{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercise 5.3: Bootstrapping \"theory\" with hacker stats\n",
    "\n",
    "<hr>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Say we have a data set with $n$ unique measurements. It can be shown that on average a fraction of $(1-1/n)^n$ of the measurements do not appear in a bootstrap sample. Note that for large samples, this is approximately $1/e \\approx 1/2.7$, since\n",
    "\n",
    "\\begin{align}\n",
    "\\lim_{n\\to\\infty} (1-1/n)^n = 1/e.\n",
    "\\end{align}\n",
    "\n",
    "Use hacker stats to show that this is, indeed true. *Hint*: Think about a convenient \"data set\" to use for drawing samples.\n",
    "\n",
    "This is kind of fun; you're investigating some theory behind hacker stats *with hacker stats*!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br />"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solution\n",
    "\n",
    "<hr>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
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       "        <span id=\"d0cc3fa3-5d42-41b4-aa46-13e905c5eef5\">Loading BokehJS ...</span>\n",
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     "output_type": "display_data"
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       "\n",
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       "     \"<p>\\n\"+\n",
       "     \"BokehJS does not appear to have successfully loaded. If loading BokehJS from CDN, this \\n\"+\n",
       "     \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n",
       "     \"</p>\\n\"+\n",
       "     \"<ul>\\n\"+\n",
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       "     \"<li>use INLINE resources instead, as so:</li>\\n\"+\n",
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       "  }\n",
       "\n",
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       "\n",
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       "    }\n",
       "    if (js_urls == null || js_urls.length === 0) {\n",
       "      run_callbacks();\n",
       "      return null;\n",
       "    }\n",
       "    console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n",
       "    root._bokeh_is_loading = css_urls.length + js_urls.length;\n",
       "\n",
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       "\n",
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       "      element.href = url;\n",
       "      console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n",
       "      document.body.appendChild(element);\n",
       "    }\n",
       "\n",
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       "      document.head.appendChild(element);\n",
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       "\n",
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       "    document.body.appendChild(element);\n",
       "  }\n",
       "\n",
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       "  const css_urls = [];\n",
       "\n",
       "  const inline_js = [    function(Bokeh) {\n",
       "      Bokeh.set_log_level(\"info\");\n",
       "    },\n",
       "function(Bokeh) {\n",
       "    }\n",
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       "    } else if (!root._bokeh_failed_load) {\n",
       "      console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n",
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       "    } else if (force !== true) {\n",
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       "      cell.output_area.append_execute_result(NB_LOAD_WARNING)\n",
       "    }\n",
       "  }\n",
       "\n",
       "  if (root._bokeh_is_loading === 0) {\n",
       "    console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n",
       "    run_inline_js();\n",
       "  } else {\n",
       "    load_libs(css_urls, js_urls, function() {\n",
       "      console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n",
       "      run_inline_js();\n",
       "    });\n",
       "  }\n",
       "}(window));"
      ],
      "application/vnd.bokehjs_load.v0+json": "(function(root) {\n  function now() {\n    return new Date();\n  }\n\n  const force = true;\n\n  if (typeof root._bokeh_onload_callbacks === \"undefined\" || force === true) {\n    root._bokeh_onload_callbacks = [];\n    root._bokeh_is_loading = undefined;\n  }\n\n\n  if (typeof (root._bokeh_timeout) === \"undefined\" || force === true) {\n    root._bokeh_timeout = Date.now() + 5000;\n    root._bokeh_failed_load = false;\n  }\n\n  const NB_LOAD_WARNING = {'data': {'text/html':\n     \"<div style='background-color: #fdd'>\\n\"+\n     \"<p>\\n\"+\n     \"BokehJS does not appear to have successfully loaded. If loading BokehJS from CDN, this \\n\"+\n     \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n     \"</p>\\n\"+\n     \"<ul>\\n\"+\n     \"<li>re-rerun `output_notebook()` to attempt to load from CDN again, or</li>\\n\"+\n     \"<li>use INLINE resources instead, as so:</li>\\n\"+\n     \"</ul>\\n\"+\n     \"<code>\\n\"+\n     \"from bokeh.resources import INLINE\\n\"+\n     \"output_notebook(resources=INLINE)\\n\"+\n     \"</code>\\n\"+\n     \"</div>\"}};\n\n  function display_loaded() {\n    const el = document.getElementById(\"d0cc3fa3-5d42-41b4-aa46-13e905c5eef5\");\n    if (el != null) {\n      el.textContent = \"BokehJS is loading...\";\n    }\n    if (root.Bokeh !== undefined) {\n      if (el != null) {\n        el.textContent = \"BokehJS \" + root.Bokeh.version + \" successfully loaded.\";\n      }\n    } else if (Date.now() < root._bokeh_timeout) {\n      setTimeout(display_loaded, 100)\n    }\n  }\n\n  function run_callbacks() {\n    try {\n      root._bokeh_onload_callbacks.forEach(function(callback) {\n        if (callback != null)\n          callback();\n      });\n    } finally {\n      delete root._bokeh_onload_callbacks\n    }\n    console.debug(\"Bokeh: all callbacks have finished\");\n  }\n\n  function load_libs(css_urls, js_urls, callback) {\n    if (css_urls == null) css_urls = [];\n    if (js_urls == null) js_urls = [];\n\n    root._bokeh_onload_callbacks.push(callback);\n    if (root._bokeh_is_loading > 0) {\n      console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n      return null;\n    }\n    if (js_urls == null || js_urls.length === 0) {\n      run_callbacks();\n      return null;\n    }\n    console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n    root._bokeh_is_loading = css_urls.length + js_urls.length;\n\n    function on_load() {\n      root._bokeh_is_loading--;\n      if (root._bokeh_is_loading === 0) {\n        console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n        run_callbacks()\n      }\n    }\n\n    function on_error(url) {\n      console.error(\"failed to load \" + url);\n    }\n\n    for (let i = 0; i < css_urls.length; i++) {\n      const url = css_urls[i];\n      const element = document.createElement(\"link\");\n      element.onload = on_load;\n      element.onerror = on_error.bind(null, url);\n      element.rel = \"stylesheet\";\n      element.type = \"text/css\";\n      element.href = url;\n      console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n      document.body.appendChild(element);\n    }\n\n    for (let i = 0; i < js_urls.length; i++) {\n      const url = js_urls[i];\n      const element = document.createElement('script');\n      element.onload = on_load;\n      element.onerror = on_error.bind(null, url);\n      element.async = false;\n      element.src = url;\n      console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n      document.head.appendChild(element);\n    }\n  };\n\n  function inject_raw_css(css) {\n    const element = document.createElement(\"style\");\n    element.appendChild(document.createTextNode(css));\n    document.body.appendChild(element);\n  }\n\n  const js_urls = [\"https://cdn.bokeh.org/bokeh/release/bokeh-3.1.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-3.1.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-3.1.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-3.1.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-mathjax-3.1.1.min.js\"];\n  const css_urls = [];\n\n  const inline_js = [    function(Bokeh) {\n      Bokeh.set_log_level(\"info\");\n    },\nfunction(Bokeh) {\n    }\n  ];\n\n  function run_inline_js() {\n    if (root.Bokeh !== undefined || force === true) {\n          for (let i = 0; i < inline_js.length; i++) {\n      inline_js[i].call(root, root.Bokeh);\n    }\nif (force === true) {\n        display_loaded();\n      }} else if (Date.now() < root._bokeh_timeout) {\n      setTimeout(run_inline_js, 100);\n    } else if (!root._bokeh_failed_load) {\n      console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n      root._bokeh_failed_load = true;\n    } else if (force !== true) {\n      const cell = $(document.getElementById(\"d0cc3fa3-5d42-41b4-aa46-13e905c5eef5\")).parents('.cell').data().cell;\n      cell.output_area.append_execute_result(NB_LOAD_WARNING)\n    }\n  }\n\n  if (root._bokeh_is_loading === 0) {\n    console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n    run_inline_js();\n  } else {\n    load_libs(css_urls, js_urls, function() {\n      console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n      run_inline_js();\n    });\n  }\n}(window));"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "import bokeh.io\n",
    "import bokeh.plotting\n",
    "\n",
    "bokeh.io.output_notebook()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will generate a data set that is just $n$ consecutive integers. Our statistic from our bootstrap sample is then\n",
    "\n",
    "\\begin{align}\n",
    "\\text{fraction omitted} = 1 - \\frac{\\text{number of unique entries in the bootstrap sample}}{n}.\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def frac_omitted(data):\n",
    "    return 1 - len(np.unique(data)) / len(data)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will draw a few thousand bootstrap replicates of this for several values of $n$ and make a plot of the mean. To facilitate, we will use the `draw_bs_reps()` function we wrote in [Exercise 8.1](exercise_8.1.ipynb)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "rng = np.random.default_rng()\n",
    "\n",
    "def draw_bs_reps(data, func, rng, size=1, args=()):\n",
    "    return np.array(\n",
    "        [\n",
    "            func(rng.choice(data, replace=True, size=len(data)), *args)\n",
    "            for _ in range(size)\n",
    "        ]\n",
    "    )\n",
    "\n",
    "\n",
    "def mean_frac_omitted(n, n_bs_reps=10000):\n",
    "    data = np.arange(n)\n",
    "    bs_reps = draw_bs_reps(data, frac_omitted, rng, size=n_bs_reps)\n",
    "\n",
    "    return np.mean(bs_reps)\n",
    "\n",
    "\n",
    "n = np.unique(np.logspace(0, 3).astype(int))\n",
    "mean_f = np.array([mean_frac_omitted(n_val) for n_val in n])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now that we have the mean fraction omitted for each value of $n$, we can plot it, together with the theoretical curve."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
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     "metadata": {},
     "output_type": "display_data"
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1050\",\"attributes\":{\"x\":{\"type\":\"field\",\"field\":\"x\"},\"y\":{\"type\":\"field\",\"field\":\"y\"},\"line_color\":\"#1f77b4\",\"line_width\":2}},\"nonselection_glyph\":{\"type\":\"object\",\"name\":\"Line\",\"id\":\"p1051\",\"attributes\":{\"x\":{\"type\":\"field\",\"field\":\"x\"},\"y\":{\"type\":\"field\",\"field\":\"y\"},\"line_color\":\"#1f77b4\",\"line_alpha\":0.1,\"line_width\":2}},\"muted_glyph\":{\"type\":\"object\",\"name\":\"Line\",\"id\":\"p1052\",\"attributes\":{\"x\":{\"type\":\"field\",\"field\":\"x\"},\"y\":{\"type\":\"field\",\"field\":\"y\"},\"line_color\":\"#1f77b4\",\"line_alpha\":0.2,\"line_width\":2}}}},{\"type\":\"object\",\"name\":\"GlyphRenderer\",\"id\":\"p1062\",\"attributes\":{\"data_source\":{\"type\":\"object\",\"name\":\"ColumnDataSource\",\"id\":\"p1056\",\"attributes\":{\"selected\":{\"type\":\"object\",\"name\":\"Selection\",\"id\":\"p1057\",\"attributes\":{\"indices\":[],\"line_indices\":[]}},\"selection_policy\":{\"type\":\"object\",\"name\":\"UnionRenderers\",\"id\":\"p1058\"},\"data\":{\"type\":\"map\",\"entries\":[[\"x\",[0.9,1100]],[\"y\",[0.36787944117144233,0.36787944117144233]]]}}},\"view\":{\"type\":\"object\",\"name\":\"CDSView\",\"id\":\"p1063\",\"attributes\":{\"filter\":{\"type\":\"object\",\"name\":\"AllIndices\",\"id\":\"p1064\"}}},\"glyph\":{\"type\":\"object\",\"name\":\"Line\",\"id\":\"p1059\",\"attributes\":{\"x\":{\"type\":\"field\",\"field\":\"x\"},\"y\":{\"type\":\"field\",\"field\":\"y\"},\"line_color\":\"gray\",\"line_dash\":[6]}},\"nonselection_glyph\":{\"type\":\"object\",\"name\":\"Line\",\"id\":\"p1060\",\"attributes\":{\"x\":{\"type\":\"field\",\"field\":\"x\"},\"y\":{\"type\":\"field\",\"field\":\"y\"},\"line_color\":\"gray\",\"line_alpha\":0.1,\"line_dash\":[6]}},\"muted_glyph\":{\"type\":\"object\",\"name\":\"Line\",\"id\":\"p1061\",\"attributes\":{\"x\":{\"type\":\"field\",\"field\":\"x\"},\"y\":{\"type\":\"field\",\"field\":\"y\"},\"line_color\":\"gray\",\"line_alpha\":0.2,\"line_dash\":[6]}}}},{\"type\":\"object\",\"name\":\"GlyphRenderer\",\"id\":\"p1071\",\"attributes\":{\"data_source\":{\"type\":\"object\",\"name\":\"ColumnDataSource\",\"id\":\"p1065\",\"attributes\":{\"selected\":{\"type\":\"object\",\"name\":\"Selection\",\"id\":\"p1066\",\"attributes\":{\"indices\":[],\"line_indices\":[]}},\"selection_policy\":{\"type\":\"object\",\"name\":\"UnionRenderers\",\"id\":\"p1067\"},\"data\":{\"type\":\"map\",\"entries\":[[\"x\",{\"type\":\"ndarray\",\"array\":{\"type\":\"bytes\",\"data\":\"AQAAAAIAAAADAAAABAAAAAUAAAAGAAAABwAAAAgAAAAJAAAACgAAAAwAAAAOAAAAEAAAABMAAAAWAAAAGQAAAB0AAAAhAAAAJwAAACwAAAAzAAAAOwAAAEQAAABPAAAAWwAAAGgAAAB4AAAAigAAAJ8AAAC4AAAA1AAAAPQAAAAZAQAAQwEAAHQBAACtAQAA7gEAADgCAACPAgAA8gIAAGQDAADoAwAA\"},\"shape\":[42],\"dtype\":\"int32\",\"order\":\"little\"}],[\"y\",{\"type\":\"ndarray\",\"array\":{\"type\":\"bytes\",\"data\":\"AAAAAAAAAABU46WbxCDQPyEcxreK2NI/woanV8oy1D/n6PF7m/7UP+Mz7AqQe9U/oZ5/PWOg1T8QWDm0yPbVPwmrYsh1BNY/n82qz9VW1j+PwvUoXI/WP4VPd9+zstY/OdbFbTTA1j+TFkNI+OHWP4M9ZDTTCNc/bJih8UQQ1z/AtA9IWijXP+f5uqiGL9c/eaL+jVBG1z8k9ymb+0HXPxYacH6rXdc/cPz/Rfda1z8nrsSdbGTXP6gmPxQhbNc/wy3eDWVn1z8u/yH99nXXP4bEPZY+dNc/zEug0fNz1z8Es9IeLXvXPzWTWg6Wetc/AlH2gP2G1z+9bPaZS3nXP8nKu8Dyf9c/YZR/NmB71z87pPNNEYPXP3YgHNHChNc/dE5QQxCH1z9deeX5rYXXP7jPJVGFhtc/mARqtHGJ1z87u8/DlofXP2qWuhqPiNc/\"},\"shape\":[42],\"dtype\":\"float64\",\"order\":\"little\"}]]}}},\"view\":{\"type\":\"object\",\"name\":\"CDSView\",\"id\":\"p1072\",\"attributes\":{\"filter\":{\"type\":\"object\",\"name\":\"AllIndices\",\"id\":\"p1073\"}}},\"glyph\":{\"type\":\"object\",\"name\":\"Circle\",\"id\":\"p1068\",\"attributes\":{\"x\":{\"type\":\"field\",\"field\":\"x\"},\"y\":{\"type\":\"field\",\"field\":\"y\"},\"line_color\":{\"type\":\"value\",\"value\":\"orange\"},\"fill_color\":{\"type\":\"value\",\"value\":\"orange\"},\"hatch_color\":{\"type\":\"value\",\"value\":\"orange\"}}},\"nonselection_glyph\":{\"type\":\"object\",\"name\":\"Circle\",\"id\":\"p1069\",\"attributes\":{\"x\":{\"type\":\"field\",\"field\":\"x\"},\"y\":{\"type\":\"field\",\"field\":\"y\"},\"line_color\":{\"type\":\"value\",\"value\":\"orange\"},\"line_alpha\":{\"type\":\"value\",\"value\":0.1},\"fill_color\":{\"type\":\"value\",\"value\":\"orange\"},\"fill_alpha\":{\"type\":\"value\",\"value\":0.1},\"hatch_color\":{\"type\":\"value\",\"value\":\"orange\"},\"hatch_alpha\":{\"type\":\"value\",\"value\":0.1}}},\"muted_glyph\":{\"type\":\"object\",\"name\":\"Circle\",\"id\":\"p1070\",\"attributes\":{\"x\":{\"type\":\"field\",\"field\":\"x\"},\"y\":{\"type\":\"field\",\"field\":\"y\"},\"line_color\":{\"type\":\"value\",\"value\":\"orange\"},\"line_alpha\":{\"type\":\"value\",\"value\":0.2},\"fill_color\":{\"type\":\"value\",\"value\":\"orange\"},\"fill_alpha\":{\"type\":\"value\",\"value\":0.2},\"hatch_color\":{\"type\":\"value\",\"value\":\"orange\"},\"hatch_alpha\":{\"type\":\"value\",\"value\":0.2}}}}}],\"toolbar\":{\"type\":\"object\",\"name\":\"Toolbar\",\"id\":\"p1008\",\"attributes\":{\"tools\":[{\"type\":\"object\",\"name\":\"PanTool\",\"id\":\"p1032\"},{\"type\":\"object\",\"name\":\"WheelZoomTool\",\"id\":\"p1033\"},{\"type\":\"object\",\"name\":\"BoxZoomTool\",\"id\":\"p1034\",\"attributes\":{\"overlay\":{\"type\":\"object\",\"name\":\"BoxAnnotation\",\"id\":\"p1035\",\"attributes\":{\"syncable\":false,\"level\":\"overlay\",\"visible\":false,\"left_units\":\"canvas\",\"right_units\":\"canvas\",\"bottom_units\":\"canvas\",\"top_units\":\"canvas\",\"line_color\":\"black\",\"line_alpha\":1.0,\"line_width\":2,\"line_dash\":[4,4],\"fill_color\":\"lightgrey\",\"fill_alpha\":0.5}}}},{\"type\":\"object\",\"name\":\"SaveTool\",\"id\":\"p1036\"},{\"type\":\"object\",\"name\":\"ResetTool\",\"id\":\"p1037\"},{\"type\":\"object\",\"name\":\"HelpTool\",\"id\":\"p1038\"}]}},\"left\":[{\"type\":\"object\",\"name\":\"LinearAxis\",\"id\":\"p1025\",\"attributes\":{\"ticker\":{\"type\":\"object\",\"name\":\"BasicTicker\",\"id\":\"p1026\",\"attributes\":{\"mantissas\":[1,2,5]}},\"formatter\":{\"type\":\"object\",\"name\":\"BasicTickFormatter\",\"id\":\"p1027\"},\"axis_label\":\"mean fraction omitted\",\"major_label_policy\":{\"type\":\"object\",\"name\":\"AllLabels\",\"id\":\"p1028\"}}}],\"below\":[{\"type\":\"object\",\"name\":\"LogAxis\",\"id\":\"p1018\",\"attributes\":{\"ticker\":{\"type\":\"object\",\"name\":\"LogTicker\",\"id\":\"p1019\",\"attributes\":{\"num_minor_ticks\":10,\"mantissas\":[1,5]}},\"formatter\":{\"type\":\"object\",\"name\":\"LogTickFormatter\",\"id\":\"p1020\"},\"axis_label\":\"n\",\"major_label_policy\":{\"type\":\"object\",\"name\":\"AllLabels\",\"id\":\"p1021\"}}}],\"center\":[{\"type\":\"object\",\"name\":\"Grid\",\"id\":\"p1024\",\"attributes\":{\"axis\":{\"id\":\"p1018\"}}},{\"type\":\"object\",\"name\":\"Grid\",\"id\":\"p1031\",\"attributes\":{\"dimension\":1,\"axis\":{\"id\":\"p1025\"}}}],\"frame_width\":350,\"frame_height\":250}}],\"callbacks\":{\"type\":\"map\"}}};\n",
       "  const render_items = [{\"docid\":\"6c931081-72ac-407d-a39e-693f0be225ba\",\"roots\":{\"p1001\":\"dd4a9b34-c373-43bb-8acb-5cf8921d8d49\"},\"root_ids\":[\"p1001\"]}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    let attempts = 0;\n",
       "    const timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        clearInterval(timer);\n",
       "        embed_document(root);\n",
       "      } else {\n",
       "        attempts++;\n",
       "        if (attempts > 100) {\n",
       "          clearInterval(timer);\n",
       "          console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        }\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "p1001"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Compute theoretical curve\n",
    "n_theor = np.logspace(0, 3, 400)\n",
    "mean_f_theor = (1 - 1/n_theor)**n_theor\n",
    "\n",
    "# Set up figure\n",
    "p = bokeh.plotting.figure(\n",
    "    frame_height=250,\n",
    "    frame_width=350,\n",
    "    x_axis_label='n',\n",
    "    y_axis_label='mean fraction omitted',\n",
    "    x_axis_type='log',\n",
    "    x_range=[0.9, 1100],\n",
    ")\n",
    "\n",
    "# Theoretical curve\n",
    "p.line(\n",
    "    x=n_theor,\n",
    "    y=mean_f_theor,\n",
    "    line_width=2\n",
    ")\n",
    "\n",
    "# Asymptotic result\n",
    "p.line(\n",
    "    x=[0.9, 1100],\n",
    "    y=[1 / np.exp(1)] * 2,\n",
    "    line_dash='dashed',\n",
    "    line_color='gray',\n",
    ")\n",
    "\n",
    "# Result from hacker stats\n",
    "p.circle(\n",
    "    x=n,\n",
    "    y=mean_f,\n",
    "    color='orange'\n",
    ")\n",
    "\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The result closely matches theory!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Computing environment"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Python implementation: CPython\n",
      "Python version       : 3.11.3\n",
      "IPython version      : 8.12.0\n",
      "\n",
      "numpy     : 1.24.3\n",
      "bokeh     : 3.1.1\n",
      "jupyterlab: 3.6.3\n",
      "\n"
     ]
    }
   ],
   "source": [
    "%load_ext watermark\n",
    "%watermark -v -p numpy,bokeh,jupyterlab"
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.11.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}