{
 "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": [
    {
     "data": {
      "text/html": [
       "\n",
       "    <div class=\"bk-root\">\n",
       "        <a href=\"https://bokeh.org\" target=\"_blank\" class=\"bk-logo bk-logo-small bk-logo-notebook\"></a>\n",
       "        <span id=\"1002\">Loading BokehJS ...</span>\n",
       "    </div>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "\n",
       "(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",
       "  const JS_MIME_TYPE = 'application/javascript';\n",
       "  const HTML_MIME_TYPE = 'text/html';\n",
       "  const EXEC_MIME_TYPE = 'application/vnd.bokehjs_exec.v0+json';\n",
       "  const CLASS_NAME = 'output_bokeh rendered_html';\n",
       "\n",
       "  /**\n",
       "   * Render data to the DOM node\n",
       "   */\n",
       "  function render(props, node) {\n",
       "    const script = document.createElement(\"script\");\n",
       "    node.appendChild(script);\n",
       "  }\n",
       "\n",
       "  /**\n",
       "   * Handle when an output is cleared or removed\n",
       "   */\n",
       "  function handleClearOutput(event, handle) {\n",
       "    const cell = handle.cell;\n",
       "\n",
       "    const id = cell.output_area._bokeh_element_id;\n",
       "    const server_id = cell.output_area._bokeh_server_id;\n",
       "    // Clean up Bokeh references\n",
       "    if (id != null && id in Bokeh.index) {\n",
       "      Bokeh.index[id].model.document.clear();\n",
       "      delete Bokeh.index[id];\n",
       "    }\n",
       "\n",
       "    if (server_id !== undefined) {\n",
       "      // Clean up Bokeh references\n",
       "      const cmd_clean = \"from bokeh.io.state import curstate; print(curstate().uuid_to_server['\" + server_id + \"'].get_sessions()[0].document.roots[0]._id)\";\n",
       "      cell.notebook.kernel.execute(cmd_clean, {\n",
       "        iopub: {\n",
       "          output: function(msg) {\n",
       "            const id = msg.content.text.trim();\n",
       "            if (id in Bokeh.index) {\n",
       "              Bokeh.index[id].model.document.clear();\n",
       "              delete Bokeh.index[id];\n",
       "            }\n",
       "          }\n",
       "        }\n",
       "      });\n",
       "      // Destroy server and session\n",
       "      const cmd_destroy = \"import bokeh.io.notebook as ion; ion.destroy_server('\" + server_id + \"')\";\n",
       "      cell.notebook.kernel.execute(cmd_destroy);\n",
       "    }\n",
       "  }\n",
       "\n",
       "  /**\n",
       "   * Handle when a new output is added\n",
       "   */\n",
       "  function handleAddOutput(event, handle) {\n",
       "    const output_area = handle.output_area;\n",
       "    const output = handle.output;\n",
       "\n",
       "    // limit handleAddOutput to display_data with EXEC_MIME_TYPE content only\n",
       "    if ((output.output_type != \"display_data\") || (!Object.prototype.hasOwnProperty.call(output.data, EXEC_MIME_TYPE))) {\n",
       "      return\n",
       "    }\n",
       "\n",
       "    const toinsert = output_area.element.find(\".\" + CLASS_NAME.split(' ')[0]);\n",
       "\n",
       "    if (output.metadata[EXEC_MIME_TYPE][\"id\"] !== undefined) {\n",
       "      toinsert[toinsert.length - 1].firstChild.textContent = output.data[JS_MIME_TYPE];\n",
       "      // store reference to embed id on output_area\n",
       "      output_area._bokeh_element_id = output.metadata[EXEC_MIME_TYPE][\"id\"];\n",
       "    }\n",
       "    if (output.metadata[EXEC_MIME_TYPE][\"server_id\"] !== undefined) {\n",
       "      const bk_div = document.createElement(\"div\");\n",
       "      bk_div.innerHTML = output.data[HTML_MIME_TYPE];\n",
       "      const script_attrs = bk_div.children[0].attributes;\n",
       "      for (let i = 0; i < script_attrs.length; i++) {\n",
       "        toinsert[toinsert.length - 1].firstChild.setAttribute(script_attrs[i].name, script_attrs[i].value);\n",
       "        toinsert[toinsert.length - 1].firstChild.textContent = bk_div.children[0].textContent\n",
       "      }\n",
       "      // store reference to server id on output_area\n",
       "      output_area._bokeh_server_id = output.metadata[EXEC_MIME_TYPE][\"server_id\"];\n",
       "    }\n",
       "  }\n",
       "\n",
       "  function register_renderer(events, OutputArea) {\n",
       "\n",
       "    function append_mime(data, metadata, element) {\n",
       "      // create a DOM node to render to\n",
       "      const toinsert = this.create_output_subarea(\n",
       "        metadata,\n",
       "        CLASS_NAME,\n",
       "        EXEC_MIME_TYPE\n",
       "      );\n",
       "      this.keyboard_manager.register_events(toinsert);\n",
       "      // Render to node\n",
       "      const props = {data: data, metadata: metadata[EXEC_MIME_TYPE]};\n",
       "      render(props, toinsert[toinsert.length - 1]);\n",
       "      element.append(toinsert);\n",
       "      return toinsert\n",
       "    }\n",
       "\n",
       "    /* Handle when an output is cleared or removed */\n",
       "    events.on('clear_output.CodeCell', handleClearOutput);\n",
       "    events.on('delete.Cell', handleClearOutput);\n",
       "\n",
       "    /* Handle when a new output is added */\n",
       "    events.on('output_added.OutputArea', handleAddOutput);\n",
       "\n",
       "    /**\n",
       "     * Register the mime type and append_mime function with output_area\n",
       "     */\n",
       "    OutputArea.prototype.register_mime_type(EXEC_MIME_TYPE, append_mime, {\n",
       "      /* Is output safe? */\n",
       "      safe: true,\n",
       "      /* Index of renderer in `output_area.display_order` */\n",
       "      index: 0\n",
       "    });\n",
       "  }\n",
       "\n",
       "  // register the mime type if in Jupyter Notebook environment and previously unregistered\n",
       "  if (root.Jupyter !== undefined) {\n",
       "    const events = require('base/js/events');\n",
       "    const OutputArea = require('notebook/js/outputarea').OutputArea;\n",
       "\n",
       "    if (OutputArea.prototype.mime_types().indexOf(EXEC_MIME_TYPE) == -1) {\n",
       "      register_renderer(events, OutputArea);\n",
       "    }\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(\"1002\");\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",
       "\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",
       "  \n",
       "  const js_urls = [\"https://cdn.bokeh.org/bokeh/release/bokeh-2.4.2.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-2.4.2.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-2.4.2.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-2.4.2.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-mathjax-2.4.2.min.js\"];\n",
       "  const css_urls = [];\n",
       "  \n",
       "\n",
       "  const inline_js = [\n",
       "    function(Bokeh) {\n",
       "      Bokeh.set_log_level(\"info\");\n",
       "    },\n",
       "    function(Bokeh) {\n",
       "    \n",
       "    \n",
       "    }\n",
       "  ];\n",
       "\n",
       "  function run_inline_js() {\n",
       "    \n",
       "    if (root.Bokeh !== undefined || force === true) {\n",
       "      \n",
       "    for (let i = 0; i < inline_js.length; i++) {\n",
       "      inline_js[i].call(root, root.Bokeh);\n",
       "    }\n",
       "    if (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(\"1002\")).parents('.cell').data().cell;\n",
       "      cell.output_area.append_execute_result(NB_LOAD_WARNING)\n",
       "    }\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": "\n(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\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(\"1002\");\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\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  \n  const js_urls = [\"https://cdn.bokeh.org/bokeh/release/bokeh-2.4.2.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-2.4.2.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-2.4.2.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-2.4.2.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-mathjax-2.4.2.min.js\"];\n  const css_urls = [];\n  \n\n  const inline_js = [\n    function(Bokeh) {\n      Bokeh.set_log_level(\"info\");\n    },\n    function(Bokeh) {\n    \n    \n    }\n  ];\n\n  function run_inline_js() {\n    \n    if (root.Bokeh !== undefined || force === true) {\n      \n    for (let i = 0; i < inline_js.length; i++) {\n      inline_js[i].call(root, root.Bokeh);\n    }\n    if (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(\"1002\")).parents('.cell').data().cell;\n      cell.output_area.append_execute_result(NB_LOAD_WARNING)\n    }\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": [
    "rg = np.random.default_rng()\n",
    "\n",
    "def draw_bs_reps(data, func, rg, size=1, args=()):\n",
    "    return np.array(\n",
    "        [\n",
    "            func(rg.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, rg, 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": [
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"3ae7d31d-1b9f-4bc6-979b-612ba4a626a8\" data-root-id=\"1003\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "    \n",
       "  const docs_json = {\"d154ec69-61b2-4bb5-9b6c-628898c3ef65\":{\"defs\":[],\"roots\":{\"references\":[{\"attributes\":{\"below\":[{\"id\":\"1012\"}],\"center\":[{\"id\":\"1015\"},{\"id\":\"1019\"}],\"frame_height\":250,\"frame_width\":350,\"left\":[{\"id\":\"1016\"}],\"renderers\":[{\"id\":\"1038\"},{\"id\":\"1044\"},{\"id\":\"1050\"}],\"title\":{\"id\":\"1052\"},\"toolbar\":{\"id\":\"1027\"},\"x_range\":{\"id\":\"1004\"},\"x_scale\":{\"id\":\"1008\"},\"y_range\":{\"id\":\"1006\"},\"y_scale\":{\"id\":\"1010\"}},\"id\":\"1003\",\"subtype\":\"Figure\",\"type\":\"Plot\"},{\"attributes\":{},\"id\":\"1056\",\"type\":\"AllLabels\"},{\"attributes\":{},\"id\":\"1025\",\"type\":\"HelpTool\"},{\"attributes\":{},\"id\":\"1060\",\"type\":\"UnionRenderers\"},{\"attributes\":{},\"id\":\"1064\",\"type\":\"UnionRenderers\"},{\"attributes\":{\"end\":1100,\"start\":0.9},\"id\":\"1004\",\"type\":\"Range1d\"},{\"attributes\":{},\"id\":\"1024\",\"type\":\"ResetTool\"},{\"attributes\":{},\"id\":\"1065\",\"type\":\"Selection\"},{\"attributes\":{},\"id\":\"1021\",\"type\":\"WheelZoomTool\"},{\"attributes\":{\"line_color\":\"#1f77b4\",\"line_width\":2,\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"y\"}},\"id\":\"1035\",\"type\":\"Line\"},{\"attributes\":{\"source\":{\"id\":\"1046\"}},\"id\":\"1051\",\"type\":\"CDSView\"},{\"attributes\":{},\"id\":\"1061\",\"type\":\"Selection\"},{\"attributes\":{},\"id\":\"1055\",\"type\":\"BasicTickFormatter\"},{\"attributes\":{},\"id\":\"1008\",\"type\":\"LogScale\"},{\"attributes\":{\"tools\":[{\"id\":\"1020\"},{\"id\":\"1021\"},{\"id\":\"1022\"},{\"id\":\"1023\"},{\"id\":\"1024\"},{\"id\":\"1025\"}]},\"id\":\"1027\",\"type\":\"Toolbar\"},{\"attributes\":{\"fill_color\":{\"value\":\"orange\"},\"hatch_color\":{\"value\":\"orange\"},\"line_color\":{\"value\":\"orange\"},\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"y\"}},\"id\":\"1047\",\"type\":\"Circle\"},{\"attributes\":{\"axis_label\":\"mean fraction omitted\",\"coordinates\":null,\"formatter\":{\"id\":\"1055\"},\"group\":null,\"major_label_policy\":{\"id\":\"1056\"},\"ticker\":{\"id\":\"1017\"}},\"id\":\"1016\",\"type\":\"LinearAxis\"},{\"attributes\":{\"line_alpha\":0.2,\"line_color\":\"gray\",\"line_dash\":[6],\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"y\"}},\"id\":\"1043\",\"type\":\"Line\"},{\"attributes\":{},\"id\":\"1020\",\"type\":\"PanTool\"},{\"attributes\":{\"line_alpha\":0.1,\"line_color\":\"gray\",\"line_dash\":[6],\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"y\"}},\"id\":\"1042\",\"type\":\"Line\"},{\"attributes\":{},\"id\":\"1063\",\"type\":\"Selection\"},{\"attributes\":{\"overlay\":{\"id\":\"1026\"}},\"id\":\"1022\",\"type\":\"BoxZoomTool\"},{\"attributes\":{},\"id\":\"1023\",\"type\":\"SaveTool\"},{\"attributes\":{\"coordinates\":null,\"data_source\":{\"id\":\"1040\"},\"glyph\":{\"id\":\"1041\"},\"group\":null,\"hover_glyph\":null,\"muted_glyph\":{\"id\":\"1043\"},\"nonselection_glyph\":{\"id\":\"1042\"},\"view\":{\"id\":\"1045\"}},\"id\":\"1044\",\"type\":\"GlyphRenderer\"},{\"attributes\":{\"num_minor_ticks\":10},\"id\":\"1013\",\"type\":\"LogTicker\"},{\"attributes\":{\"data\":{\"x\":[0.9,1100],\"y\":[0.36787944117144233,0.36787944117144233]},\"selected\":{\"id\":\"1063\"},\"selection_policy\":{\"id\":\"1062\"}},\"id\":\"1040\",\"type\":\"ColumnDataSource\"},{\"attributes\":{\"data\":{\"x\":{\"__ndarray__\":\"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\",\"dtype\":\"float64\",\"order\":\"little\",\"shape\":[400]},\"y\":{\"__ndarray__\":\"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\",\"dtype\":\"float64\",\"order\":\"little\",\"shape\":[400]}},\"selected\":{\"id\":\"1061\"},\"selection_policy\":{\"id\":\"1060\"}},\"id\":\"1034\",\"type\":\"ColumnDataSource\"},{\"attributes\":{\"bottom_units\":\"screen\",\"coordinates\":null,\"fill_alpha\":0.5,\"fill_color\":\"lightgrey\",\"group\":null,\"left_units\":\"screen\",\"level\":\"overlay\",\"line_alpha\":1.0,\"line_color\":\"black\",\"line_dash\":[4,4],\"line_width\":2,\"right_units\":\"screen\",\"syncable\":false,\"top_units\":\"screen\"},\"id\":\"1026\",\"type\":\"BoxAnnotation\"},{\"attributes\":{\"coordinates\":null,\"data_source\":{\"id\":\"1034\"},\"glyph\":{\"id\":\"1035\"},\"group\":null,\"hover_glyph\":null,\"muted_glyph\":{\"id\":\"1037\"},\"nonselection_glyph\":{\"id\":\"1036\"},\"view\":{\"id\":\"1039\"}},\"id\":\"1038\",\"type\":\"GlyphRenderer\"},{\"attributes\":{\"line_alpha\":0.2,\"line_color\":\"#1f77b4\",\"line_width\":2,\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"y\"}},\"id\":\"1037\",\"type\":\"Line\"},{\"attributes\":{\"coordinates\":null,\"data_source\":{\"id\":\"1046\"},\"glyph\":{\"id\":\"1047\"},\"group\":null,\"hover_glyph\":null,\"muted_glyph\":{\"id\":\"1049\"},\"nonselection_glyph\":{\"id\":\"1048\"},\"view\":{\"id\":\"1051\"}},\"id\":\"1050\",\"type\":\"GlyphRenderer\"},{\"attributes\":{\"fill_alpha\":{\"value\":0.1},\"fill_color\":{\"value\":\"orange\"},\"hatch_alpha\":{\"value\":0.1},\"hatch_color\":{\"value\":\"orange\"},\"line_alpha\":{\"value\":0.1},\"line_color\":{\"value\":\"orange\"},\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"y\"}},\"id\":\"1048\",\"type\":\"Circle\"},{\"attributes\":{},\"id\":\"1059\",\"type\":\"AllLabels\"},{\"attributes\":{},\"id\":\"1062\",\"type\":\"UnionRenderers\"},{\"attributes\":{\"fill_alpha\":{\"value\":0.2},\"fill_color\":{\"value\":\"orange\"},\"hatch_alpha\":{\"value\":0.2},\"hatch_color\":{\"value\":\"orange\"},\"line_alpha\":{\"value\":0.2},\"line_color\":{\"value\":\"orange\"},\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"y\"}},\"id\":\"1049\",\"type\":\"Circle\"},{\"attributes\":{\"axis_label\":\"n\",\"coordinates\":null,\"formatter\":{\"id\":\"1058\"},\"group\":null,\"major_label_policy\":{\"id\":\"1059\"},\"ticker\":{\"id\":\"1013\"}},\"id\":\"1012\",\"type\":\"LogAxis\"},{\"attributes\":{\"line_alpha\":0.1,\"line_color\":\"#1f77b4\",\"line_width\":2,\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"y\"}},\"id\":\"1036\",\"type\":\"Line\"},{\"attributes\":{\"data\":{\"x\":[1,2,3,4,5,6,7,8,9,10,12,14,16,19,22,25,29,33,39,44,51,59,68,79,91,104,120,138,159,184,212,244,281,323,372,429,494,568,655,754,868,1000],\"y\":{\"__ndarray__\":\"AAAAAAAAAACGONbFbTTQP54zorQ3+NI/l/+Qfvs61D8vxVVl3xXVPw5Pr5RliNU/ujNUR8vC1T9+jLlrCfnVP0tWWIMKIdY/Iuo+AKlN1j8YJlMFo5LWP00f/8gzrdY/ArwFEhS/1j9qIeaPb+PWP2aWe+ukCdc/Ft7lIr4T1z8gGKQWsR7XPzuRG6N7J9c/B8AMzgtE1z9AuQOUO0DXP+OQ2Lk0Sdc/ncR9NMVe1z8g2iMwMl3XP8WqqLP7Ytc/L6vT6h9n1z+si9toAG/XP84kxG8ddtc/Qc1TDwJx1z8t0yVrKW7XP/D7e4I3ddc//fOppfR41z9dBI6q6H7XP/tql+omhtc/l9hcANaE1z9tu14nKYDXP9EN2EVAh9c/VTubMSqE1z8phRpt4IbXP4wzGeyOhtc/59Qh1R+F1z/5cRI01YjXP0BbGDPTh9c/\",\"dtype\":\"float64\",\"order\":\"little\",\"shape\":[42]}},\"selected\":{\"id\":\"1065\"},\"selection_policy\":{\"id\":\"1064\"}},\"id\":\"1046\",\"type\":\"ColumnDataSource\"},{\"attributes\":{\"coordinates\":null,\"group\":null},\"id\":\"1052\",\"type\":\"Title\"},{\"attributes\":{},\"id\":\"1010\",\"type\":\"LinearScale\"},{\"attributes\":{\"source\":{\"id\":\"1040\"}},\"id\":\"1045\",\"type\":\"CDSView\"},{\"attributes\":{\"axis\":{\"id\":\"1012\"},\"coordinates\":null,\"group\":null,\"ticker\":null},\"id\":\"1015\",\"type\":\"Grid\"},{\"attributes\":{\"line_color\":\"gray\",\"line_dash\":[6],\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"y\"}},\"id\":\"1041\",\"type\":\"Line\"},{\"attributes\":{\"source\":{\"id\":\"1034\"}},\"id\":\"1039\",\"type\":\"CDSView\"},{\"attributes\":{\"ticker\":null},\"id\":\"1058\",\"type\":\"LogTickFormatter\"},{\"attributes\":{},\"id\":\"1006\",\"type\":\"DataRange1d\"},{\"attributes\":{},\"id\":\"1017\",\"type\":\"BasicTicker\"},{\"attributes\":{\"axis\":{\"id\":\"1016\"},\"coordinates\":null,\"dimension\":1,\"group\":null,\"ticker\":null},\"id\":\"1019\",\"type\":\"Grid\"}],\"root_ids\":[\"1003\"]},\"title\":\"Bokeh Application\",\"version\":\"2.4.2\"}};\n",
       "  const render_items = [{\"docid\":\"d154ec69-61b2-4bb5-9b6c-628898c3ef65\",\"root_ids\":[\"1003\"],\"roots\":{\"1003\":\"3ae7d31d-1b9f-4bc6-979b-612ba4a626a8\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\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": "1003"
      }
     },
     "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       : 3.9.12\n",
      "IPython version      : 8.3.0\n",
      "\n",
      "numpy     : 1.21.5\n",
      "bokeh     : 2.4.2\n",
      "jupyterlab: 3.3.2\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": "3.9.12"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}