58 844.527,1275.82 845.523,1274.06 846.518,1272.28 847.513,1270.5 848.508,1268.71 849.504,1266.92 850.499,1265.12 \n",
       "  851.494,1263.31 852.489,1261.51 853.485,1259.7 854.48,1257.9 855.475,1256.09 856.47,1254.29 857.466,1252.49 858.461,1250.69 859.456,1248.89 860.451,1247.1 \n",
       "  861.447,1245.31 862.442,1243.52 863.437,1241.73 864.432,1239.95 865.428,1238.17 866.423,1236.4 867.418,1234.62 868.414,1232.85 869.409,1231.08 870.404,1229.31 \n",
       "  871.399,1227.55 872.395,1225.78 873.39,1224.01 874.385,1222.25 875.38,1220.48 876.376,1218.71 877.371,1216.93 878.366,1215.16 879.361,1213.38 880.357,1211.6 \n",
       "  881.352,1209.81 882.347,1208.01 883.342,1206.21 884.338,1204.4 885.333,1202.58 886.328,1200.76 887.323,1198.92 888.319,1197.07 889.314,1195.21 890.309,1193.33 \n",
       "  891.305,1191.44 892.3,1189.54 893.295,1187.61 894.29,1185.67 895.286,1183.71 896.281,1181.73 897.276,1179.73 898.271,1177.71 899.267,1175.66 900.262,1173.58 \n",
       "  901.257,1171.48 902.252,1169.36 903.248,1167.2 904.243,1165.02 905.238,1162.8 906.233,1160.56 907.229,1158.28 908.224,1155.98 909.219,1153.64 910.214,1151.27 \n",
       "  911.21,1148.87 912.205,1146.44 913.2,1143.97 914.196,1141.48 915.191,1138.95 916.186,1136.39 917.181,1133.81 918.177,1131.19 919.172,1128.55 920.167,1125.88 \n",
       "  921.162,1123.18 922.158,1120.46 923.153,1117.72 924.148,1114.96 925.143,1112.17 926.139,1109.37 927.134,1106.54 928.129,1103.7 929.124,1100.85 930.12,1097.98 \n",
       "  931.115,1095.1 932.11,1092.21 933.105,1089.31 934.101,1086.39 935.096,1083.47 936.091,1080.55 937.087,1077.61 938.082,1074.67 939.077,1071.72 940.072,1068.77 \n",
       "  941.068,1065.81 942.063,1062.85 943.058,1059.88 944.053,1056.91 945.049,1053.94 946.044,1050.95 947.039,1047.97 948.034,1044.97 949.03,1041.97 950.025,1038.97 \n",
       "  951.02,1035.96 952.015,1032.94 953.011,1029.92 954.006,1026.88 955.001,1023.85 955.996,1020.8 956.992,1017.75 957.987,1014.69 958.982,1011.63 959.978,1008.56 \n",
       "  960.973,1005.48 961.968,1002.4 962.963,999.314 963.959,996.225 964.954,993.132 965.949,990.037 966.944,986.941 967.94,983.844 968.935,980.746 969.93,977.649 \n",
       "  970.925,974.554 971.921,971.46 972.916,968.369 973.911,965.281 974.906,962.196 975.902,959.116 976.897,956.039 977.892,952.966 978.887,949.897 979.883,946.832 \n",
       "  980.878,943.77 981.873,940.71 982.869,937.654 983.864,934.599 984.859,931.544 985.854,928.489 986.85,925.434 987.845,922.375 988.84,919.314 989.835,916.247 \n",
       "  990.831,913.174 991.826,910.095 992.821,907.006 993.816,903.907 994.812,900.797 995.807,897.675 996.802,894.539 997.797,891.387 998.793,888.22 999.788,885.035 \n",
       "  1000.78,881.832 1001.78,878.61 1002.77,875.367 1003.77,872.104 1004.76,868.82 1005.76,865.513 1006.75,862.183 1007.75,858.831 1008.75,855.455 1009.74,852.055 \n",
       "  1010.74,848.632 1011.73,845.184 1012.73,841.711 1013.72,838.215 1014.72,834.694 1015.71,831.149 1016.71,827.58 1017.7,823.987 1018.7,820.37 1019.69,816.729 \n",
       "  1020.69,813.066 1021.68,809.38 1022.68,805.671 1023.67,801.941 1024.67,798.189 1025.66,794.417 1026.66,790.624 1027.66,786.812 1028.65,782.981 1029.65,779.132 \n",
       "  1030.64,775.265 1031.64,771.382 1032.63,767.483 1033.63,763.57 1034.62,759.642 1035.62,755.7 1036.61,751.747 1037.61,747.782 1038.6,743.806 1039.6,739.819 \n",
       "  1040.59,735.824 1041.59,731.82 1042.58,727.809 1043.58,723.79 1044.57,719.765 1045.57,715.733 1046.57,711.697 1047.56,707.656 1048.56,703.611 1049.55,699.562 \n",
       "  1050.55,695.51 1051.54,691.456 1052.54,687.399 1053.53,683.341 1054.53,679.282 1055.52,675.223 1056.52,671.164 1057.51,667.106 1058.51,663.051 1059.5,658.998 \n",
       "  1060.5,654.949 1061.49,650.904 1062.49,646.866 1063.48,642.834 1064.48,638.811 1065.48,634.798 1066.47,630.795 1067.47,626.805 1068.46,622.829 1069.46,618.869 \n",
       "  1070.45,614.925 1071.45,611.001 1072.44,607.096 1073.44,603.213 1074.43,599.353 1075.43,595.518 1076.42,591.709 1077.42,587.927 1078.41,584.174 1079.41,580.451 \n",
       "  1080.4,576.758 1081.4,573.098 1082.39,569.469 1083.39,565.874 1084.39,562.313 1085.38,558.786 1086.38,555.294 1087.37,551.836 1088.37,548.414 1089.36,545.026 \n",
       "  1090.36,541.672 1091.35,538.353 1092.35,535.068 1093.34,531.816 1094.34,528.597 1095.33,525.41 1096.33,522.254 1097.32,519.128 1098.32,516.032 1099.31,512.964 \n",
       "  1100.31,509.924 1101.3,506.909 1102.3,503.919 1103.3,500.953 1104.29,498.008 1105.29,495.083 1106.28,492.178 1107.28,489.289 1108.27,486.415 1109.27,483.555 \n",
       "  1110.26,480.706 1111.26,477.866 1112.25,475.034 1113.25,472.207 1114.24,469.382 1115.24,466.559 1116.23,463.733 1117.23,460.904 1118.22,458.068 1119.22,455.223 \n",
       "  1120.21,452.366 1121.21,449.496 1122.21,446.609 1123.2,443.703 1124.2,440.777 1125.19,437.827 1126.19,434.852 1127.18,431.849 1128.18,428.816 1129.17,425.753 \n",
       "  1130.17,422.657 1131.16,419.527 1132.16,416.362 1133.15,413.161 1134.15,409.923 1135.14,406.649 1136.14,403.338 1137.13,399.99 1138.13,396.606 1139.12,393.187 \n",
       "  1140.12,389.735 1141.12,386.25 1142.11,382.734 1143.11,379.191 1144.1,375.622 1145.1,372.029 1146.09,368.417 1147.09,364.788 1148.08,361.146 1149.08,357.494 \n",
       "  1150.07,353.836 1151.07,350.177 1152.06,346.52 1153.06,342.869 1154.05,339.229 1155.05,335.605 1156.04,331.999 1157.04,328.418 1158.03,324.863 1159.03,321.34 \n",
       "  1160.03,317.853 1161.02,314.405 1162.02,310.999 1163.01,307.639 1164.01,304.329 1165,301.07 1166,297.865 1166.99,294.717 1167.99,291.627 1168.98,288.597 \n",
       "  1169.98,285.629 1170.97,282.723 1171.97,279.881 1172.96,277.102 1173.96,274.388 1174.95,271.737 1175.95,269.149 1176.94,266.624 1177.94,264.16 1178.93,261.757 \n",
       "  1179.93,259.413 1180.93,257.127 1181.92,254.896 1182.92,252.719 1183.91,250.593 1184.91,248.517 1185.9,246.488 1186.9,244.504 1187.89,242.561 1188.89,240.658 \n",
       "  1189.88,238.792 1190.88,236.961 1191.87,235.161 1192.87,233.39 1193.86,231.645 1194.86,229.925 1195.85,228.227 1196.85,226.547 1197.84,224.885 1198.84,223.238 \n",
       "  1199.84,221.605 1200.83,219.982 1201.83,218.369 1202.82,216.764 1203.82,215.166 1204.81,213.574 1205.81,211.986 1206.8,210.402 1207.8,208.821 1208.79,207.243 \n",
       "  1209.79,205.666 1210.78,204.092 1211.78,202.521 1212.77,200.951 1213.77,199.384 1214.76,197.82 1215.76,196.26 1216.75,194.704 1217.75,193.152 1218.75,191.606 \n",
       "  1219.74,190.066 1220.74,188.533 1221.73,187.008 1222.73,185.492 1223.72,183.984 1224.72,182.487 1225.71,181 1226.71,179.524 1227.7,178.059 1228.7,176.606 \n",
       "  1229.69,175.165 1230.69,173.737 1231.68,172.32 1232.68,170.916 1233.67,169.523 1234.67,168.143 1235.66,166.774 1236.66,165.416 1237.66,164.068 1238.65,162.731 \n",
       "  1239.65,161.404 1240.64,160.086 1241.64,158.777 1242.63,157.476 1243.63,156.183 1244.62,154.898 1245.62,153.621 1246.61,152.351 1247.61,151.088 1248.6,149.833 \n",
       "  1249.6,148.585 1250.59,147.345 1251.59,146.114 1252.58,144.892 1253.58,143.68 1254.57,142.479 1255.57,141.29 1256.57,140.114 1257.56,138.952 1258.56,137.807 \n",
       "  1259.55,136.68 1260.55,135.572 1261.54,134.485 1262.54,133.421 1263.53,132.383 1264.53,131.372 1265.52,130.391 1266.52,129.44 1267.51,128.524 1268.51,127.643 \n",
       "  1269.5,126.799 1270.5,125.995 1271.49,125.233 1272.49,124.514 1273.48,123.841 1274.48,123.213 1275.48,122.634 1276.47,122.104 1277.47,121.625 1278.46,121.197 \n",
       "  1279.46,120.82 1280.45,120.497 1281.45,120.225 1282.44,120.007 1283.44,119.842 1284.43,119.728 1285.43,119.667 1286.42,119.657 1287.42,119.697 1288.41,119.787 \n",
       "  1289.41,119.924 1290.4,120.109 1291.4,120.339 1292.39,120.613 1293.39,120.929 1294.39,121.287 1295.38,121.684 1296.38,122.119 1297.37,122.59 1298.37,123.097 \n",
       "  1299.36,123.637 1300.36,124.21 1301.35,124.814 1302.35,125.45 1303.34,126.115 1304.34,126.809 1305.33,127.532 1306.33,128.283 1307.32,129.063 1308.32,129.872 \n",
       "  1309.31,130.709 1310.31,131.575 1311.3,132.471 1312.3,133.397 1313.3,134.355 1314.29,135.345 1315.29,136.367 1316.28,137.424 1317.28,138.517 1318.27,139.646 \n",
       "  1319.27,140.813 1320.26,142.02 1321.26,143.266 1322.25,144.555 1323.25,145.886 1324.24,147.261 1325.24,148.681 1326.23,150.147 1327.23,151.66 1328.22,153.221 \n",
       "  1329.22,154.829 1330.21,156.487 1331.21,158.193 1332.21,159.949 1333.2,161.754 1334.2,163.608 1335.19,165.512 1336.19,167.464 1337.18,169.465 1338.18,171.513 \n",
       "  1339.17,173.608 1340.17,175.75 1341.16,177.936 1342.16,180.167 1343.15,182.44 1344.15,184.756 1345.14,187.112 1346.14,189.507 1347.13,191.939 1348.13,194.408 \n",
       "  1349.12,196.912 1350.12,199.449 1351.12,202.018 1352.11,204.618 1353.11,207.247 1354.1,209.903 1355.1,212.587 1356.09,215.295 1357.09,218.027 1358.08,220.783 \n",
       "  1359.08,223.56 1360.07,226.358 1361.07,229.176 1362.06,232.013 1363.06,234.868 1364.05,237.741 1365.05,240.63 1366.04,243.536 1367.04,246.458 1368.03,249.395 \n",
       "  1369.03,252.346 1370.03,255.312 1371.02,258.292 1372.02,261.285 1373.01,264.29 1374.01,267.308 1375,270.338 1376,273.379 1376.99,276.431 1377.99,279.493 \n",
       "  1378.98,282.564 1379.98,285.643 1380.97,288.731 1381.97,291.825 1382.96,294.925 1383.96,298.031 1384.95,301.14 1385.95,304.252 1386.94,307.365 1387.94,310.479 \n",
       "  1388.94,313.593 1389.93,316.704 1390.93,319.812 1391.92,322.916 1392.92,326.014 1393.91,329.106 1394.91,332.189 1395.9,335.263 1396.9,338.327 1397.89,341.38 \n",
       "  1398.89,344.421 1399.88,347.449 1400.88,350.464 1401.87,353.465 1402.87,356.451 1403.86,359.423 1404.86,362.38 1405.85,365.322 1406.85,368.249 1407.85,371.161 \n",
       "  1408.84,374.06 1409.84,376.945 1410.83,379.817 1411.83,382.678 1412.82,385.527 1413.82,388.366 1414.81,391.196 1415.81,394.019 1416.8,396.836 1417.8,399.647 \n",
       "  1418.79,402.455 1419.79,405.261 1420.78,408.067 1421.78,410.874 1422.77,413.683 1423.77,416.496 1424.76,419.314 1425.76,422.14 1426.76,424.973 1427.75,427.816 \n",
       "  1428.75,430.67 1429.74,433.535 1430.74,436.412 1431.73,439.304 1432.73,442.21 1433.72,445.131 1434.72,448.068 1435.71,451.021 1436.71,453.991 1437.7,456.978 \n",
       "  1438.7,459.983 1439.69,463.006 1440.69,466.047 1441.68,469.106 1442.68,472.183 1443.67,475.279 1444.67,478.393 1445.66,481.525 1446.66,484.676 1447.66,487.845 \n",
       "  1448.65,491.033 1449.65,494.24 1450.64,497.464 1451.64,500.708 1452.63,503.969 1453.63,507.25 1454.62,510.548 1455.62,513.865 1456.61,517.2 1457.61,520.554 \n",
       "  1458.6,523.925 1459.6,527.315 1460.59,530.722 1461.59,534.147 1462.58,537.589 1463.58,541.048 1464.57,544.524 1465.57,548.016 1466.57,551.523 1467.56,555.046 \n",
       "  1468.56,558.583 1469.55,562.134 1470.55,565.698 1471.54,569.274 1472.54,572.862 1473.53,576.46 1474.53,580.068 1475.52,583.685 1476.52,587.31 1477.51,590.942 \n",
       "  1478.51,594.58 1479.5,598.222 1480.5,601.869 1481.49,605.52 1482.49,609.172 1483.48,612.826 1484.48,616.48 1485.48,620.133 1486.47,623.785 1487.47,627.435 \n",
       "  1488.46,631.082 1489.46,634.724 1490.45,638.362 1491.45,641.995 1492.44,645.621 1493.44,649.239 1494.43,652.85 1495.43,656.452 1496.42,660.044 1497.42,663.626 \n",
       "  1498.41,667.197 1499.41,670.756 1500.4,674.303 1501.4,677.836 1502.39,681.354 1503.39,684.858 1504.39,688.347 1505.38,691.82 1506.38,695.276 1507.37,698.715 \n",
       "  1508.37,702.137 1509.36,705.541 1510.36,708.927 1511.35,712.296 1512.35,715.647 1513.34,718.98 1514.34,722.297 1515.33,725.597 1516.33,728.881 1517.32,732.151 \n",
       "  1518.32,735.408 1519.31,738.652 1520.31,741.886 1521.3,745.11 1522.3,748.328 1523.3,751.54 1524.29,754.75 1525.29,757.959 1526.28,761.17 1527.28,764.386 \n",
       "  1528.27,767.608 1529.27,770.841 1530.26,774.085 1531.26,777.345 1532.25,780.623 1533.25,783.92 1534.24,787.241 1535.24,790.587 1536.23,793.961 1537.23,797.365 \n",
       "  1538.22,800.8 1539.22,804.269 1540.21,807.773 1541.21,811.313 1542.21,814.891 1543.2,818.507 1544.2,822.161 1545.19,825.855 1546.19,829.587 1547.18,833.358 \n",
       "  1548.18,837.167 1549.17,841.012 1550.17,844.894 1551.16,848.809 1552.16,852.757 1553.15,856.735 1554.15,860.741 1555.14,864.772 1556.14,868.827 1557.13,872.901 \n",
       "  1558.13,876.992 1559.12,881.097 1560.12,885.212 1561.12,889.334 1562.11,893.459 1563.11,897.585 1564.1,901.706 1565.1,905.821 1566.09,909.926 1567.09,914.016 \n",
       "  1568.08,918.089 1569.08,922.142 1570.07,926.171 1571.07,930.173 1572.06,934.147 1573.06,938.088 1574.05,941.995 1575.05,945.866 1576.04,949.698 1577.04,953.49 \n",
       "  1578.03,957.24 1579.03,960.948 1580.03,964.611 1581.02,968.23 1582.02,971.803 1583.01,975.331 1584.01,978.813 1585,982.249 1586,985.641 1586.99,988.987 \n",
       "  1587.99,992.29 1588.98,995.551 1589.98,998.769 1590.97,1001.95 1591.97,1005.09 1592.96,1008.19 1593.96,1011.26 1594.95,1014.29 1595.95,1017.29 1596.94,1020.27 \n",
       "  1597.94,1023.21 1598.94,1026.14 1599.93,1029.03 1600.93,1031.91 1601.92,1034.77 1602.92,1037.62 1603.91,1040.44 1604.91,1043.26 1605.9,1046.07 1606.9,1048.87 \n",
       "  1607.89,1051.66 1608.89,1054.45 1609.88,1057.23 1610.88,1060.01 1611.87,1062.79 1612.87,1065.57 1613.86,1068.35 1614.86,1071.13 1615.85,1073.91 1616.85,1076.7 \n",
       "  1617.85,1079.48 1618.84,1082.26 1619.84,1085.05 1620.83,1087.83 1621.83,1090.61 1622.82,1093.39 1623.82,1096.17 1624.81,1098.94 1625.81,1101.7 1626.8,1104.46 \n",
       "  1627.8,1107.21 1628.79,1109.94 1629.79,1112.66 1630.78,1115.37 1631.78,1118.06 1632.77,1120.73 1633.77,1123.39 1634.76,1126.02 1635.76,1128.63 1636.76,1131.22 \n",
       "  1637.75,1133.78 1638.75,1136.31 1639.74,1138.82 1640.74,1141.3 1641.73,1143.75 1642.73,1146.17 1643.72,1148.57 1644.72,1150.93 1645.71,1153.26 1646.71,1155.56 \n",
       "  1647.7,1157.83 1648.7,1160.07 1649.69,1162.28 1650.69,1164.46 1651.68,1166.61 1652.68,1168.74 1653.67,1170.83 1654.67,1172.9 1655.67,1174.94 1656.66,1176.96 \n",
       "  1657.66,1178.95 1658.65,1180.92 1659.65,1182.87 1660.64,1184.8 1661.64,1186.71 1662.63,1188.6 1663.63,1190.48 1664.62,1192.33 1665.62,1194.18 1666.61,1196.01 \n",
       "  1667.61,1197.83 1668.6,1199.64 1669.6,1201.44 1670.59,1203.24 1671.59,1205.02 1672.58,1206.81 1673.58,1208.59 1674.58,1210.36 1675.57,1212.14 1676.57,1213.92 \n",
       "  1677.56,1215.7 1678.56,1217.47 1679.55,1219.26 1680.55,1221.04 1681.54,1222.83 1682.54,1224.62 1683.53,1226.42 1684.53,1228.23 1685.52,1230.03 1686.52,1231.85 \n",
       "  1687.51,1233.66 1688.51,1235.48 1689.5,1237.31 1690.5,1239.14 1691.49,1240.97 1692.49,1242.81 1693.48,1244.64 1694.48,1246.48 1695.48,1248.31 1696.47,1250.14 \n",
       "  1697.47,1251.97 1698.46,1253.8 1699.46,1255.62 1700.45,1257.43 1701.45,1259.24 1702.44,1261.04 1703.44,1262.82 1704.43,1264.6 1705.43,1266.37 1706.42,1268.12 \n",
       "  1707.42,1269.86 1708.41,1271.59 1709.41,1273.3 1710.4,1275 1711.4,1276.68 1712.39,1278.34 1713.39,1279.99 1714.39,1281.62 1715.38,1283.23 1716.38,1284.83 \n",
       "  1717.37,1286.41 1718.37,1287.97 1719.36,1289.52 1720.36,1291.05 1721.35,1292.56 1722.35,1294.06 1723.34,1295.54 1724.34,1297 1725.33,1298.46 1726.33,1299.89 \n",
       "  1727.32,1301.31 1728.32,1302.72 1729.31,1304.12 1730.31,1305.5 1731.3,1306.87 1732.3,1308.23 1733.3,1309.57 1734.29,1310.91 1735.29,1312.23 1736.28,1313.54 \n",
       "  1737.28,1314.84 1738.27,1316.13 1739.27,1317.41 1740.26,1318.68 1741.26,1319.94 1742.25,1321.19 1743.25,1322.43 1744.24,1323.67 1745.24,1324.89 1746.23,1326.1 \n",
       "  1747.23,1327.3 1748.22,1328.5 1749.22,1329.69 1750.21,1330.87 1751.21,1332.04 1752.21,1333.2 1753.2,1334.36 1754.2,1335.51 1755.19,1336.65 1756.19,1337.79 \n",
       "  1757.18,1338.92 1758.18,1340.05 1759.17,1341.17 1760.17,1342.29 1761.16,1343.4 1762.16,1344.51 1763.15,1345.62 1764.15,1346.73 1765.14,1347.84 1766.14,1348.94 \n",
       "  1767.13,1350.05 1768.13,1351.15 1769.12,1352.25 1770.12,1353.36 1771.12,1354.46 1772.11,1355.56 1773.11,1356.66 1774.1,1357.76 1775.1,1358.86 1776.09,1359.95 \n",
       "  1777.09,1361.05 1778.08,1362.14 1779.08,1363.23 1780.07,1364.31 1781.07,1365.39 1782.06,1366.47 1783.06,1367.53 1784.05,1368.59 1785.05,1369.64 1786.04,1370.69 \n",
       "  1787.04,1371.72 1788.03,1372.73 1789.03,1373.74 1790.03,1374.73 1791.02,1375.71 1792.02,1376.68 1793.01,1377.63 1794.01,1378.56 1795,1379.47 1796,1380.37 \n",
       "  1796.99,1381.25 1797.99,1382.12 1798.98,1382.96 1799.98,1383.79 1800.97,1384.59 1801.97,1385.38 1802.96,1386.16 1803.96,1386.91 1804.95,1387.65 1805.95,1388.37 \n",
       "  1806.94,1389.07 1807.94,1389.76 1808.94,1390.43 1809.93,1391.09 1810.93,1391.74 1811.92,1392.37 1812.92,1392.98 1813.91,1393.59 1814.91,1394.18 1815.9,1394.76 \n",
       "  1816.9,1395.34 1817.89,1395.9 1818.89,1396.45 1819.88,1396.99 1820.88,1397.53 1821.87,1398.05 1822.87,1398.57 1823.86,1399.08 1824.86,1399.59 1825.85,1400.09 \n",
       "  1826.85,1400.58 1827.85,1401.06 1828.84,1401.53 1829.84,1402 1830.83,1402.47 1831.83,1402.92 1832.82,1403.37 1833.82,1403.81 1834.81,1404.25 1835.81,1404.68 \n",
       "  1836.8,1405.1 1837.8,1405.51 1838.79,1405.92 1839.79,1406.32 1840.78,1406.71 1841.78,1407.09 1842.77,1407.47 1843.77,1407.84 1844.76,1408.2 1845.76,1408.56 \n",
       "  1846.76,1408.91 1847.75,1409.26 1848.75,1409.6 1849.74,1409.93 1850.74,1410.26 1851.73,1410.59 1852.73,1410.91 1853.72,1411.23 1854.72,1411.55 1855.71,1411.86 \n",
       "  1856.71,1412.18 1857.7,1412.49 1858.7,1412.81 1859.69,1413.12 1860.69,1413.44 1861.68,1413.76 1862.68,1414.09 1863.67,1414.42 1864.67,1414.75 1865.67,1415.09 \n",
       "  1866.66,1415.43 1867.66,1415.77 1868.65,1416.13 1869.65,1416.49 1870.64,1416.85 1871.64,1417.22 1872.63,1417.6 1873.63,1417.98 1874.62,1418.36 1875.62,1418.75 \n",
       "  1876.61,1419.15 1877.61,1419.55 1878.6,1419.95 1879.6,1420.35 1880.59,1420.76 1881.59,1421.17 1882.58,1421.58 1883.58,1421.98 1884.58,1422.39 1885.57,1422.79 \n",
       "  1886.57,1423.19 1887.56,1423.59 1888.56,1423.98 1889.55,1424.37 1890.55,1424.75 1891.54,1425.13 1892.54,1425.49 1893.53,1425.85 1894.53,1426.2 1895.52,1426.55 \n",
       "  1896.52,1426.88 1897.51,1427.2 1898.51,1427.52 1899.5,1427.82 1900.5,1428.12 1901.49,1428.4 1902.49,1428.68 1903.49,1428.94 1904.48,1429.2 1905.48,1429.44 \n",
       "  1906.47,1429.68 1907.47,1429.91 1908.46,1430.13 1909.46,1430.34 1910.45,1430.55 1911.45,1430.74 1912.44,1430.93 1913.44,1431.12 1914.43,1431.3 1915.43,1431.47 \n",
       "  1916.42,1431.64 1917.42,1431.8 1918.41,1431.96 1919.41,1432.11 1920.4,1432.26 1921.4,1432.41 1922.4,1432.55 1923.39,1432.7 1924.39,1432.84 1925.38,1432.97 \n",
       "  1926.38,1433.11 1927.37,1433.24 1928.37,1433.38 1929.36,1433.51 1930.36,1433.64 1931.35,1433.77 1932.35,1433.9 1933.34,1434.03 1934.34,1434.15 1935.33,1434.28 \n",
       "  1936.33,1434.41 1937.32,1434.53 1938.32,1434.66 1939.31,1434.78 1940.31,1434.91 1941.31,1435.03 1942.3,1435.15 1943.3,1435.27 1944.29,1435.4 1945.29,1435.52 \n",
       "  1946.28,1435.64 1947.28,1435.76 1948.27,1435.88 1949.27,1436.01 1950.26,1436.13 1951.26,1436.25 1952.25,1436.37 1953.25,1436.49 1954.24,1436.61 1955.24,1436.74 \n",
       "  1956.23,1436.86 1957.23,1436.98 1958.22,1437.11 1959.22,1437.23 1960.21,1437.36 1961.21,1437.48 1962.21,1437.61 1963.2,1437.74 1964.2,1437.87 1965.19,1438 \n",
       "  1966.19,1438.13 1967.18,1438.26 1968.18,1438.39 1969.17,1438.53 1970.17,1438.66 1971.16,1438.8 1972.16,1438.94 1973.15,1439.08 1974.15,1439.21 1975.14,1439.35 \n",
       "  1976.14,1439.5 1977.13,1439.64 1978.13,1439.78 1979.12,1439.92 1980.12,1440.06 1981.12,1440.21 1982.11,1440.35 1983.11,1440.49 1984.1,1440.64 1985.1,1440.78 \n",
       "  1986.09,1440.92 1987.09,1441.07 1988.08,1441.21 1989.08,1441.35 1990.07,1441.49 1991.07,1441.63 1992.06,1441.76 1993.06,1441.9 1994.05,1442.03 1995.05,1442.16 \n",
       "  1996.04,1442.29 1997.04,1442.42 1998.03,1442.54 1999.03,1442.67 2000.03,1442.79 2001.02,1442.9 2002.02,1443.02 2003.01,1443.13 2004.01,1443.23 2005,1443.34 \n",
       "  2006,1443.44 2006.99,1443.53 2007.99,1443.63 2008.98,1443.72 2009.98,1443.8 2010.97,1443.88 2011.97,1443.96 2012.96,1444.04 2013.96,1444.11 2014.95,1444.17 \n",
       "  2015.95,1444.23 2016.94,1444.29 2017.94,1444.35 2018.94,1444.4 2019.93,1444.45 2020.93,1444.49 2021.92,1444.54 2022.92,1444.57 2023.91,1444.61 2024.91,1444.64 \n",
       "  2025.9,1444.67 2026.9,1444.7 2027.89,1444.72 2028.89,1444.74 2029.88,1444.76 2030.88,1444.77 2031.87,1444.79 2032.87,1444.8 2033.86,1444.8 2034.86,1444.81 \n",
       "  2035.85,1444.82 2036.85,1444.82 2037.85,1444.82 2038.84,1444.82 2039.84,1444.82 2040.83,1444.81 2041.83,1444.81 2042.82,1444.8 2043.82,1444.79 2044.81,1444.79 \n",
       "  2045.81,1444.78 2046.8,1444.76 2047.8,1444.75 2048.79,1444.74 2049.79,1444.73 2050.78,1444.71 2051.78,1444.7 2052.77,1444.69 2053.77,1444.67 2054.76,1444.66 \n",
       "  2055.76,1444.64 2056.76,1444.63 2057.75,1444.61 2058.75,1444.6 2059.74,1444.59 2060.74,1444.57 2061.73,1444.56 2062.73,1444.55 2063.72,1444.54 2064.72,1444.53 \n",
       "  2065.71,1444.52 2066.71,1444.51 2067.7,1444.5 2068.7,1444.5 2069.69,1444.49 2070.69,1444.49 2071.68,1444.49 2072.68,1444.49 2073.67,1444.49 2074.67,1444.49 \n",
       "  2075.67,1444.49 2076.66,1444.5 2077.66,1444.5 2078.65,1444.51 2079.65,1444.52 2080.64,1444.53 2081.64,1444.54 2082.63,1444.56 2083.63,1444.57 2084.62,1444.59 \n",
       "  2085.62,1444.6 2086.61,1444.62 2087.61,1444.64 2088.6,1444.66 2089.6,1444.68 2090.59,1444.7 2091.59,1444.72 2092.58,1444.74 2093.58,1444.76 2094.58,1444.78 \n",
       "  2095.57,1444.81 2096.57,1444.83 2097.56,1444.85 2098.56,1444.87 2099.55,1444.89 2100.55,1444.91 2101.54,1444.93 2102.54,1444.95 2103.53,1444.98 2104.53,1444.99 \n",
       "  2105.52,1445.01 2106.52,1445.03 2107.51,1445.05 2108.51,1445.07 2109.5,1445.09 2110.5,1445.1 2111.49,1445.12 2112.49,1445.14 2113.49,1445.15 2114.48,1445.17 \n",
       "  2115.48,1445.18 2116.47,1445.2 2117.47,1445.22 2118.46,1445.23 2119.46,1445.24 2120.45,1445.26 2121.45,1445.27 2122.44,1445.29 2123.44,1445.3 2124.43,1445.32 \n",
       "  2125.43,1445.33 2126.42,1445.34 2127.42,1445.36 2128.41,1445.37 2129.41,1445.39 2130.4,1445.4 2131.4,1445.41 2132.4,1445.43 2133.39,1445.44 2134.39,1445.45 \n",
       "  2135.38,1445.46 2136.38,1445.48 2137.37,1445.49 2138.37,1445.5 2139.36,1445.51 2140.36,1445.53 2141.35,1445.54 2142.35,1445.55 2143.34,1445.56 2144.34,1445.57 \n",
       "  2145.33,1445.58 2146.33,1445.59 2147.32,1445.6 2148.32,1445.61 2149.31,1445.62 2150.31,1445.62 2151.31,1445.63 2152.3,1445.64 2153.3,1445.64 2154.29,1445.65 \n",
       "  2155.29,1445.66 2156.28,1445.66 2157.28,1445.67 2158.27,1445.67 2159.27,1445.67 2160.26,1445.68 2161.26,1445.68 2162.25,1445.68 2163.25,1445.69 2164.24,1445.69 \n",
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      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "randomVector = randn(100000) # Randomly generate n values \n",
    "                             # from a distribution N(0,1)\n",
    "histogram(randomVector)\n",
    "p = kde(randomVector)\n",
    "plot!(p.x, p.density .* length(randomVector) .* 0.1, linewidth=3, label=\"kde fit\")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4. Probability distributions\n",
    "\n",
    "The Distributions.jl package provides everything needeed to work with probability distributions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "params(d) = (0.0, 1.0)\n",
      "mean(d) = 0.0\n",
      "std(d) = 1.0\n"
     ]
    },
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       "  314.053,1445.62 315.048,1445.61 316.044,1445.6 317.039,1445.6 318.034,1445.59 319.029,1445.58 320.025,1445.58 321.02,1445.57 322.015,1445.56 323.01,1445.56 \n",
       "  324.006,1445.55 325.001,1445.54 325.996,1445.54 326.991,1445.53 327.987,1445.52 328.982,1445.51 329.977,1445.51 330.973,1445.5 331.968,1445.49 332.963,1445.49 \n",
       "  333.958,1445.48 334.954,1445.47 335.949,1445.46 336.944,1445.46 337.939,1445.45 338.935,1445.44 339.93,1445.43 340.925,1445.43 341.92,1445.42 342.916,1445.41 \n",
       "  343.911,1445.4 344.906,1445.4 345.901,1445.39 346.897,1445.38 347.892,1445.37 348.887,1445.36 349.882,1445.35 350.878,1445.34 351.873,1445.33 352.868,1445.32 \n",
       "  353.864,1445.31 354.859,1445.3 355.854,1445.29 356.849,1445.28 357.845,1445.27 358.84,1445.26 359.835,1445.25 360.83,1445.23 361.826,1445.22 362.821,1445.21 \n",
       "  363.816,1445.2 364.811,1445.18 365.807,1445.17 366.802,1445.15 367.797,1445.14 368.792,1445.12 369.788,1445.11 370.783,1445.09 371.778,1445.08 372.773,1445.06 \n",
       "  373.769,1445.05 374.764,1445.03 375.759,1445.02 376.755,1445 377.75,1444.99 378.745,1444.97 379.74,1444.96 380.736,1444.94 381.731,1444.92 382.726,1444.91 \n",
       "  383.721,1444.9 384.717,1444.88 385.712,1444.87 386.707,1444.85 387.702,1444.84 388.698,1444.83 389.693,1444.81 390.688,1444.8 391.683,1444.79 392.679,1444.78 \n",
       "  393.674,1444.76 394.669,1444.75 395.664,1444.74 396.66,1444.73 397.655,1444.72 398.65,1444.71 399.646,1444.69 400.641,1444.68 401.636,1444.67 402.631,1444.66 \n",
       "  403.627,1444.65 404.622,1444.64 405.617,1444.63 406.612,1444.61 407.608,1444.6 408.603,1444.59 409.598,1444.57 410.593,1444.56 411.589,1444.54 412.584,1444.53 \n",
       "  413.579,1444.51 414.574,1444.49 415.57,1444.47 416.565,1444.45 417.56,1444.43 418.555,1444.41 419.551,1444.39 420.546,1444.36 421.541,1444.34 422.537,1444.31 \n",
       "  423.532,1444.28 424.527,1444.25 425.522,1444.22 426.518,1444.19 427.513,1444.15 428.508,1444.12 429.503,1444.08 430.499,1444.04 431.494,1444 432.489,1443.96 \n",
       "  433.484,1443.92 434.48,1443.88 435.475,1443.83 436.47,1443.79 437.465,1443.74 438.461,1443.7 439.456,1443.65 440.451,1443.61 441.446,1443.56 442.442,1443.52 \n",
       "  443.437,1443.48 444.432,1443.43 445.428,1443.39 446.423,1443.35 447.418,1443.31 448.413,1443.27 449.409,1443.23 450.404,1443.2 451.399,1443.16 452.394,1443.13 \n",
       "  453.39,1443.1 454.385,1443.07 455.38,1443.04 456.375,1443.02 457.371,1443 458.366,1442.98 459.361,1442.96 460.356,1442.94 461.352,1442.93 462.347,1442.92 \n",
       "  463.342,1442.9 464.337,1442.89 465.333,1442.89 466.328,1442.88 467.323,1442.87 468.319,1442.86 469.314,1442.86 470.309,1442.85 471.304,1442.85 472.3,1442.84 \n",
       "  473.295,1442.83 474.29,1442.82 475.285,1442.81 476.281,1442.8 477.276,1442.79 478.271,1442.77 479.266,1442.76 480.262,1442.74 481.257,1442.72 482.252,1442.69 \n",
       "  483.247,1442.66 484.243,1442.63 485.238,1442.6 486.233,1442.56 487.228,1442.52 488.224,1442.48 489.219,1442.43 490.214,1442.38 491.21,1442.33 492.205,1442.27 \n",
       "  493.2,1442.21 494.195,1442.15 495.191,1442.08 496.186,1442.01 497.181,1441.94 498.176,1441.87 499.172,1441.79 500.167,1441.71 501.162,1441.63 502.157,1441.54 \n",
       "  503.153,1441.45 504.148,1441.36 505.143,1441.27 506.138,1441.18 507.134,1441.08 508.129,1440.98 509.124,1440.88 510.12,1440.78 511.115,1440.67 512.11,1440.57 \n",
       "  513.105,1440.46 514.101,1440.35 515.096,1440.24 516.091,1440.13 517.086,1440.02 518.082,1439.9 519.077,1439.79 520.072,1439.67 521.067,1439.55 522.063,1439.43 \n",
       "  523.058,1439.31 524.053,1439.19 525.048,1439.07 526.044,1438.95 527.039,1438.82 528.034,1438.7 529.029,1438.57 530.025,1438.45 531.02,1438.32 532.015,1438.2 \n",
       "  533.011,1438.07 534.006,1437.95 535.001,1437.82 535.996,1437.7 536.992,1437.57 537.987,1437.45 538.982,1437.33 539.977,1437.2 540.973,1437.08 541.968,1436.96 \n",
       "  542.963,1436.84 543.958,1436.72 544.954,1436.6 545.949,1436.48 546.944,1436.36 547.939,1436.24 548.935,1436.12 549.93,1436 550.925,1435.89 551.92,1435.77 \n",
       "  552.916,1435.65 553.911,1435.54 554.906,1435.42 555.902,1435.31 556.897,1435.19 557.892,1435.07 558.887,1434.95 559.883,1434.83 560.878,1434.71 561.873,1434.59 \n",
       "  562.868,1434.47 563.864,1434.34 564.859,1434.21 565.854,1434.08 566.849,1433.95 567.845,1433.82 568.84,1433.68 569.835,1433.54 570.83,1433.4 571.826,1433.25 \n",
       "  572.821,1433.11 573.816,1432.95 574.811,1432.8 575.807,1432.64 576.802,1432.47 577.797,1432.31 578.793,1432.13 579.788,1431.96 580.783,1431.78 581.778,1431.59 \n",
       "  582.774,1431.41 583.769,1431.21 584.764,1431.01 585.759,1430.81 586.755,1430.6 587.75,1430.39 588.745,1430.17 589.74,1429.95 590.736,1429.72 591.731,1429.48 \n",
       "  592.726,1429.24 593.721,1429 594.717,1428.75 595.712,1428.49 596.707,1428.22 597.702,1427.96 598.698,1427.68 599.693,1427.4 600.688,1427.11 601.684,1426.81 \n",
       "  602.679,1426.51 603.674,1426.2 604.669,1425.89 605.665,1425.57 606.66,1425.24 607.655,1424.91 608.65,1424.57 609.646,1424.22 610.641,1423.86 611.636,1423.5 \n",
       "  612.631,1423.14 613.627,1422.77 614.622,1422.39 615.617,1422.01 616.612,1421.62 617.608,1421.22 618.603,1420.82 619.598,1420.42 620.593,1420.01 621.589,1419.6 \n",
       "  622.584,1419.19 623.579,1418.77 624.575,1418.35 625.57,1417.93 626.565,1417.5 627.56,1417.08 628.556,1416.65 629.551,1416.22 630.546,1415.79 631.541,1415.36 \n",
       "  632.537,1414.93 633.532,1414.51 634.527,1414.08 635.522,1413.66 636.518,1413.24 637.513,1412.82 638.508,1412.4 639.503,1411.99 640.499,1411.58 641.494,1411.17 \n",
       "  642.489,1410.77 643.484,1410.37 644.48,1409.97 645.475,1409.58 646.47,1409.19 647.466,1408.81 648.461,1408.42 649.456,1408.05 650.451,1407.67 651.447,1407.3 \n",
       "  652.442,1406.93 653.437,1406.56 654.432,1406.19 655.428,1405.83 656.423,1405.47 657.418,1405.11 658.413,1404.74 659.409,1404.38 660.404,1404.02 661.399,1403.66 \n",
       "  662.394,1403.29 663.39,1402.93 664.385,1402.56 665.38,1402.19 666.376,1401.82 667.371,1401.44 668.366,1401.07 669.361,1400.68 670.357,1400.3 671.352,1399.91 \n",
       "  672.347,1399.51 673.342,1399.11 674.338,1398.71 675.333,1398.3 676.328,1397.89 677.323,1397.47 678.319,1397.04 679.314,1396.62 680.309,1396.18 681.304,1395.74 \n",
       "  682.3,1395.3 683.295,1394.85 684.29,1394.4 685.285,1393.94 686.281,1393.48 687.276,1393.01 688.271,1392.54 689.267,1392.06 690.262,1391.58 691.257,1391.09 \n",
       "  692.252,1390.59 693.248,1390.09 694.243,1389.59 695.238,1389.08 696.233,1388.56 697.229,1388.04 698.224,1387.51 699.219,1386.97 700.214,1386.43 701.21,1385.87 \n",
       "  702.205,1385.31 703.2,1384.74 704.195,1384.16 705.191,1383.57 706.186,1382.97 707.181,1382.35 708.176,1381.73 709.172,1381.09 710.167,1380.44 711.162,1379.77 \n",
       "  712.158,1379.09 713.153,1378.4 714.148,1377.69 715.143,1376.96 716.139,1376.22 717.134,1375.45 718.129,1374.67 719.124,1373.87 720.12,1373.06 721.115,1372.22 \n",
       "  722.11,1371.36 723.105,1370.49 724.101,1369.59 725.096,1368.68 726.091,1367.74 727.086,1366.79 728.082,1365.82 729.077,1364.82 730.072,1363.81 731.067,1362.79 \n",
       "  732.063,1361.74 733.058,1360.68 734.053,1359.6 735.049,1358.5 736.044,1357.39 737.039,1356.27 738.034,1355.13 739.03,1353.99 740.025,1352.83 741.02,1351.65 \n",
       "  742.015,1350.47 743.011,1349.28 744.006,1348.09 745.001,1346.88 745.996,1345.67 746.992,1344.45 747.987,1343.23 748.982,1342 749.977,1340.77 750.973,1339.54 \n",
       "  751.968,1338.3 752.963,1337.06 753.958,1335.82 754.954,1334.58 755.949,1333.33 756.944,1332.08 757.94,1330.83 758.935,1329.57 759.93,1328.31 760.925,1327.05 \n",
       "  761.921,1325.79 762.916,1324.52 763.911,1323.25 764.906,1321.97 765.902,1320.69 766.897,1319.4 767.892,1318.11 768.887,1316.8 769.883,1315.49 770.878,1314.17 \n",
       "  771.873,1312.85 772.868,1311.51 773.864,1310.16 774.859,1308.81 775.854,1307.44 776.849,1306.06 777.845,1304.67 778.84,1303.27 779.835,1301.86 780.831,1300.44 \n",
       "  781.826,1299 782.821,1297.55 783.816,1296.09 784.812,1294.62 785.807,1293.14 786.802,1291.65 787.797,1290.14 788.793,1288.63 789.788,1287.1 790.783,1285.57 \n",
       "  791.778,1284.02 792.774,1282.46 793.769,1280.9 794.764,1279.32 795.759,1277.74 796.755,1276.15 797.75,1274.55 798.745,1272.94 799.74,1271.32 800.736,1269.69 \n",
       "  801.731,1268.05 802.726,1266.41 803.722,1264.76 804.717,1263.09 805.712,1261.42 806.707,1259.74 807.703,1258.04 808.698,1256.34 809.693,1254.63 810.688,1252.9 \n",
       "  811.684,1251.16 812.679,1249.41 813.674,1247.65 814.669,1245.88 815.665,1244.09 816.66,1242.29 817.655,1240.47 818.65,1238.64 819.646,1236.8 820.641,1234.95 \n",
       "  821.636,1233.08 822.631,1231.2 823.627,1229.31 824.622,1227.4 825.617,1225.48 826.613,1223.56 827.608,1221.62 828.603,1219.67 829.598,1217.72 830.594,1215.75 \n",
       "  831.589,1213.78 832.584,1211.81 833.579,1209.83 834.575,1207.85 835.57,1205.86 836.565,1203.87 837.56,1201.88 838.556,1199.89 839.551,1197.9 840.546,1195.92 \n",
       "  841.541,1193.93 842.537,1191.95 843.532,1189.96 844.527,1187.98 845.523,1186 846.518,1184.03 847.513,1182.05 848.508,1180.07 849.504,1178.1 850.499,1176.12 \n",
       "  851.494,1174.14 852.489,1172.16 853.485,1170.17 854.48,1168.18 855.475,1166.18 856.47,1164.17 857.466,1162.15 858.461,1160.12 859.456,1158.08 860.451,1156.02 \n",
       "  861.447,1153.95 862.442,1151.86 863.437,1149.76 864.432,1147.63 865.428,1145.49 866.423,1143.32 867.418,1141.14 868.414,1138.93 869.409,1136.7 870.404,1134.45 \n",
       "  871.399,1132.18 872.395,1129.88 873.39,1127.57 874.385,1125.23 875.38,1122.88 876.376,1120.5 877.371,1118.11 878.366,1115.7 879.361,1113.27 880.357,1110.83 \n",
       "  881.352,1108.37 882.347,1105.91 883.342,1103.43 884.338,1100.94 885.333,1098.44 886.328,1095.94 887.323,1093.43 888.319,1090.92 889.314,1088.4 890.309,1085.87 \n",
       "  891.305,1083.35 892.3,1080.82 893.295,1078.29 894.29,1075.76 895.286,1073.22 896.281,1070.68 897.276,1068.14 898.271,1065.6 899.267,1063.06 900.262,1060.51 \n",
       "  901.257,1057.95 902.252,1055.4 903.248,1052.83 904.243,1050.26 905.238,1047.68 906.233,1045.09 907.229,1042.48 908.224,1039.87 909.219,1037.25 910.214,1034.61 \n",
       "  911.21,1031.96 912.205,1029.3 913.2,1026.62 914.196,1023.93 915.191,1021.22 916.186,1018.49 917.181,1015.75 918.177,1012.99 919.172,1010.22 920.167,1007.43 \n",
       "  921.162,1004.63 922.158,1001.81 923.153,998.977 924.148,996.132 925.143,993.274 926.139,990.405 927.134,987.524 928.129,984.634 929.124,981.733 930.12,978.824 \n",
       "  931.115,975.907 932.11,972.983 933.105,970.051 934.101,967.113 935.096,964.17 936.091,961.22 937.087,958.265 938.082,955.305 939.077,952.339 940.072,949.367 \n",
       "  941.068,946.389 942.063,943.405 943.058,940.415 944.053,937.417 945.049,934.411 946.044,931.397 947.039,928.373 948.034,925.339 949.03,922.294 950.025,919.238 \n",
       "  951.02,916.169 952.015,913.088 953.011,909.993 954.006,906.884 955.001,903.761 955.996,900.622 956.992,897.469 957.987,894.3 958.982,891.116 959.978,887.917 \n",
       "  960.973,884.703 961.968,881.474 962.963,878.232 963.959,874.975 964.954,871.706 965.949,868.425 966.944,865.133 967.94,861.83 968.935,858.517 969.93,855.196 \n",
       "  970.925,851.867 971.921,848.53 972.916,845.188 973.911,841.84 974.906,838.488 975.902,835.131 976.897,831.771 977.892,828.407 978.887,825.04 979.883,821.671 \n",
       "  980.878,818.298 981.873,814.922 982.869,811.543 983.864,808.161 984.859,804.774 985.854,801.382 986.85,797.985 987.845,794.582 988.84,791.172 989.835,787.754 \n",
       "  990.831,784.328 991.826,780.893 992.821,777.447 993.816,773.991 994.812,770.523 995.807,767.043 996.802,763.55 997.797,760.044 998.793,756.524 999.788,752.991 \n",
       "  1000.78,749.445 1001.78,745.884 1002.77,742.311 1003.77,738.724 1004.76,735.124 1005.76,731.512 1006.75,727.889 1007.75,724.255 1008.75,720.612 1009.74,716.959 \n",
       "  1010.74,713.298 1011.73,709.63 1012.73,705.956 1013.72,702.278 1014.72,698.595 1015.71,694.909 1016.71,691.221 1017.7,687.532 1018.7,683.843 1019.69,680.154 \n",
       "  1020.69,676.466 1021.68,672.78 1022.68,669.096 1023.67,665.414 1024.67,661.736 1025.66,658.06 1026.66,654.388 1027.66,650.719 1028.65,647.053 1029.65,643.39 \n",
       "  1030.64,639.73 1031.64,636.073 1032.63,632.419 1033.63,628.766 1034.62,625.116 1035.62,621.467 1036.61,617.819 1037.61,614.172 1038.6,610.526 1039.6,606.88 \n",
       "  1040.59,603.234 1041.59,599.589 1042.58,595.943 1043.58,592.297 1044.57,588.65 1045.57,585.004 1046.57,581.358 1047.56,577.712 1048.56,574.067 1049.55,570.423 \n",
       "  1050.55,566.781 1051.54,563.141 1052.54,559.504 1053.53,555.871 1054.53,552.242 1055.52,548.618 1056.52,545.001 1057.51,541.391 1058.51,537.789 1059.5,534.195 \n",
       "  1060.5,530.612 1061.49,527.041 1062.49,523.481 1063.48,519.934 1064.48,516.401 1065.48,512.883 1066.47,509.381 1067.47,505.894 1068.46,502.425 1069.46,498.973 \n",
       "  1070.45,495.539 1071.45,492.123 1072.44,488.726 1073.44,485.346 1074.43,481.985 1075.43,478.642 1076.42,475.316 1077.42,472.007 1078.41,468.715 1079.41,465.438 \n",
       "  1080.4,462.176 1081.4,458.928 1082.39,455.693 1083.39,452.469 1084.39,449.256 1085.38,446.053 1086.38,442.857 1087.37,439.669 1088.37,436.486 1089.36,433.307 \n",
       "  1090.36,430.132 1091.35,426.959 1092.35,423.788 1093.34,420.617 1094.34,417.446 1095.33,414.274 1096.33,411.101 1097.32,407.926 1098.32,404.749 1099.31,401.571 \n",
       "  1100.31,398.392 1101.3,395.211 1102.3,392.029 1103.3,388.848 1104.29,385.668 1105.29,382.489 1106.28,379.313 1107.28,376.142 1108.27,372.976 1109.27,369.818 \n",
       "  1110.26,366.668 1111.26,363.529 1112.25,360.402 1113.25,357.29 1114.24,354.194 1115.24,351.116 1116.23,348.058 1117.23,345.023 1118.22,342.012 1119.22,339.028 \n",
       "  1120.21,336.072 1121.21,333.147 1122.21,330.255 1123.2,327.398 1124.2,324.577 1125.19,321.796 1126.19,319.055 1127.18,316.356 1128.18,313.701 1129.17,311.092 \n",
       "  1130.17,308.531 1131.16,306.018 1132.16,303.555 1133.15,301.144 1134.15,298.784 1135.14,296.477 1136.14,294.224 1137.13,292.025 1138.13,289.881 1139.12,287.79 \n",
       "  1140.12,285.754 1141.12,283.772 1142.11,281.842 1143.11,279.965 1144.1,278.14 1145.1,276.364 1146.09,274.636 1147.09,272.955 1148.08,271.319 1149.08,269.724 \n",
       "  1150.07,268.169 1151.07,266.651 1152.06,265.167 1153.06,263.713 1154.05,262.287 1155.05,260.885 1156.04,259.504 1157.04,258.14 1158.03,256.79 1159.03,255.449 \n",
       "  1160.03,254.114 1161.02,252.781 1162.02,251.447 1163.01,250.108 1164.01,248.76 1165,247.4 1166,246.024 1166.99,244.63 1167.99,243.215 1168.98,241.776 \n",
       "  1169.98,240.31 1170.97,238.815 1171.97,237.289 1172.96,235.73 1173.96,234.137 1174.95,232.509 1175.95,230.845 1176.94,229.144 1177.94,227.406 1178.93,225.631 \n",
       "  1179.93,223.819 1180.93,221.971 1181.92,220.088 1182.92,218.17 1183.91,216.219 1184.91,214.236 1185.9,212.224 1186.9,210.184 1187.89,208.118 1188.89,206.029 \n",
       "  1189.88,203.919 1190.88,201.79 1191.87,199.645 1192.87,197.486 1193.86,195.317 1194.86,193.139 1195.85,190.956 1196.85,188.769 1197.84,186.582 1198.84,184.397 \n",
       "  1199.84,182.217 1200.83,180.043 1201.83,177.877 1202.82,175.723 1203.82,173.581 1204.81,171.454 1205.81,169.344 1206.8,167.251 1207.8,165.177 1208.79,163.124 \n",
       "  1209.79,161.094 1210.78,159.086 1211.78,157.102 1212.77,155.143 1213.77,153.21 1214.76,151.304 1215.76,149.426 1216.75,147.577 1217.75,145.757 1218.75,143.968 \n",
       "  1219.74,142.21 1220.74,140.485 1221.73,138.793 1222.73,137.136 1223.72,135.514 1224.72,133.93 1225.71,132.385 1226.71,130.879 1227.7,129.415 1228.7,127.994 \n",
       "  1229.69,126.617 1230.69,125.288 1231.68,124.007 1232.68,122.776 1233.67,121.597 1234.67,120.472 1235.66,119.403 1236.66,118.392 1237.66,117.439 1238.65,116.547 \n",
       "  1239.65,115.718 1240.64,114.952 1241.64,114.25 1242.63,113.615 1243.63,113.046 1244.62,112.544 1245.62,112.109 1246.61,111.743 1247.61,111.443 1248.6,111.211 \n",
       "  1249.6,111.045 1250.59,110.944 1251.59,110.908 1252.58,110.935 1253.58,111.022 1254.57,111.169 1255.57,111.373 1256.57,111.631 1257.56,111.941 1258.56,112.301 \n",
       "  1259.55,112.707 1260.55,113.157 1261.54,113.647 1262.54,114.175 1263.53,114.737 1264.53,115.33 1265.52,115.952 1266.52,116.598 1267.51,117.268 1268.51,117.958 \n",
       "  1269.5,118.665 1270.5,119.388 1271.49,120.124 1272.49,120.872 1273.48,121.631 1274.48,122.399 1275.48,123.175 1276.47,123.96 1277.47,124.753 1278.46,125.554 \n",
       "  1279.46,126.364 1280.45,127.184 1281.45,128.015 1282.44,128.859 1283.44,129.717 1284.43,130.591 1285.43,131.484 1286.42,132.398 1287.42,133.336 1288.41,134.3 \n",
       "  1289.41,135.292 1290.4,136.316 1291.4,137.375 1292.39,138.471 1293.39,139.606 1294.39,140.783 1295.38,142.005 1296.38,143.273 1297.37,144.589 1298.37,145.955 \n",
       "  1299.36,147.372 1300.36,148.84 1301.35,150.362 1302.35,151.936 1303.34,153.563 1304.34,155.242 1305.33,156.973 1306.33,158.754 1307.32,160.584 1308.32,162.461 \n",
       "  1309.31,164.384 1310.31,166.349 1311.3,168.354 1312.3,170.397 1313.3,172.475 1314.29,174.584 1315.29,176.721 1316.28,178.883 1317.28,181.066 1318.27,183.267 \n",
       "  1319.27,185.484 1320.26,187.712 1321.26,189.948 1322.25,192.19 1323.25,194.435 1324.24,196.679 1325.24,198.921 1326.23,201.158 1327.23,203.389 1328.22,205.61 \n",
       "  1329.22,207.822 1330.21,210.022 1331.21,212.209 1332.21,214.384 1333.2,216.545 1334.2,218.693 1335.19,220.827 1336.19,222.947 1337.18,225.054 1338.18,227.15 \n",
       "  1339.17,229.233 1340.17,231.306 1341.16,233.37 1342.16,235.425 1343.15,237.473 1344.15,239.516 1345.14,241.554 1346.14,243.589 1347.13,245.623 1348.13,247.656 \n",
       "  1349.12,249.689 1350.12,251.725 1351.12,253.763 1352.11,255.806 1353.11,257.853 1354.1,259.905 1355.1,261.964 1356.09,264.03 1357.09,266.103 1358.08,268.183 \n",
       "  1359.08,270.271 1360.07,272.368 1361.07,274.472 1362.06,276.584 1363.06,278.705 1364.05,280.833 1365.05,282.97 1366.04,285.114 1367.04,287.267 1368.03,289.427 \n",
       "  1369.03,291.596 1370.03,293.773 1371.02,295.958 1372.02,298.153 1373.01,300.356 1374.01,302.569 1375,304.792 1376,307.026 1376.99,309.271 1377.99,311.529 \n",
       "  1378.98,313.801 1379.98,316.086 1380.97,318.387 1381.97,320.705 1382.96,323.04 1383.96,325.394 1384.95,327.767 1385.95,330.162 1386.94,332.579 1387.94,335.019 \n",
       "  1388.94,337.484 1389.93,339.973 1390.93,342.489 1391.92,345.032 1392.92,347.602 1393.91,350.201 1394.91,352.829 1395.9,355.487 1396.9,358.174 1397.89,360.891 \n",
       "  1398.89,363.639 1399.88,366.417 1400.88,369.225 1401.87,372.064 1402.87,374.933 1403.86,377.832 1404.86,380.761 1405.85,383.72 1406.85,386.707 1407.85,389.724 \n",
       "  1408.84,392.77 1409.84,395.844 1410.83,398.946 1411.83,402.075 1412.82,405.231 1413.82,408.414 1414.81,411.623 1415.81,414.858 1416.8,418.118 1417.8,421.403 \n",
       "  1418.79,424.712 1419.79,428.046 1420.78,431.402 1421.78,434.781 1422.77,438.181 1423.77,441.603 1424.76,445.045 1425.76,448.507 1426.76,451.987 1427.75,455.485 \n",
       "  1428.75,458.998 1429.74,462.527 1430.74,466.07 1431.73,469.625 1432.73,473.191 1433.72,476.766 1434.72,480.349 1435.71,483.938 1436.71,487.531 1437.7,491.126 \n",
       "  1438.7,494.721 1439.69,498.315 1440.69,501.906 1441.68,505.491 1442.68,509.069 1443.67,512.637 1444.67,516.194 1445.66,519.738 1446.66,523.268 1447.66,526.782 \n",
       "  1448.65,530.277 1449.65,533.754 1450.64,537.211 1451.64,540.647 1452.63,544.06 1453.63,547.452 1454.62,550.82 1455.62,554.166 1456.61,557.489 1457.61,560.789 \n",
       "  1458.6,564.068 1459.6,567.326 1460.59,570.563 1461.59,573.782 1462.58,576.984 1463.58,580.17 1464.57,583.342 1465.57,586.503 1466.57,589.654 1467.56,592.798 \n",
       "  1468.56,595.937 1469.55,599.074 1470.55,602.212 1471.54,605.351 1472.54,608.497 1473.53,611.65 1474.53,614.813 1475.52,617.989 1476.52,621.179 1477.51,624.387 \n",
       "  1478.51,627.613 1479.5,630.861 1480.5,634.131 1481.49,637.425 1482.49,640.744 1483.48,644.09 1484.48,647.463 1485.48,650.863 1486.47,654.293 1487.47,657.751 \n",
       "  1488.46,661.238 1489.46,664.754 1490.45,668.299 1491.45,671.871 1492.44,675.472 1493.44,679.1 1494.43,682.753 1495.43,686.433 1496.42,690.136 1497.42,693.864 \n",
       "  1498.41,697.613 1499.41,701.383 1500.4,705.173 1501.4,708.982 1502.39,712.808 1503.39,716.65 1504.39,720.506 1505.38,724.376 1506.38,728.258 1507.37,732.15 \n",
       "  1508.37,736.052 1509.36,739.961 1510.36,743.877 1511.35,747.798 1512.35,751.723 1513.34,755.651 1514.34,759.581 1515.33,763.51 1516.33,767.438 1517.32,771.364 \n",
       "  1518.32,775.286 1519.31,779.203 1520.31,783.114 1521.3,787.018 1522.3,790.914 1523.3,794.8 1524.29,798.676 1525.29,802.54 1526.28,806.392 1527.28,810.23 \n",
       "  1528.27,814.053 1529.27,817.862 1530.26,821.654 1531.26,825.43 1532.25,829.188 1533.25,832.927 1534.24,836.648 1535.24,840.349 1536.23,844.031 1537.23,847.692 \n",
       "  1538.22,851.332 1539.22,854.951 1540.21,858.549 1541.21,862.125 1542.21,865.678 1543.2,869.21 1544.2,872.719 1545.19,876.206 1546.19,879.669 1547.18,883.11 \n",
       "  1548.18,886.528 1549.17,889.923 1550.17,893.295 1551.16,896.644 1552.16,899.97 1553.15,903.273 1554.15,906.553 1555.14,909.809 1556.14,913.043 1557.13,916.254 \n",
       "  1558.13,919.442 1559.12,922.607 1560.12,925.75 1561.12,928.87 1562.11,931.968 1563.11,935.044 1564.1,938.099 1565.1,941.132 1566.09,944.144 1567.09,947.136 \n",
       "  1568.08,950.107 1569.08,953.058 1570.07,955.99 1571.07,958.904 1572.06,961.799 1573.06,964.677 1574.05,967.538 1575.05,970.383 1576.04,973.212 1577.04,976.028 \n",
       "  1578.03,978.83 1579.03,981.619 1580.03,984.397 1581.02,987.164 1582.02,989.922 1583.01,992.671 1584.01,995.413 1585,998.149 1586,1000.88 1586.99,1003.61 \n",
       "  1587.99,1006.33 1588.98,1009.05 1589.98,1011.78 1590.97,1014.5 1591.97,1017.23 1592.96,1019.96 1593.96,1022.69 1594.95,1025.43 1595.95,1028.18 1596.94,1030.93 \n",
       "  1597.94,1033.69 1598.94,1036.47 1599.93,1039.25 1600.93,1042.04 1601.92,1044.84 1602.92,1047.65 1603.91,1050.48 1604.91,1053.31 1605.9,1056.16 1606.9,1059.02 \n",
       "  1607.89,1061.89 1608.89,1064.76 1609.88,1067.65 1610.88,1070.55 1611.87,1073.45 1612.87,1076.36 1613.86,1079.28 1614.86,1082.19 1615.85,1085.11 1616.85,1088.03 \n",
       "  1617.85,1090.95 1618.84,1093.87 1619.84,1096.78 1620.83,1099.68 1621.83,1102.57 1622.82,1105.45 1623.82,1108.31 1624.81,1111.16 1625.81,1113.99 1626.8,1116.8 \n",
       "  1627.8,1119.58 1628.79,1122.34 1629.79,1125.07 1630.78,1127.78 1631.78,1130.45 1632.77,1133.09 1633.77,1135.69 1634.76,1138.26 1635.76,1140.8 1636.76,1143.29 \n",
       "  1637.75,1145.74 1638.75,1148.16 1639.74,1150.53 1640.74,1152.86 1641.73,1155.15 1642.73,1157.4 1643.72,1159.6 1644.72,1161.77 1645.71,1163.89 1646.71,1165.97 \n",
       "  1647.7,1168.01 1648.7,1170.01 1649.69,1171.98 1650.69,1173.91 1651.68,1175.8 1652.68,1177.66 1653.67,1179.49 1654.67,1181.28 1655.67,1183.05 1656.66,1184.8 \n",
       "  1657.66,1186.51 1658.65,1188.21 1659.65,1189.89 1660.64,1191.55 1661.64,1193.19 1662.63,1194.82 1663.63,1196.44 1664.62,1198.05 1665.62,1199.65 1666.61,1201.25 \n",
       "  1667.61,1202.85 1668.6,1204.44 1669.6,1206.04 1670.59,1207.64 1671.59,1209.24 1672.58,1210.85 1673.58,1212.46 1674.58,1214.09 1675.57,1215.72 1676.57,1217.36 \n",
       "  1677.56,1219.01 1678.56,1220.67 1679.55,1222.35 1680.55,1224.03 1681.54,1225.73 1682.54,1227.44 1683.53,1229.16 1684.53,1230.89 1685.52,1232.63 1686.52,1234.39 \n",
       "  1687.51,1236.15 1688.51,1237.93 1689.5,1239.71 1690.5,1241.5 1691.49,1243.3 1692.49,1245.11 1693.48,1246.92 1694.48,1248.74 1695.48,1250.56 1696.47,1252.38 \n",
       "  1697.47,1254.21 1698.46,1256.04 1699.46,1257.87 1700.45,1259.7 1701.45,1261.53 1702.44,1263.35 1703.44,1265.18 1704.43,1267 1705.43,1268.81 1706.42,1270.62 \n",
       "  1707.42,1272.42 1708.41,1274.22 1709.41,1276 1710.4,1277.78 1711.4,1279.55 1712.39,1281.3 1713.39,1283.04 1714.39,1284.77 1715.38,1286.49 1716.38,1288.19 \n",
       "  1717.37,1289.87 1718.37,1291.54 1719.36,1293.19 1720.36,1294.82 1721.35,1296.44 1722.35,1298.03 1723.34,1299.6 1724.34,1301.15 1725.33,1302.68 1726.33,1304.18 \n",
       "  1727.32,1305.66 1728.32,1307.12 1729.31,1308.56 1730.31,1309.97 1731.3,1311.35 1732.3,1312.71 1733.3,1314.05 1734.29,1315.35 1735.29,1316.64 1736.28,1317.9 \n",
       "  1737.28,1319.13 1738.27,1320.34 1739.27,1321.53 1740.26,1322.69 1741.26,1323.82 1742.25,1324.94 1743.25,1326.03 1744.24,1327.1 1745.24,1328.15 1746.23,1329.17 \n",
       "  1747.23,1330.18 1748.22,1331.17 1749.22,1332.14 1750.21,1333.09 1751.21,1334.03 1752.21,1334.95 1753.2,1335.85 1754.2,1336.74 1755.19,1337.62 1756.19,1338.49 \n",
       "  1757.18,1339.35 1758.18,1340.2 1759.17,1341.04 1760.17,1341.87 1761.16,1342.69 1762.16,1343.51 1763.15,1344.33 1764.15,1345.14 1765.14,1345.95 1766.14,1346.76 \n",
       "  1767.13,1347.57 1768.13,1348.38 1769.12,1349.19 1770.12,1350 1771.12,1350.81 1772.11,1351.63 1773.11,1352.45 1774.1,1353.27 1775.1,1354.1 1776.09,1354.94 \n",
       "  1777.09,1355.78 1778.08,1356.62 1779.08,1357.47 1780.07,1358.33 1781.07,1359.19 1782.06,1360.06 1783.06,1360.94 1784.05,1361.82 1785.05,1362.7 1786.04,1363.59 \n",
       "  1787.04,1364.49 1788.03,1365.39 1789.03,1366.29 1790.03,1367.19 1791.02,1368.1 1792.02,1369.01 1793.01,1369.92 1794.01,1370.83 1795,1371.74 1796,1372.65 \n",
       "  1796.99,1373.55 1797.99,1374.45 1798.98,1375.35 1799.98,1376.24 1800.97,1377.13 1801.97,1378.01 1802.96,1378.88 1803.96,1379.74 1804.95,1380.59 1805.95,1381.43 \n",
       "  1806.94,1382.26 1807.94,1383.08 1808.94,1383.89 1809.93,1384.68 1810.93,1385.46 1811.92,1386.22 1812.92,1386.97 1813.91,1387.7 1814.91,1388.42 1815.9,1389.12 \n",
       "  1816.9,1389.8 1817.89,1390.47 1818.89,1391.13 1819.88,1391.76 1820.88,1392.38 1821.87,1392.99 1822.87,1393.57 1823.86,1394.15 1824.86,1394.71 1825.85,1395.25 \n",
       "  1826.85,1395.78 1827.85,1396.29 1828.84,1396.8 1829.84,1397.29 1830.83,1397.77 1831.83,1398.23 1832.82,1398.69 1833.82,1399.14 1834.81,1399.58 1835.81,1400.01 \n",
       "  1836.8,1400.44 1837.8,1400.86 1838.79,1401.27 1839.79,1401.68 1840.78,1402.09 1841.78,1402.49 1842.77,1402.89 1843.77,1403.28 1844.76,1403.68 1845.76,1404.07 \n",
       "  1846.76,1404.46 1847.75,1404.85 1848.75,1405.24 1849.74,1405.63 1850.74,1406.02 1851.73,1406.41 1852.73,1406.8 1853.72,1407.19 1854.72,1407.58 1855.71,1407.97 \n",
       "  1856.71,1408.36 1857.7,1408.75 1858.7,1409.13 1859.69,1409.52 1860.69,1409.9 1861.68,1410.29 1862.68,1410.67 1863.67,1411.05 1864.67,1411.42 1865.67,1411.8 \n",
       "  1866.66,1412.17 1867.66,1412.54 1868.65,1412.91 1869.65,1413.27 1870.64,1413.63 1871.64,1413.99 1872.63,1414.34 1873.63,1414.69 1874.62,1415.04 1875.62,1415.39 \n",
       "  1876.61,1415.73 1877.61,1416.07 1878.6,1416.4 1879.6,1416.73 1880.59,1417.06 1881.59,1417.39 1882.58,1417.72 1883.58,1418.04 1884.58,1418.36 1885.57,1418.68 \n",
       "  1886.57,1419 1887.56,1419.32 1888.56,1419.63 1889.55,1419.95 1890.55,1420.26 1891.54,1420.58 1892.54,1420.89 1893.53,1421.21 1894.53,1421.52 1895.52,1421.83 \n",
       "  1896.52,1422.15 1897.51,1422.46 1898.51,1422.77 1899.5,1423.08 1900.5,1423.4 1901.49,1423.71 1902.49,1424.02 1903.49,1424.33 1904.48,1424.64 1905.48,1424.95 \n",
       "  1906.47,1425.25 1907.47,1425.56 1908.46,1425.86 1909.46,1426.16 1910.45,1426.46 1911.45,1426.76 1912.44,1427.05 1913.44,1427.34 1914.43,1427.63 1915.43,1427.91 \n",
       "  1916.42,1428.19 1917.42,1428.46 1918.41,1428.73 1919.41,1429 1920.4,1429.26 1921.4,1429.52 1922.4,1429.77 1923.39,1430.02 1924.39,1430.26 1925.38,1430.5 \n",
       "  1926.38,1430.73 1927.37,1430.96 1928.37,1431.18 1929.36,1431.4 1930.36,1431.61 1931.35,1431.82 1932.35,1432.03 1933.34,1432.23 1934.34,1432.42 1935.33,1432.61 \n",
       "  1936.33,1432.8 1937.32,1432.98 1938.32,1433.17 1939.31,1433.34 1940.31,1433.52 1941.31,1433.69 1942.3,1433.85 1943.3,1434.02 1944.29,1434.18 1945.29,1434.34 \n",
       "  1946.28,1434.5 1947.28,1434.66 1948.27,1434.81 1949.27,1434.96 1950.26,1435.11 1951.26,1435.26 1952.25,1435.41 1953.25,1435.55 1954.24,1435.7 1955.24,1435.84 \n",
       "  1956.23,1435.97 1957.23,1436.11 1958.22,1436.25 1959.22,1436.38 1960.21,1436.51 1961.21,1436.64 1962.21,1436.77 1963.2,1436.89 1964.2,1437.02 1965.19,1437.14 \n",
       "  1966.19,1437.26 1967.18,1437.37 1968.18,1437.49 1969.17,1437.6 1970.17,1437.71 1971.16,1437.82 1972.16,1437.93 1973.15,1438.03 1974.15,1438.13 1975.14,1438.23 \n",
       "  1976.14,1438.33 1977.13,1438.43 1978.13,1438.53 1979.12,1438.62 1980.12,1438.72 1981.12,1438.81 1982.11,1438.9 1983.11,1438.99 1984.1,1439.08 1985.1,1439.16 \n",
       "  1986.09,1439.25 1987.09,1439.34 1988.08,1439.42 1989.08,1439.51 1990.07,1439.59 1991.07,1439.68 1992.06,1439.76 1993.06,1439.84 1994.05,1439.92 1995.05,1440 \n",
       "  1996.04,1440.08 1997.04,1440.16 1998.03,1440.24 1999.03,1440.32 2000.03,1440.4 2001.02,1440.47 2002.02,1440.55 2003.01,1440.62 2004.01,1440.7 2005,1440.77 \n",
       "  2006,1440.84 2006.99,1440.91 2007.99,1440.98 2008.98,1441.05 2009.98,1441.12 2010.97,1441.18 2011.97,1441.25 2012.96,1441.31 2013.96,1441.37 2014.95,1441.43 \n",
       "  2015.95,1441.49 2016.94,1441.55 2017.94,1441.61 2018.94,1441.67 2019.93,1441.72 2020.93,1441.78 2021.92,1441.83 2022.92,1441.88 2023.91,1441.93 2024.91,1441.99 \n",
       "  2025.9,1442.04 2026.9,1442.08 2027.89,1442.13 2028.89,1442.18 2029.88,1442.23 2030.88,1442.27 2031.87,1442.32 2032.87,1442.36 2033.86,1442.41 2034.86,1442.45 \n",
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      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d = Distributions.Normal(0.0, 1.0) # Normal distribution with μ=0 y σ=0\n",
    "@show params(d)\n",
    "@show mean(d)\n",
    "@show std(d);\n",
    "\n",
    "randomVector = rand(d, 100000)\n",
    "histogram(randomVector, label=\"N(0,1)\")\n",
    "p = kde(randomVector)\n",
    "plot!(p.x, p.density .* length(randomVector) .* 0.1, linewidth=3, label=\"fit\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "params(d) = (3.4877830882352936, 1.1392712102257678)\n"
     ]
    },
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      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d = fit(Normal, eruptions)\n",
    "@show params(d)\n",
    "\n",
    "randomVector= rand(d, 1000)\n",
    "histogram(randomVector, nbins=20, fillalpha=0.3, label=\"fit\")\n",
    "histogram!(eruptions, nbins=20, linecolor=:red, fillalpha=0.3, label=\"my vector\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5. Hypothesis testing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The package *HypothesisTesting.jl* implements several hypothesis tests in Julia."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "One sample t-test\n",
       "-----------------\n",
       "Population details:\n",
       "    parameter of interest:   Mean\n",
       "    value under h_0:         0\n",
       "    point estimate:          3.4877830882352936\n",
       "    95% confidence interval: (3.3515, 3.624)\n",
       "\n",
       "Test summary:\n",
       "    outcome with 95% confidence: reject h_0\n",
       "    two-sided p-value:           <1e-99\n",
       "\n",
       "Details:\n",
       "    number of observations:   272\n",
       "    t-statistic:              50.39726752574924\n",
       "    degrees of freedom:       271\n",
       "    empirical standard error: 0.06920579744632577\n"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "test = OneSampleTTest(eruptions)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 6. MLBase\n",
    "\n",
    "The MLBase package provides useful tools for machine leaning applications (data preprocessing, performance evaluation, cross validation)\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Confusion matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "k = 2 # no. of classes\n",
    "gt = [1, 1, 1, 1, 1, 1, 1, 2]  # ground truths\n",
    "pred = [1, 1, 2, 2, 1, 1, 1, 1]; # predictions "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute the confusion matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×2 Array{Int64,2}:\n",
       " 5  2\n",
       " 1  0"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cm = confusmat(k, gt, pred)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.625"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "correctrate(gt, pred)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.375"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "errorrate(gt, pred)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### ROC curve"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "gt = [1, 1, 1, 1, 1, 1, 1, 0];\n",
    "pred = [1, 1, 0, 0, 1, 1, 1, 1];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ROCNums{Int64}\n",
       "  p = 7\n",
       "  n = 1\n",
       "  tp = 5\n",
       "  tn = 0\n",
       "  fp = 1\n",
       "  fn = 2\n"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ROC = MLBase.roc(gt, pred)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.7142857142857143"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "recall(ROC)    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.8333333333333334"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "precision(ROC)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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