5. Collecting experiments data in a data frame.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Collecting experiments data in a data frame\n",
    "\n",
    "### Bogumił Kamiński"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\u001b[32m\u001b[1m  Activating\u001b[22m\u001b[39m project at `~/JuliaAcademy-DataFrames`\n"
     ]
    }
   ],
   "source": [
    "] activate ."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "using DataFrames"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Statistics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "using PyPlot"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-info\">\n",
    "    <p><b>PyPlot.jl configuration:</b></p>\n",
    "    <p>In some environments automatic installation of PyPlot.jl might fail. If you encounter this ussue please refer to <a href=\"https://github.com/JuliaPy/PyPlot.jl#installation\">the PyPlot.jl installation instructions</a>. </p>\n",
    "</div>\n",
    "\n",
    "In particular typically executing the following commands:\n",
    "\n",
    "```\n",
    "using Pkg\n",
    "ENV[\"PYTHON\"]=\"\"\n",
    "Pkg.build(\"PyCall\")\n",
    "```\n",
    "\n",
    "should resolve the PyPlot.jl installation issues. However, on OS X sometimes more configuration steps are required. You can find the detailed instructions [here](https://github.com/JuliaPy/PyPlot.jl#os-x)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Random"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Pipe"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this part we will run a simple Monte Carlo simulation so show examples how one can work with data frames."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the following puzzle.\n",
    "\n",
    "We draw independent random numbers from $U(0,1)$ distribution. On the average, how many draws do we need, till the sum of these numbers exceeds $1$?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here is the code that runs this experiment once. For tutorial reasons we keep all the generated random numbers and recalculate their sum in each iteration (you can try to improve the efficiency of this code as an exercise)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sim_e (generic function with 1 method)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function sim_e()\n",
    "    draw = Float64[]\n",
    "    while true\n",
    "        push!(draw, rand())\n",
    "        sum(draw) > 1.0 && return draw\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "Random.seed!(1234); # just to make sure we get the same results if we are on the same version of Julia"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us run our simulation several times:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5-element Vector{Vector{Float64}}:\n",
       " [0.32597672886359486, 0.5490511363155669, 0.21858665481883066]\n",
       " [0.8942454282009883, 0.35311164439921205]\n",
       " [0.39425536741585077, 0.9531246272848422]\n",
       " [0.7955469475347194, 0.4942498668904206]\n",
       " [0.7484150218874741, 0.5782319465613976]"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res = [sim_e() for _ in 1:5]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and check that each time we finished just when we exceeded $1$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5-element Vector{Float64}:\n",
       " 1.0936145199979923\n",
       " 1.2473570726002003\n",
       " 1.347379994700693\n",
       " 1.2897968144251402\n",
       " 1.3266469684488718"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum.(res)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5-element Vector{Float64}:\n",
       " 0.8750278651791616\n",
       " 0.8942454282009883\n",
       " 0.3942553674158509\n",
       " 0.7955469475347196\n",
       " 0.7484150218874741"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@. sum(res) - last(res)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "All looks good so far! (and as a bonus we have just made a small exercise in broadcasting)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let us populate a data frame with the results of our experiments"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " 13.492488 seconds (120.34 M allocations: 3.966 GiB, 50.07% gc time, 2.38% compilation time)\n"
     ]
    }
   ],
   "source": [
    "df = DataFrame()\n",
    "\n",
    "@time for i in 1:10^7\n",
    "    push!(df, (id=i, pos=sim_e()))\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As you can see the process was quite fast, `push!`-ing data to a `DataFrame` is efficient."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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text-align: right;\">8</td><td style = \"text-align: right;\">8</td><td style = \"text-align: left;\">[0.365854, 0.131026, 0.946453]</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9</td><td style = \"text-align: right;\">9</td><td style = \"text-align: left;\">[0.574323, 0.67765]</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">10</td><td style = \"text-align: right;\">10</td><td style = \"text-align: left;\">[0.571586, 0.0727161, 0.701116]</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">11</td><td style = \"text-align: right;\">11</td><td style = \"text-align: left;\">[0.0952175, 0.845515, 0.348995]</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">12</td><td style = \"text-align: right;\">12</td><td style = \"text-align: left;\">[0.768308, 0.26906]</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">13</td><td style = \"text-align: right;\">13</td><td style = \"text-align: left;\">[0.539631, 0.293905, 0.242195]</td></tr><tr><td style = \"text-align: right;\">&vellip;</td><td style = \"text-align: right;\">&vellip;</td><td style = \"text-align: right;\">&vellip;</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; 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text-align: right;\">9999994</td><td style = \"text-align: right;\">9999994</td><td style = \"text-align: left;\">[0.846778, 0.328575]</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999995</td><td style = \"text-align: right;\">9999995</td><td style = \"text-align: left;\">[0.251749, 0.28344, 0.16416, 0.618446]</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999996</td><td style = \"text-align: right;\">9999996</td><td style = \"text-align: left;\">[0.383316, 0.352557, 0.470778]</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999997</td><td style = \"text-align: right;\">9999997</td><td style = \"text-align: left;\">[0.362197, 0.639499]</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999998</td><td style = \"text-align: right;\">9999998</td><td style = \"text-align: left;\">[0.046897, 0.552855, 0.172669, 0.156109, 0.799254]</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999999</td><td style = \"text-align: right;\">9999999</td><td style = \"text-align: left;\">[0.815608, 0.663247]</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">10000000</td><td style = \"text-align: right;\">10000000</td><td style = \"text-align: left;\">[0.61177, 0.5798]</td></tr></tbody></table></div>"
      ],
      "text/latex": [
       "\\begin{tabular}{r|cc}\n",
       "\t& id & pos\\\\\n",
       "\t\\hline\n",
       "\t& Int64 & Array…\\\\\n",
       "\t\\hline\n",
       "\t1 & 1 & [0.727935, 0.00744801, 0.199377, 0.439243] \\\\\n",
       "\t2 & 2 & [0.682533, 0.956741] \\\\\n",
       "\t3 & 3 & [0.647855, 0.996665] \\\\\n",
       "\t4 & 4 & [0.749194, 0.110084, 0.491383] \\\\\n",
       "\t5 & 5 & [0.565145, 0.253812, 0.626794] \\\\\n",
       "\t6 & 6 & [0.234105, 0.124792, 0.609875, 0.672793] \\\\\n",
       "\t7 & 7 & [0.761916, 0.588872] \\\\\n",
       "\t8 & 8 & [0.365854, 0.131026, 0.946453] \\\\\n",
       "\t9 & 9 & [0.574323, 0.67765] \\\\\n",
       "\t10 & 10 & [0.571586, 0.0727161, 0.701116] \\\\\n",
       "\t11 & 11 & [0.0952175, 0.845515, 0.348995] \\\\\n",
       "\t12 & 12 & [0.768308, 0.26906] \\\\\n",
       "\t13 & 13 & [0.539631, 0.293905, 0.242195] \\\\\n",
       "\t14 & 14 & [0.97807, 0.853242] \\\\\n",
       "\t15 & 15 & [0.706065, 0.442139] \\\\\n",
       "\t16 & 16 & [0.884577, 0.520741] \\\\\n",
       "\t17 & 17 & [0.941831, 0.810699] \\\\\n",
       "\t18 & 18 & [0.464849, 0.977012] \\\\\n",
       "\t19 & 19 & [0.500161, 0.715846] \\\\\n",
       "\t20 & 20 & [0.18985, 0.376441, 0.565864] \\\\\n",
       "\t21 & 21 & [0.172236, 0.327351, 0.632108] \\\\\n",
       "\t22 & 22 & [0.374141, 0.728543] \\\\\n",
       "\t23 & 23 & [0.00277646, 0.134685, 0.531899, 0.0731709, 0.501756] \\\\\n",
       "\t24 & 24 & [0.906939, 0.116769] \\\\\n",
       "\t25 & 25 & [0.367198, 0.881163] \\\\\n",
       "\t26 & 26 & [0.67189, 0.479862] \\\\\n",
       "\t27 & 27 & [0.949968, 0.0262331, 0.347189] \\\\\n",
       "\t28 & 28 & [0.819254, 0.792831] \\\\\n",
       "\t29 & 29 & [0.96723, 0.471452] \\\\\n",
       "\t30 & 30 & [0.19378, 0.941781] \\\\\n",
       "\t$\\dots$ & $\\dots$ & $\\dots$ \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "\u001b[1m10000000×2 DataFrame\u001b[0m\n",
       "\u001b[1m      Row \u001b[0m│\u001b[1m id       \u001b[0m\u001b[1m pos                               \u001b[0m\n",
       "          │\u001b[90m Int64    \u001b[0m\u001b[90m Array…                            \u001b[0m\n",
       "──────────┼─────────────────────────────────────────────\n",
       "        1 │        1  [0.727935, 0.00744801, 0.199377,…\n",
       "        2 │        2  [0.682533, 0.956741]\n",
       "        3 │        3  [0.647855, 0.996665]\n",
       "        4 │        4  [0.749194, 0.110084, 0.491383]\n",
       "        5 │        5  [0.565145, 0.253812, 0.626794]\n",
       "        6 │        6  [0.234105, 0.124792, 0.609875, 0…\n",
       "        7 │        7  [0.761916, 0.588872]\n",
       "        8 │        8  [0.365854, 0.131026, 0.946453]\n",
       "        9 │        9  [0.574323, 0.67765]\n",
       "       10 │       10  [0.571586, 0.0727161, 0.701116]\n",
       "       11 │       11  [0.0952175, 0.845515, 0.348995]\n",
       "    ⋮     │    ⋮                      ⋮\n",
       "  9999991 │  9999991  [0.382258, 0.901623]\n",
       "  9999992 │  9999992  [0.640595, 0.916399]\n",
       "  9999993 │  9999993  [0.963508, 0.301685]\n",
       "  9999994 │  9999994  [0.846778, 0.328575]\n",
       "  9999995 │  9999995  [0.251749, 0.28344, 0.16416, 0.6…\n",
       "  9999996 │  9999996  [0.383316, 0.352557, 0.470778]\n",
       "  9999997 │  9999997  [0.362197, 0.639499]\n",
       "  9999998 │  9999998  [0.046897, 0.552855, 0.172669, 0…\n",
       "  9999999 │  9999999  [0.815608, 0.663247]\n",
       " 10000000 │ 10000000  [0.61177, 0.5798]\n",
       "\u001b[36m                                    9999979 rows omitted\u001b[0m"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us count the number of jumps we have made in each step using the `transform!` function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div><div style = \"float: left;\"><span>10000000×3 DataFrame</span></div><div style = \"float: right;\"><span style = \"font-style: italic;\">9999975 rows omitted</span></div><div style = \"clear: both;\"></div></div><div class = \"data-frame\" style = \"overflow-x: scroll;\"><table class = \"data-frame\" style = \"margin-bottom: 6px;\"><thead><tr class = \"header\"><th class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">Row</th><th style = \"text-align: left;\">id</th><th style = \"text-align: left;\">pos</th><th style = \"text-align: left;\">jumps</th></tr><tr class = \"subheader headerLastRow\"><th class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\"></th><th title = \"Int64\" style = \"text-align: left;\">Int64</th><th title = \"Vector{Float64}\" style = \"text-align: left;\">Array…</th><th title = \"Int64\" style = \"text-align: left;\">Int64</th></tr></thead><tbody><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">1</td><td style = \"text-align: right;\">1</td><td style = \"text-align: left;\">[0.727935, 0.00744801, 0.199377, 0.439243]</td><td style = \"text-align: right;\">4</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">2</td><td style = \"text-align: right;\">2</td><td style = \"text-align: left;\">[0.682533, 0.956741]</td><td style = \"text-align: right;\">2</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">3</td><td style = \"text-align: right;\">3</td><td style = \"text-align: left;\">[0.647855, 0.996665]</td><td style = \"text-align: right;\">2</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">4</td><td style = \"text-align: right;\">4</td><td style = \"text-align: left;\">[0.749194, 0.110084, 0.491383]</td><td style = \"text-align: right;\">3</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">5</td><td style = \"text-align: right;\">5</td><td style = \"text-align: left;\">[0.565145, 0.253812, 0.626794]</td><td style = \"text-align: right;\">3</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">6</td><td style = \"text-align: right;\">6</td><td style = \"text-align: left;\">[0.234105, 0.124792, 0.609875, 0.672793]</td><td style = \"text-align: right;\">4</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">7</td><td style = \"text-align: right;\">7</td><td style = \"text-align: left;\">[0.761916, 0.588872]</td><td style = \"text-align: right;\">2</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">8</td><td style = \"text-align: right;\">8</td><td style = \"text-align: left;\">[0.365854, 0.131026, 0.946453]</td><td style = \"text-align: right;\">3</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9</td><td style = \"text-align: right;\">9</td><td style = \"text-align: left;\">[0.574323, 0.67765]</td><td style = \"text-align: right;\">2</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">10</td><td style = \"text-align: right;\">10</td><td style = \"text-align: left;\">[0.571586, 0.0727161, 0.701116]</td><td style = \"text-align: right;\">3</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">11</td><td style = \"text-align: right;\">11</td><td style = \"text-align: left;\">[0.0952175, 0.845515, 0.348995]</td><td style = \"text-align: right;\">3</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">12</td><td style = \"text-align: right;\">12</td><td style = \"text-align: left;\">[0.768308, 0.26906]</td><td style = \"text-align: right;\">2</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">13</td><td style = \"text-align: right;\">13</td><td style = \"text-align: left;\">[0.539631, 0.293905, 0.242195]</td><td style = \"text-align: right;\">3</td></tr><tr><td style = \"text-align: right;\">&vellip;</td><td style = \"text-align: right;\">&vellip;</td><td style = \"text-align: right;\">&vellip;</td><td style = \"text-align: right;\">&vellip;</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999989</td><td style = \"text-align: right;\">9999989</td><td style = \"text-align: left;\">[0.383955, 0.70686]</td><td style = \"text-align: right;\">2</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999990</td><td style = \"text-align: right;\">9999990</td><td style = \"text-align: left;\">[0.193813, 0.649544, 0.539589]</td><td style = \"text-align: right;\">3</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999991</td><td style = \"text-align: right;\">9999991</td><td style = \"text-align: left;\">[0.382258, 0.901623]</td><td style = \"text-align: right;\">2</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999992</td><td style = \"text-align: right;\">9999992</td><td style = \"text-align: left;\">[0.640595, 0.916399]</td><td style = \"text-align: right;\">2</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999993</td><td style = \"text-align: right;\">9999993</td><td style = \"text-align: left;\">[0.963508, 0.301685]</td><td style = \"text-align: right;\">2</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999994</td><td style = \"text-align: right;\">9999994</td><td style = \"text-align: left;\">[0.846778, 0.328575]</td><td style = \"text-align: right;\">2</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999995</td><td style = \"text-align: right;\">9999995</td><td style = \"text-align: left;\">[0.251749, 0.28344, 0.16416, 0.618446]</td><td style = \"text-align: right;\">4</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999996</td><td style = \"text-align: right;\">9999996</td><td style = \"text-align: left;\">[0.383316, 0.352557, 0.470778]</td><td style = \"text-align: right;\">3</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999997</td><td style = \"text-align: right;\">9999997</td><td style = \"text-align: left;\">[0.362197, 0.639499]</td><td style = \"text-align: right;\">2</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999998</td><td style = \"text-align: right;\">9999998</td><td style = \"text-align: left;\">[0.046897, 0.552855, 0.172669, 0.156109, 0.799254]</td><td style = \"text-align: right;\">5</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999999</td><td style = \"text-align: right;\">9999999</td><td style = \"text-align: left;\">[0.815608, 0.663247]</td><td style = \"text-align: right;\">2</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">10000000</td><td style = \"text-align: right;\">10000000</td><td style = \"text-align: left;\">[0.61177, 0.5798]</td><td style = \"text-align: right;\">2</td></tr></tbody></table></div>"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ccc}\n",
       "\t& id & pos & jumps\\\\\n",
       "\t\\hline\n",
       "\t& Int64 & Array… & Int64\\\\\n",
       "\t\\hline\n",
       "\t1 & 1 & [0.727935, 0.00744801, 0.199377, 0.439243] & 4 \\\\\n",
       "\t2 & 2 & [0.682533, 0.956741] & 2 \\\\\n",
       "\t3 & 3 & [0.647855, 0.996665] & 2 \\\\\n",
       "\t4 & 4 & [0.749194, 0.110084, 0.491383] & 3 \\\\\n",
       "\t5 & 5 & [0.565145, 0.253812, 0.626794] & 3 \\\\\n",
       "\t6 & 6 & [0.234105, 0.124792, 0.609875, 0.672793] & 4 \\\\\n",
       "\t7 & 7 & [0.761916, 0.588872] & 2 \\\\\n",
       "\t8 & 8 & [0.365854, 0.131026, 0.946453] & 3 \\\\\n",
       "\t9 & 9 & [0.574323, 0.67765] & 2 \\\\\n",
       "\t10 & 10 & [0.571586, 0.0727161, 0.701116] & 3 \\\\\n",
       "\t11 & 11 & [0.0952175, 0.845515, 0.348995] & 3 \\\\\n",
       "\t12 & 12 & [0.768308, 0.26906] & 2 \\\\\n",
       "\t13 & 13 & [0.539631, 0.293905, 0.242195] & 3 \\\\\n",
       "\t14 & 14 & [0.97807, 0.853242] & 2 \\\\\n",
       "\t15 & 15 & [0.706065, 0.442139] & 2 \\\\\n",
       "\t16 & 16 & [0.884577, 0.520741] & 2 \\\\\n",
       "\t17 & 17 & [0.941831, 0.810699] & 2 \\\\\n",
       "\t18 & 18 & [0.464849, 0.977012] & 2 \\\\\n",
       "\t19 & 19 & [0.500161, 0.715846] & 2 \\\\\n",
       "\t20 & 20 & [0.18985, 0.376441, 0.565864] & 3 \\\\\n",
       "\t21 & 21 & [0.172236, 0.327351, 0.632108] & 3 \\\\\n",
       "\t22 & 22 & [0.374141, 0.728543] & 2 \\\\\n",
       "\t23 & 23 & [0.00277646, 0.134685, 0.531899, 0.0731709, 0.501756] & 5 \\\\\n",
       "\t24 & 24 & [0.906939, 0.116769] & 2 \\\\\n",
       "\t25 & 25 & [0.367198, 0.881163] & 2 \\\\\n",
       "\t26 & 26 & [0.67189, 0.479862] & 2 \\\\\n",
       "\t27 & 27 & [0.949968, 0.0262331, 0.347189] & 3 \\\\\n",
       "\t28 & 28 & [0.819254, 0.792831] & 2 \\\\\n",
       "\t29 & 29 & [0.96723, 0.471452] & 2 \\\\\n",
       "\t30 & 30 & [0.19378, 0.941781] & 2 \\\\\n",
       "\t$\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "\u001b[1m10000000×3 DataFrame\u001b[0m\n",
       "\u001b[1m      Row \u001b[0m│\u001b[1m id       \u001b[0m\u001b[1m pos                               \u001b[0m\u001b[1m jumps \u001b[0m\n",
       "          │\u001b[90m Int64    \u001b[0m\u001b[90m Array…                            \u001b[0m\u001b[90m Int64 \u001b[0m\n",
       "──────────┼────────────────────────────────────────────────────\n",
       "        1 │        1  [0.727935, 0.00744801, 0.199377,…      4\n",
       "        2 │        2  [0.682533, 0.956741]                   2\n",
       "        3 │        3  [0.647855, 0.996665]                   2\n",
       "        4 │        4  [0.749194, 0.110084, 0.491383]         3\n",
       "        5 │        5  [0.565145, 0.253812, 0.626794]         3\n",
       "        6 │        6  [0.234105, 0.124792, 0.609875, 0…      4\n",
       "        7 │        7  [0.761916, 0.588872]                   2\n",
       "        8 │        8  [0.365854, 0.131026, 0.946453]         3\n",
       "        9 │        9  [0.574323, 0.67765]                    2\n",
       "       10 │       10  [0.571586, 0.0727161, 0.701116]        3\n",
       "       11 │       11  [0.0952175, 0.845515, 0.348995]        3\n",
       "    ⋮     │    ⋮                      ⋮                    ⋮\n",
       "  9999991 │  9999991  [0.382258, 0.901623]                   2\n",
       "  9999992 │  9999992  [0.640595, 0.916399]                   2\n",
       "  9999993 │  9999993  [0.963508, 0.301685]                   2\n",
       "  9999994 │  9999994  [0.846778, 0.328575]                   2\n",
       "  9999995 │  9999995  [0.251749, 0.28344, 0.16416, 0.6…      4\n",
       "  9999996 │  9999996  [0.383316, 0.352557, 0.470778]         3\n",
       "  9999997 │  9999997  [0.362197, 0.639499]                   2\n",
       "  9999998 │  9999998  [0.046897, 0.552855, 0.172669, 0…      5\n",
       "  9999999 │  9999999  [0.815608, 0.663247]                   2\n",
       " 10000000 │ 10000000  [0.61177, 0.5798]                      2\n",
       "\u001b[36m                                           9999979 rows omitted\u001b[0m"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "transform!(df, :pos => ByRow(length) => :jumps)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us dissect what we have written above:\n",
    "* `transform!` adds columns to a data frame in-place\n",
    "* `:pos` is a source column\n",
    "* `ByRow(length)` tells us that we want to apply `length` function to each element for `:pos` column (without it `length` would be applied to the whole column - can you guess what would be the result?)\n",
    "* `:jumps` is the name of the column that should be created"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we are ready to find the average number of jumps that are made:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.7183826"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mean(df.jumps)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "or"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div><div style = \"float: left;\"><span>1×1 DataFrame</span></div><div style = \"clear: both;\"></div></div><div class = \"data-frame\" style = \"overflow-x: scroll;\"><table class = \"data-frame\" style = \"margin-bottom: 6px;\"><thead><tr class = \"header\"><th class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">Row</th><th style = \"text-align: left;\">jumps_mean</th></tr><tr class = \"subheader headerLastRow\"><th class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\"></th><th title = \"Float64\" style = \"text-align: left;\">Float64</th></tr></thead><tbody><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">1</td><td style = \"text-align: right;\">2.71838</td></tr></tbody></table></div>"
      ],
      "text/latex": [
       "\\begin{tabular}{r|c}\n",
       "\t& jumps\\_mean\\\\\n",
       "\t\\hline\n",
       "\t& Float64\\\\\n",
       "\t\\hline\n",
       "\t1 & 2.71838 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "\u001b[1m1×1 DataFrame\u001b[0m\n",
       "\u001b[1m Row \u001b[0m│\u001b[1m jumps_mean \u001b[0m\n",
       "     │\u001b[90m Float64    \u001b[0m\n",
       "─────┼────────────\n",
       "   1 │    2.71838"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "combine(df, :jumps => mean)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "which happens to be very close to:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ℯ = 2.7182818284590..."
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "MathConstants.e"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us now find a distribution of number of jumps:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div><div style = \"float: left;\"><span>10×2 DataFrame</span></div><div style = \"clear: both;\"></div></div><div class = \"data-frame\" style = \"overflow-x: scroll;\"><table class = \"data-frame\" style = \"margin-bottom: 6px;\"><thead><tr class = \"header\"><th class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">Row</th><th style = \"text-align: left;\">jumps</th><th style = \"text-align: left;\">jumps_length</th></tr><tr class = \"subheader headerLastRow\"><th class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\"></th><th title = \"Int64\" style = \"text-align: left;\">Int64</th><th title = \"Int64\" style = \"text-align: left;\">Int64</th></tr></thead><tbody><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">1</td><td style = \"text-align: right;\">2</td><td style = \"text-align: right;\">5000265</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">2</td><td style = \"text-align: right;\">3</td><td style = \"text-align: right;\">3332702</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">3</td><td style = \"text-align: right;\">4</td><td style = \"text-align: right;\">1249879</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">4</td><td style = \"text-align: right;\">5</td><td style = \"text-align: right;\">333412</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">5</td><td style = \"text-align: right;\">6</td><td style = \"text-align: right;\">69869</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">6</td><td style = \"text-align: right;\">7</td><td style = \"text-align: right;\">11866</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">7</td><td style = \"text-align: right;\">8</td><td style = \"text-align: right;\">1754</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">8</td><td style = \"text-align: right;\">9</td><td style = \"text-align: right;\">227</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9</td><td style = \"text-align: right;\">10</td><td style = \"text-align: right;\">23</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">10</td><td style = \"text-align: right;\">11</td><td style = \"text-align: right;\">3</td></tr></tbody></table></div>"
      ],
      "text/latex": [
       "\\begin{tabular}{r|cc}\n",
       "\t& jumps & jumps\\_length\\\\\n",
       "\t\\hline\n",
       "\t& Int64 & Int64\\\\\n",
       "\t\\hline\n",
       "\t1 & 2 & 5000265 \\\\\n",
       "\t2 & 3 & 3332702 \\\\\n",
       "\t3 & 4 & 1249879 \\\\\n",
       "\t4 & 5 & 333412 \\\\\n",
       "\t5 & 6 & 69869 \\\\\n",
       "\t6 & 7 & 11866 \\\\\n",
       "\t7 & 8 & 1754 \\\\\n",
       "\t8 & 9 & 227 \\\\\n",
       "\t9 & 10 & 23 \\\\\n",
       "\t10 & 11 & 3 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "\u001b[1m10×2 DataFrame\u001b[0m\n",
       "\u001b[1m Row \u001b[0m│\u001b[1m jumps \u001b[0m\u001b[1m jumps_length \u001b[0m\n",
       "     │\u001b[90m Int64 \u001b[0m\u001b[90m Int64        \u001b[0m\n",
       "─────┼─────────────────────\n",
       "   1 │     2       5000265\n",
       "   2 │     3       3332702\n",
       "   3 │     4       1249879\n",
       "   4 │     5        333412\n",
       "   5 │     6         69869\n",
       "   6 │     7         11866\n",
       "   7 │     8          1754\n",
       "   8 │     9           227\n",
       "   9 │    10            23\n",
       "  10 │    11             3"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "jumps_agg = @pipe df |>\n",
    "                  groupby(_, :jumps, sort=true) |>\n",
    "                  combine(_, :jumps => length)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and normalize it as a fraction (and at the same time calculate some theoretical result that we have *guessed* :)):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div><div style = \"float: left;\"><span>10×4 DataFrame</span></div><div style = \"clear: both;\"></div></div><div class = \"data-frame\" style = \"overflow-x: scroll;\"><table class = \"data-frame\" style = \"margin-bottom: 6px;\"><thead><tr class = \"header\"><th class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">Row</th><th style = \"text-align: left;\">jumps</th><th style = \"text-align: left;\">jumps_length</th><th style = \"text-align: left;\">simulation</th><th style = \"text-align: left;\">theory</th></tr><tr class = \"subheader headerLastRow\"><th class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\"></th><th title = \"Int64\" style = \"text-align: left;\">Int64</th><th title = \"Int64\" style = \"text-align: left;\">Int64</th><th title = \"Float64\" style = \"text-align: left;\">Float64</th><th title = \"Float64\" style = \"text-align: left;\">Float64</th></tr></thead><tbody><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">1</td><td style = \"text-align: right;\">2</td><td style = \"text-align: right;\">5000265</td><td style = \"text-align: right;\">0.500027</td><td style = \"text-align: right;\">0.5</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">2</td><td style = \"text-align: right;\">3</td><td style = \"text-align: right;\">3332702</td><td style = \"text-align: right;\">0.33327</td><td style = \"text-align: right;\">0.333333</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">3</td><td style = \"text-align: right;\">4</td><td style = \"text-align: right;\">1249879</td><td style = \"text-align: right;\">0.124988</td><td style = \"text-align: right;\">0.125</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">4</td><td style = \"text-align: right;\">5</td><td style = \"text-align: right;\">333412</td><td style = \"text-align: right;\">0.0333412</td><td style = \"text-align: right;\">0.0333333</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">5</td><td style = \"text-align: right;\">6</td><td style = \"text-align: right;\">69869</td><td style = \"text-align: right;\">0.0069869</td><td style = \"text-align: right;\">0.00694444</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">6</td><td style = \"text-align: right;\">7</td><td style = \"text-align: right;\">11866</td><td style = \"text-align: right;\">0.0011866</td><td style = \"text-align: right;\">0.00119048</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">7</td><td style = \"text-align: right;\">8</td><td style = \"text-align: right;\">1754</td><td style = \"text-align: right;\">0.0001754</td><td style = \"text-align: right;\">0.000173611</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">8</td><td style = \"text-align: right;\">9</td><td style = \"text-align: right;\">227</td><td style = \"text-align: right;\">2.27e-5</td><td style = \"text-align: right;\">2.20459e-5</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9</td><td style = \"text-align: right;\">10</td><td style = \"text-align: right;\">23</td><td style = \"text-align: right;\">2.3e-6</td><td style = \"text-align: right;\">2.48016e-6</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">10</td><td style = \"text-align: right;\">11</td><td style = \"text-align: right;\">3</td><td style = \"text-align: right;\">3.0e-7</td><td style = \"text-align: right;\">2.50521e-7</td></tr></tbody></table></div>"
      ],
      "text/latex": [
       "\\begin{tabular}{r|cccc}\n",
       "\t& jumps & jumps\\_length & simulation & theory\\\\\n",
       "\t\\hline\n",
       "\t& Int64 & Int64 & Float64 & Float64\\\\\n",
       "\t\\hline\n",
       "\t1 & 2 & 5000265 & 0.500027 & 0.5 \\\\\n",
       "\t2 & 3 & 3332702 & 0.33327 & 0.333333 \\\\\n",
       "\t3 & 4 & 1249879 & 0.124988 & 0.125 \\\\\n",
       "\t4 & 5 & 333412 & 0.0333412 & 0.0333333 \\\\\n",
       "\t5 & 6 & 69869 & 0.0069869 & 0.00694444 \\\\\n",
       "\t6 & 7 & 11866 & 0.0011866 & 0.00119048 \\\\\n",
       "\t7 & 8 & 1754 & 0.0001754 & 0.000173611 \\\\\n",
       "\t8 & 9 & 227 & 2.27e-5 & 2.20459e-5 \\\\\n",
       "\t9 & 10 & 23 & 2.3e-6 & 2.48016e-6 \\\\\n",
       "\t10 & 11 & 3 & 3.0e-7 & 2.50521e-7 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "\u001b[1m10×4 DataFrame\u001b[0m\n",
       "\u001b[1m Row \u001b[0m│\u001b[1m jumps \u001b[0m\u001b[1m jumps_length \u001b[0m\u001b[1m simulation \u001b[0m\u001b[1m theory      \u001b[0m\n",
       "     │\u001b[90m Int64 \u001b[0m\u001b[90m Int64        \u001b[0m\u001b[90m Float64    \u001b[0m\u001b[90m Float64     \u001b[0m\n",
       "─────┼──────────────────────────────────────────────\n",
       "   1 │     2       5000265   0.500027   0.5\n",
       "   2 │     3       3332702   0.33327    0.333333\n",
       "   3 │     4       1249879   0.124988   0.125\n",
       "   4 │     5        333412   0.0333412  0.0333333\n",
       "   5 │     6         69869   0.0069869  0.00694444\n",
       "   6 │     7         11866   0.0011866  0.00119048\n",
       "   7 │     8          1754   0.0001754  0.000173611\n",
       "   8 │     9           227   2.27e-5    2.20459e-5\n",
       "   9 │    10            23   2.3e-6     2.48016e-6\n",
       "  10 │    11             3   3.0e-7     2.50521e-7"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "transform!(jumps_agg,\n",
    "           :jumps_length => (x -> x ./ sum(x)) => :simulation,\n",
    "           :jumps => ByRow(x -> (x-1) / factorial(x)) => :theory)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us briefly justify how we have guessed it (you can safely skip the derivation):\n",
    "\n",
    "Formula\n",
    "$$\n",
    "p_n = \\frac{n-1}{n!}\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\sum_{n=2}^{+\\infty}p_n=\\sum_{n=2}^{+\\infty} \\frac{n-1}{n!} = \\sum_{n=1}^{+\\infty} \\frac{1}{n!} - \\sum_{n=2}^{+\\infty} \\frac{1}{n!} = 1\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\sum_{n=2}^{+\\infty}n\\cdot p_n=\\sum_{n=2}^{+\\infty} n\\frac{n-1}{n!} = \\sum_{n=2}^{+\\infty} \\frac{1}{(n-2)!} = e\n",
    "$$\n",
    "\n",
    "Now we note that:\n",
    "\n",
    "$$\n",
    "1-\\sum_{n=2}^k p_n = \\frac{1}{k!}\n",
    "$$\n",
    "which can be most easily justified by a geometric argument."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To finish this section of the tutorial let us check if random numbers generated using `rand()` were indeed $U(0,1)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To do this we will add some columns to `df` data frame."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div><div style = \"float: left;\"><span>10000000×3 DataFrame</span></div><div style = \"float: right;\"><span style = \"font-style: italic;\">9999975 rows omitted</span></div><div style = \"clear: both;\"></div></div><div class = \"data-frame\" style = \"overflow-x: scroll;\"><table class = \"data-frame\" style = \"margin-bottom: 6px;\"><thead><tr class = \"header\"><th class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">Row</th><th style = \"text-align: left;\">id</th><th style = \"text-align: left;\">pos</th><th style = \"text-align: left;\">jumps</th></tr><tr class = \"subheader headerLastRow\"><th class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\"></th><th title = \"Int64\" style = \"text-align: left;\">Int64</th><th title = \"Vector{Float64}\" style = \"text-align: left;\">Array…</th><th title = \"Int64\" style = \"text-align: left;\">Int64</th></tr></thead><tbody><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">1</td><td style = \"text-align: right;\">1</td><td style = \"text-align: left;\">[0.727935, 0.00744801, 0.199377, 0.439243]</td><td style = \"text-align: right;\">4</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">2</td><td style = \"text-align: right;\">2</td><td style = \"text-align: left;\">[0.682533, 0.956741]</td><td style = \"text-align: right;\">2</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">3</td><td style = \"text-align: right;\">3</td><td style = \"text-align: left;\">[0.647855, 0.996665]</td><td style = \"text-align: right;\">2</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">4</td><td style = \"text-align: right;\">4</td><td style = \"text-align: left;\">[0.749194, 0.110084, 0.491383]</td><td style = \"text-align: right;\">3</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">5</td><td style = \"text-align: right;\">5</td><td style = \"text-align: left;\">[0.565145, 0.253812, 0.626794]</td><td style = \"text-align: right;\">3</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">6</td><td style = \"text-align: right;\">6</td><td style = \"text-align: left;\">[0.234105, 0.124792, 0.609875, 0.672793]</td><td style = \"text-align: right;\">4</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">7</td><td style = \"text-align: right;\">7</td><td style = \"text-align: left;\">[0.761916, 0.588872]</td><td style = \"text-align: right;\">2</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">8</td><td style = \"text-align: right;\">8</td><td style = \"text-align: left;\">[0.365854, 0.131026, 0.946453]</td><td style = \"text-align: right;\">3</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9</td><td style = \"text-align: right;\">9</td><td style = \"text-align: left;\">[0.574323, 0.67765]</td><td style = \"text-align: right;\">2</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">10</td><td style = \"text-align: right;\">10</td><td style = \"text-align: left;\">[0.571586, 0.0727161, 0.701116]</td><td style = \"text-align: right;\">3</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">11</td><td style = \"text-align: right;\">11</td><td style = \"text-align: left;\">[0.0952175, 0.845515, 0.348995]</td><td style = \"text-align: right;\">3</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">12</td><td style = \"text-align: right;\">12</td><td style = \"text-align: left;\">[0.768308, 0.26906]</td><td style = \"text-align: right;\">2</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">13</td><td style = \"text-align: right;\">13</td><td style = \"text-align: left;\">[0.539631, 0.293905, 0.242195]</td><td style = \"text-align: right;\">3</td></tr><tr><td style = \"text-align: right;\">&vellip;</td><td style = \"text-align: right;\">&vellip;</td><td style = \"text-align: right;\">&vellip;</td><td style = \"text-align: right;\">&vellip;</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999989</td><td style = \"text-align: right;\">9999989</td><td style = \"text-align: left;\">[0.383955, 0.70686]</td><td style = \"text-align: right;\">2</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999990</td><td style = \"text-align: right;\">9999990</td><td style = \"text-align: left;\">[0.193813, 0.649544, 0.539589]</td><td style = \"text-align: right;\">3</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999991</td><td style = \"text-align: right;\">9999991</td><td style = \"text-align: left;\">[0.382258, 0.901623]</td><td style = \"text-align: right;\">2</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999992</td><td style = \"text-align: right;\">9999992</td><td style = \"text-align: left;\">[0.640595, 0.916399]</td><td style = \"text-align: right;\">2</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999993</td><td style = \"text-align: right;\">9999993</td><td style = \"text-align: left;\">[0.963508, 0.301685]</td><td style = \"text-align: right;\">2</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999994</td><td style = \"text-align: right;\">9999994</td><td style = \"text-align: left;\">[0.846778, 0.328575]</td><td style = \"text-align: right;\">2</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999995</td><td style = \"text-align: right;\">9999995</td><td style = \"text-align: left;\">[0.251749, 0.28344, 0.16416, 0.618446]</td><td style = \"text-align: right;\">4</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999996</td><td style = \"text-align: right;\">9999996</td><td style = \"text-align: left;\">[0.383316, 0.352557, 0.470778]</td><td style = \"text-align: right;\">3</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999997</td><td style = \"text-align: right;\">9999997</td><td style = \"text-align: left;\">[0.362197, 0.639499]</td><td style = \"text-align: right;\">2</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999998</td><td style = \"text-align: right;\">9999998</td><td style = \"text-align: left;\">[0.046897, 0.552855, 0.172669, 0.156109, 0.799254]</td><td style = \"text-align: right;\">5</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999999</td><td style = \"text-align: right;\">9999999</td><td style = \"text-align: left;\">[0.815608, 0.663247]</td><td style = \"text-align: right;\">2</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">10000000</td><td style = \"text-align: right;\">10000000</td><td style = \"text-align: left;\">[0.61177, 0.5798]</td><td style = \"text-align: right;\">2</td></tr></tbody></table></div>"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ccc}\n",
       "\t& id & pos & jumps\\\\\n",
       "\t\\hline\n",
       "\t& Int64 & Array… & Int64\\\\\n",
       "\t\\hline\n",
       "\t1 & 1 & [0.727935, 0.00744801, 0.199377, 0.439243] & 4 \\\\\n",
       "\t2 & 2 & [0.682533, 0.956741] & 2 \\\\\n",
       "\t3 & 3 & [0.647855, 0.996665] & 2 \\\\\n",
       "\t4 & 4 & [0.749194, 0.110084, 0.491383] & 3 \\\\\n",
       "\t5 & 5 & [0.565145, 0.253812, 0.626794] & 3 \\\\\n",
       "\t6 & 6 & [0.234105, 0.124792, 0.609875, 0.672793] & 4 \\\\\n",
       "\t7 & 7 & [0.761916, 0.588872] & 2 \\\\\n",
       "\t8 & 8 & [0.365854, 0.131026, 0.946453] & 3 \\\\\n",
       "\t9 & 9 & [0.574323, 0.67765] & 2 \\\\\n",
       "\t10 & 10 & [0.571586, 0.0727161, 0.701116] & 3 \\\\\n",
       "\t11 & 11 & [0.0952175, 0.845515, 0.348995] & 3 \\\\\n",
       "\t12 & 12 & [0.768308, 0.26906] & 2 \\\\\n",
       "\t13 & 13 & [0.539631, 0.293905, 0.242195] & 3 \\\\\n",
       "\t14 & 14 & [0.97807, 0.853242] & 2 \\\\\n",
       "\t15 & 15 & [0.706065, 0.442139] & 2 \\\\\n",
       "\t16 & 16 & [0.884577, 0.520741] & 2 \\\\\n",
       "\t17 & 17 & [0.941831, 0.810699] & 2 \\\\\n",
       "\t18 & 18 & [0.464849, 0.977012] & 2 \\\\\n",
       "\t19 & 19 & [0.500161, 0.715846] & 2 \\\\\n",
       "\t20 & 20 & [0.18985, 0.376441, 0.565864] & 3 \\\\\n",
       "\t21 & 21 & [0.172236, 0.327351, 0.632108] & 3 \\\\\n",
       "\t22 & 22 & [0.374141, 0.728543] & 2 \\\\\n",
       "\t23 & 23 & [0.00277646, 0.134685, 0.531899, 0.0731709, 0.501756] & 5 \\\\\n",
       "\t24 & 24 & [0.906939, 0.116769] & 2 \\\\\n",
       "\t25 & 25 & [0.367198, 0.881163] & 2 \\\\\n",
       "\t26 & 26 & [0.67189, 0.479862] & 2 \\\\\n",
       "\t27 & 27 & [0.949968, 0.0262331, 0.347189] & 3 \\\\\n",
       "\t28 & 28 & [0.819254, 0.792831] & 2 \\\\\n",
       "\t29 & 29 & [0.96723, 0.471452] & 2 \\\\\n",
       "\t30 & 30 & [0.19378, 0.941781] & 2 \\\\\n",
       "\t$\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "\u001b[1m10000000×3 DataFrame\u001b[0m\n",
       "\u001b[1m      Row \u001b[0m│\u001b[1m id       \u001b[0m\u001b[1m pos                               \u001b[0m\u001b[1m jumps \u001b[0m\n",
       "          │\u001b[90m Int64    \u001b[0m\u001b[90m Array…                            \u001b[0m\u001b[90m Int64 \u001b[0m\n",
       "──────────┼────────────────────────────────────────────────────\n",
       "        1 │        1  [0.727935, 0.00744801, 0.199377,…      4\n",
       "        2 │        2  [0.682533, 0.956741]                   2\n",
       "        3 │        3  [0.647855, 0.996665]                   2\n",
       "        4 │        4  [0.749194, 0.110084, 0.491383]         3\n",
       "        5 │        5  [0.565145, 0.253812, 0.626794]         3\n",
       "        6 │        6  [0.234105, 0.124792, 0.609875, 0…      4\n",
       "        7 │        7  [0.761916, 0.588872]                   2\n",
       "        8 │        8  [0.365854, 0.131026, 0.946453]         3\n",
       "        9 │        9  [0.574323, 0.67765]                    2\n",
       "       10 │       10  [0.571586, 0.0727161, 0.701116]        3\n",
       "       11 │       11  [0.0952175, 0.845515, 0.348995]        3\n",
       "    ⋮     │    ⋮                      ⋮                    ⋮\n",
       "  9999991 │  9999991  [0.382258, 0.901623]                   2\n",
       "  9999992 │  9999992  [0.640595, 0.916399]                   2\n",
       "  9999993 │  9999993  [0.963508, 0.301685]                   2\n",
       "  9999994 │  9999994  [0.846778, 0.328575]                   2\n",
       "  9999995 │  9999995  [0.251749, 0.28344, 0.16416, 0.6…      4\n",
       "  9999996 │  9999996  [0.383316, 0.352557, 0.470778]         3\n",
       "  9999997 │  9999997  [0.362197, 0.639499]                   2\n",
       "  9999998 │  9999998  [0.046897, 0.552855, 0.172669, 0…      5\n",
       "  9999999 │  9999999  [0.815608, 0.663247]                   2\n",
       " 10000000 │ 10000000  [0.61177, 0.5798]                      2\n",
       "\u001b[36m                                           9999979 rows omitted\u001b[0m"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div><div style = \"float: left;\"><span>10000000×2 DataFrame</span></div><div style = \"float: right;\"><span style = \"font-style: italic;\">9999975 rows omitted</span></div><div style = \"clear: both;\"></div></div><div class = \"data-frame\" style = \"overflow-x: scroll;\"><table class = \"data-frame\" style = \"margin-bottom: 6px;\"><thead><tr class = \"header\"><th class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">Row</th><th style = \"text-align: left;\">first</th><th style = \"text-align: left;\">last</th></tr><tr class = \"subheader headerLastRow\"><th class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\"></th><th title = \"Float64\" style = \"text-align: left;\">Float64</th><th title = \"Float64\" style = \"text-align: left;\">Float64</th></tr></thead><tbody><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">1</td><td style = \"text-align: right;\">0.727935</td><td style = \"text-align: right;\">0.439243</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">2</td><td style = \"text-align: right;\">0.682533</td><td style = \"text-align: right;\">0.956741</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">3</td><td style = \"text-align: right;\">0.647855</td><td style = \"text-align: right;\">0.996665</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">4</td><td style = \"text-align: right;\">0.749194</td><td style = \"text-align: right;\">0.491383</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">5</td><td style = \"text-align: right;\">0.565145</td><td style = \"text-align: right;\">0.626794</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">6</td><td style = \"text-align: right;\">0.234105</td><td style = \"text-align: right;\">0.672793</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">7</td><td style = \"text-align: right;\">0.761916</td><td style = \"text-align: right;\">0.588872</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">8</td><td style = \"text-align: right;\">0.365854</td><td style = \"text-align: right;\">0.946453</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9</td><td style = \"text-align: right;\">0.574323</td><td style = \"text-align: right;\">0.67765</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">10</td><td style = \"text-align: right;\">0.571586</td><td style = \"text-align: right;\">0.701116</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">11</td><td style = \"text-align: right;\">0.0952175</td><td style = \"text-align: right;\">0.348995</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">12</td><td style = \"text-align: right;\">0.768308</td><td style = \"text-align: right;\">0.26906</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">13</td><td style = \"text-align: right;\">0.539631</td><td style = \"text-align: right;\">0.242195</td></tr><tr><td style = \"text-align: right;\">&vellip;</td><td style = \"text-align: right;\">&vellip;</td><td style = \"text-align: right;\">&vellip;</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999989</td><td style = \"text-align: right;\">0.383955</td><td style = \"text-align: right;\">0.70686</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999990</td><td style = \"text-align: right;\">0.193813</td><td style = \"text-align: right;\">0.539589</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999991</td><td style = \"text-align: right;\">0.382258</td><td style = \"text-align: right;\">0.901623</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999992</td><td style = \"text-align: right;\">0.640595</td><td style = \"text-align: right;\">0.916399</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999993</td><td style = \"text-align: right;\">0.963508</td><td style = \"text-align: right;\">0.301685</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999994</td><td style = \"text-align: right;\">0.846778</td><td style = \"text-align: right;\">0.328575</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999995</td><td style = \"text-align: right;\">0.251749</td><td style = \"text-align: right;\">0.618446</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999996</td><td style = \"text-align: right;\">0.383316</td><td style = \"text-align: right;\">0.470778</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999997</td><td style = \"text-align: right;\">0.362197</td><td style = \"text-align: right;\">0.639499</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999998</td><td style = \"text-align: right;\">0.046897</td><td style = \"text-align: right;\">0.799254</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9999999</td><td style = \"text-align: right;\">0.815608</td><td style = \"text-align: right;\">0.663247</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">10000000</td><td style = \"text-align: right;\">0.61177</td><td style = \"text-align: right;\">0.5798</td></tr></tbody></table></div>"
      ],
      "text/latex": [
       "\\begin{tabular}{r|cc}\n",
       "\t& first & last\\\\\n",
       "\t\\hline\n",
       "\t& Float64 & Float64\\\\\n",
       "\t\\hline\n",
       "\t1 & 0.727935 & 0.439243 \\\\\n",
       "\t2 & 0.682533 & 0.956741 \\\\\n",
       "\t3 & 0.647855 & 0.996665 \\\\\n",
       "\t4 & 0.749194 & 0.491383 \\\\\n",
       "\t5 & 0.565145 & 0.626794 \\\\\n",
       "\t6 & 0.234105 & 0.672793 \\\\\n",
       "\t7 & 0.761916 & 0.588872 \\\\\n",
       "\t8 & 0.365854 & 0.946453 \\\\\n",
       "\t9 & 0.574323 & 0.67765 \\\\\n",
       "\t10 & 0.571586 & 0.701116 \\\\\n",
       "\t11 & 0.0952175 & 0.348995 \\\\\n",
       "\t12 & 0.768308 & 0.26906 \\\\\n",
       "\t13 & 0.539631 & 0.242195 \\\\\n",
       "\t14 & 0.97807 & 0.853242 \\\\\n",
       "\t15 & 0.706065 & 0.442139 \\\\\n",
       "\t16 & 0.884577 & 0.520741 \\\\\n",
       "\t17 & 0.941831 & 0.810699 \\\\\n",
       "\t18 & 0.464849 & 0.977012 \\\\\n",
       "\t19 & 0.500161 & 0.715846 \\\\\n",
       "\t20 & 0.18985 & 0.565864 \\\\\n",
       "\t21 & 0.172236 & 0.632108 \\\\\n",
       "\t22 & 0.374141 & 0.728543 \\\\\n",
       "\t23 & 0.00277646 & 0.501756 \\\\\n",
       "\t24 & 0.906939 & 0.116769 \\\\\n",
       "\t25 & 0.367198 & 0.881163 \\\\\n",
       "\t26 & 0.67189 & 0.479862 \\\\\n",
       "\t27 & 0.949968 & 0.347189 \\\\\n",
       "\t28 & 0.819254 & 0.792831 \\\\\n",
       "\t29 & 0.96723 & 0.471452 \\\\\n",
       "\t30 & 0.19378 & 0.941781 \\\\\n",
       "\t$\\dots$ & $\\dots$ & $\\dots$ \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "\u001b[1m10000000×2 DataFrame\u001b[0m\n",
       "\u001b[1m      Row \u001b[0m│\u001b[1m first     \u001b[0m\u001b[1m last     \u001b[0m\n",
       "          │\u001b[90m Float64   \u001b[0m\u001b[90m Float64  \u001b[0m\n",
       "──────────┼─────────────────────\n",
       "        1 │ 0.727935   0.439243\n",
       "        2 │ 0.682533   0.956741\n",
       "        3 │ 0.647855   0.996665\n",
       "        4 │ 0.749194   0.491383\n",
       "        5 │ 0.565145   0.626794\n",
       "        6 │ 0.234105   0.672793\n",
       "        7 │ 0.761916   0.588872\n",
       "        8 │ 0.365854   0.946453\n",
       "        9 │ 0.574323   0.67765\n",
       "       10 │ 0.571586   0.701116\n",
       "       11 │ 0.0952175  0.348995\n",
       "    ⋮     │     ⋮         ⋮\n",
       "  9999991 │ 0.382258   0.901623\n",
       "  9999992 │ 0.640595   0.916399\n",
       "  9999993 │ 0.963508   0.301685\n",
       "  9999994 │ 0.846778   0.328575\n",
       "  9999995 │ 0.251749   0.618446\n",
       "  9999996 │ 0.383316   0.470778\n",
       "  9999997 │ 0.362197   0.639499\n",
       "  9999998 │ 0.046897   0.799254\n",
       "  9999999 │ 0.815608   0.663247\n",
       " 10000000 │ 0.61177    0.5798\n",
       "\u001b[36m            9999979 rows omitted\u001b[0m"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_test = select(df, :pos => ByRow(first) => :first, :pos => ByRow(last) => :last)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Figure(PyObject <Figure size 640x480 with 1 Axes>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "hist(df_test.first, 100);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So far all looks good. But let us look at the distribution of the last dawn random number:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Figure(PyObject <Figure size 640x480 with 1 Axes>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "hist(df_test.last, 100);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So - is the `rand()` function broken for the last generated random number in each sequence or something else has made the distribution stop being uniform?"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Julia 1.9.0",
   "language": "julia",
   "name": "julia-1.9"
  },
  "language_info": {
   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
   "version": "1.9.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}