{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lesson 23: Random number generation\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=\"1001\">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",
       "  var 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",
       "  var JS_MIME_TYPE = 'application/javascript';\n",
       "  var HTML_MIME_TYPE = 'text/html';\n",
       "  var EXEC_MIME_TYPE = 'application/vnd.bokehjs_exec.v0+json';\n",
       "  var CLASS_NAME = 'output_bokeh rendered_html';\n",
       "\n",
       "  /**\n",
       "   * Render data to the DOM node\n",
       "   */\n",
       "  function render(props, node) {\n",
       "    var 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",
       "    var cell = handle.cell;\n",
       "\n",
       "    var id = cell.output_area._bokeh_element_id;\n",
       "    var 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",
       "      var cmd = \"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, {\n",
       "        iopub: {\n",
       "          output: function(msg) {\n",
       "            var 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",
       "      var cmd = \"import bokeh.io.notebook as ion; ion.destroy_server('\" + server_id + \"')\";\n",
       "      cell.notebook.kernel.execute(cmd);\n",
       "    }\n",
       "  }\n",
       "\n",
       "  /**\n",
       "   * Handle when a new output is added\n",
       "   */\n",
       "  function handleAddOutput(event, handle) {\n",
       "    var output_area = handle.output_area;\n",
       "    var output = handle.output;\n",
       "\n",
       "    // limit handleAddOutput to display_data with EXEC_MIME_TYPE content only\n",
       "    if ((output.output_type != \"display_data\") || (!output.data.hasOwnProperty(EXEC_MIME_TYPE))) {\n",
       "      return\n",
       "    }\n",
       "\n",
       "    var 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",
       "      var bk_div = document.createElement(\"div\");\n",
       "      bk_div.innerHTML = output.data[HTML_MIME_TYPE];\n",
       "      var script_attrs = bk_div.children[0].attributes;\n",
       "      for (var 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",
       "      var 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",
       "      var 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",
       "    var events = require('base/js/events');\n",
       "    var 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",
       "  var 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",
       "    var el = document.getElementById(\"1001\");\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() {\n",
       "      console.error(\"failed to load \" + url);\n",
       "    }\n",
       "\n",
       "    for (var i = 0; i < css_urls.length; i++) {\n",
       "      var url = css_urls[i];\n",
       "      const element = document.createElement(\"link\");\n",
       "      element.onload = on_load;\n",
       "      element.onerror = on_error;\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",
       "    const hashes = {\"https://cdn.bokeh.org/bokeh/release/bokeh-2.1.0.min.js\": \"YobFyzPeVUsFQydHkJGsJL1kyfHnWxOlPc3EwaV22TmBaeGoXHLWx5aRRVPS9xlE\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-2.1.0.min.js\": \"NuAg9+TcTQQqvQCTtkCneRrpkTiMhhfiq0KHiBzx8ECiKiLWXHN6i6ia3q7b3eHu\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-2.1.0.min.js\": \"uMVqQc8JqHitD67bXTn9a06Mrk3EiHRaZ18EJENQenAKJ/KL71SakdXYomZQpGRr\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-2.1.0.min.js\": \"u+eGuEXC8aw0VSCm2mH+b/tQEAitUOYiR1H6SuIVEdUmXsf4vN8m/SmXpmjb7U/X\"};\n",
       "\n",
       "    for (var i = 0; i < js_urls.length; i++) {\n",
       "      var url = js_urls[i];\n",
       "      var element = document.createElement('script');\n",
       "      element.onload = on_load;\n",
       "      element.onerror = on_error;\n",
       "      element.async = false;\n",
       "      element.src = url;\n",
       "      if (url in hashes) {\n",
       "        element.crossOrigin = \"anonymous\";\n",
       "        element.integrity = \"sha384-\" + hashes[url];\n",
       "      }\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",
       "  var js_urls = [\"https://cdn.bokeh.org/bokeh/release/bokeh-2.1.0.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-2.1.0.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-2.1.0.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-2.1.0.min.js\"];\n",
       "  var css_urls = [];\n",
       "  \n",
       "\n",
       "  var 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 (var 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",
       "      var cell = $(document.getElementById(\"1001\")).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  var 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  var 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    var el = document.getElementById(\"1001\");\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() {\n      console.error(\"failed to load \" + url);\n    }\n\n    for (var i = 0; i < css_urls.length; i++) {\n      var url = css_urls[i];\n      const element = document.createElement(\"link\");\n      element.onload = on_load;\n      element.onerror = on_error;\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    const hashes = {\"https://cdn.bokeh.org/bokeh/release/bokeh-2.1.0.min.js\": \"YobFyzPeVUsFQydHkJGsJL1kyfHnWxOlPc3EwaV22TmBaeGoXHLWx5aRRVPS9xlE\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-2.1.0.min.js\": \"NuAg9+TcTQQqvQCTtkCneRrpkTiMhhfiq0KHiBzx8ECiKiLWXHN6i6ia3q7b3eHu\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-2.1.0.min.js\": \"uMVqQc8JqHitD67bXTn9a06Mrk3EiHRaZ18EJENQenAKJ/KL71SakdXYomZQpGRr\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-2.1.0.min.js\": \"u+eGuEXC8aw0VSCm2mH+b/tQEAitUOYiR1H6SuIVEdUmXsf4vN8m/SmXpmjb7U/X\"};\n\n    for (var i = 0; i < js_urls.length; i++) {\n      var url = js_urls[i];\n      var element = document.createElement('script');\n      element.onload = on_load;\n      element.onerror = on_error;\n      element.async = false;\n      element.src = url;\n      if (url in hashes) {\n        element.crossOrigin = \"anonymous\";\n        element.integrity = \"sha384-\" + hashes[url];\n      }\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  var js_urls = [\"https://cdn.bokeh.org/bokeh/release/bokeh-2.1.0.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-2.1.0.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-2.1.0.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-2.1.0.min.js\"];\n  var css_urls = [];\n  \n\n  var 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 (var 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      var cell = $(document.getElementById(\"1001\")).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",
    "import pandas as pd\n",
    "import scipy.stats\n",
    "\n",
    "import bokeh_catplot\n",
    "\n",
    "import bokeh.io\n",
    "import bokeh.plotting\n",
    "\n",
    "bokeh.io.output_notebook()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n",
    "\n",
    "**Random number generation** (RNG), besides being a [song in the original off-Broadway run of *Hedwig and the Angry Inch*](https://www.youtube.com/watch?v=KzGHjsMGdy0), is the process by which a string of random numbers may be drawn. Of course, the numbers are not completely random for several reasons.\n",
    "\n",
    "1. They are drawn from a probability distribution. The most common one is the uniform distribution on the domain $0 \\le x < 1$, i.e., random numbers between zero and one. (\"Completely random\" does not make sense because of the infinite magnitude of numbers.) \n",
    "2. In most computer applications, including the ones we'll use in bootcamp, the random numbers are actually pseudorandom. They depend entirely on an input **seed** and are then generated by a deterministic algorithm from that seed.\n",
    "\n",
    "This is a bit academic. Let's jump right in generating random numbers. Much of the random number generation functionality you will need is in the `np.random` module. Let's start by generating random numbers from a Uniform distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.25372623, 0.34038804, 0.6401285 , 0.7396487 , 0.67860559,\n",
       "       0.41240982, 0.62789879, 0.1591896 , 0.43698308, 0.19694275])"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.uniform(low=0, high=1, size=10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function `uniform()` in the `np.random` module generates random numbers on the interval \\[`low`, `high`) from a Uniform distribution. The `size` kwarg is how many random numbers you wish to generate, and is a kwarg in all of Numpy's random number generators. The random numbers are returned as a NumPy array.\n",
    "\n",
    "We can check to make sure it is appropriately drawing random numbers out of the uniform distribution by plotting the cumulative distribution function, just like we did last time.  We'll generate 1,000 random numbers and plot them along with the CDF of a Uniform distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"f040ab1b-ee62-4767-99ca-736aceef3080\" data-root-id=\"1002\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "    \n",
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\",\"dtype\":\"float64\",\"order\":\"little\",\"shape\":[1000]}},\"selected\":{\"id\":\"1047\"},\"selection_policy\":{\"id\":\"1046\"}},\"id\":\"1033\",\"type\":\"ColumnDataSource\"},{\"attributes\":{\"axis\":{\"id\":\"1011\"},\"ticker\":null},\"id\":\"1014\",\"type\":\"Grid\"},{\"attributes\":{},\"id\":\"1024\",\"type\":\"HelpTool\"},{\"attributes\":{\"fill_alpha\":{\"value\":0.1},\"fill_color\":{\"value\":\"#1f77b3\"},\"line_alpha\":{\"value\":0.1},\"line_color\":{\"value\":\"#1f77b3\"},\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"__ECDF\"}},\"id\":\"1036\",\"type\":\"Circle\"}],\"root_ids\":[\"1002\"]},\"title\":\"Bokeh Application\",\"version\":\"2.1.0\"}};\n",
       "  var render_items = [{\"docid\":\"6671fcef-5f86-4b3d-a05c-5320720f070a\",\"root_ids\":[\"1002\"],\"roots\":{\"1002\":\"f040ab1b-ee62-4767-99ca-736aceef3080\"}}];\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",
       "    var attempts = 0;\n",
       "    var 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": "1002"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Generate random numbers\n",
    "x = np.random.uniform(low=0, high=1, size=1000)\n",
    "\n",
    "# Plot the ECDF of randomly generated numbers\n",
    "p = bokeh_catplot.ecdf(x)\n",
    "\n",
    "p.line(\n",
    "    x=[0, 1],\n",
    "    y=[0, 1],\n",
    "    line_width=2,\n",
    "    line_color='orange',\n",
    ")\n",
    "\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So, it looks like our random number generator is doing a good job.\n",
    "\n",
    "Generating random numbers on the uniform interval is one of the most commonly used RNG techniques. In fact, many of the other contexts of RNG are derived from draws from the uniform distribution. For example, you can simulate flipping a biased (unfair) coin."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ True  True  True  True False  True  True False  True  True  True  True\n",
      " False  True False False  True False False  True]\n",
      "\n",
      "There were 13 heads.\n"
     ]
    }
   ],
   "source": [
    "# Generate 20 random numbers on uniform interval\n",
    "x = np.random.uniform(low=0, high=1, size=20)\n",
    "\n",
    "# Make the coin flips (< 0.7 means we have a 70% chance of heads)\n",
    "heads = x < 0.7\n",
    "\n",
    "# Show which were heads, and count the number of heads\n",
    "print(heads)\n",
    "print('\\nThere were', np.sum(heads), 'heads.')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Choice of generator\n",
    "\n",
    "As of version 1.17.3 of Numpy, the algorithm under the hood of calls to functions like `np.random.uniform()` is the [Mersenne Twister Algorithm](https://en.wikipedia.org/wiki/Mersenne_Twister) for generating random numbers. It is a very widely used and reliable method for generating random numbers. However, starting with version 1.17.4, the `numpy.random` module offers random number generators with better statistical performance, including the [PCG64 generator](http://www.pcg-random.org). Going forward, the preferred approach to doing random number generation is to first instantiate a generator of your choice, and then use its methods to generate numbers out of probability distributions.\n",
    "\n",
    "Let's set up a PCG64 generator, which is Numpy's default."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "rg = np.random.default_rng()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now that we have the generator, we can use it to draw numbers out of distributions. The syntax is the same as before, except `rg` replaces `np.random`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.63700832, 0.29237701, 0.75159707, 0.86642735, 0.61456861,\n",
       "       0.0661897 , 0.70930897, 0.84815804, 0.89896461, 0.2549689 ,\n",
       "       0.86330332, 0.91823482, 0.35862654, 0.69643518, 0.09490646,\n",
       "       0.2043692 , 0.3039475 , 0.22940164, 0.39408351, 0.38121526])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rg.uniform(low=0, high=1, size=20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Seeding random number generators\n",
    "\n",
    "Now, just to demonstrate that random number generation is deterministic, we will explicitly **seed** the random number generator (which is usually seeded with a number representing the date/time to avoid repeats) to show that we get the same random numbers."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.18866535, 0.04418857, 0.02961285, 0.22083971, 0.43341773,\n",
       "       0.13166813, 0.42112164, 0.43507845, 0.61380912, 0.30627603])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Instantiate generator with a seed\n",
    "rg = np.random.default_rng(seed=3252)\n",
    "\n",
    "# Generate random numbers\n",
    "rg.uniform(size=10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.18866535, 0.04418857, 0.02961285, 0.22083971, 0.43341773,\n",
       "       0.13166813, 0.42112164, 0.43507845, 0.61380912, 0.30627603])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Re-seed the RNG\n",
    "rg = np.random.default_rng(seed=3252)\n",
    "\n",
    "# Generate random numbers\n",
    "rg.uniform(size=10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The random number sequence is exactly the same. If we choose a different seed, we get totally different random numbers."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.31390226, 0.73012457, 0.05800998, 0.01557021, 0.29825701,\n",
       "       0.10106784, 0.06329107, 0.58614237, 0.52023168, 0.52779988])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Fit one more fan in the 3252\n",
    "rg = np.random.default_rng(seed=3253)\n",
    "rg.uniform(size=10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If you are writing tests, it is often useful to seed the random number generator to get reproducible results."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Drawing random numbers out of other distributions\n",
    "\n",
    "We can also draw random numbers from other probability distributions. For example, say we wanted to draw random samples from a Normal distribution with mean μ and standard deviation σ. (We already saw this example when we were looking at histograms, but we repeat it here.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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PDF\",\"formatter\":{\"id\":\"1170\"},\"ticker\":{\"id\":\"1146\"}},\"id\":\"1145\",\"type\":\"LinearAxis\"},{\"attributes\":{\"items\":[{\"id\":\"1178\"}],\"visible\":false},\"id\":\"1177\",\"type\":\"Legend\"},{\"attributes\":{},\"id\":\"1193\",\"type\":\"UnionRenderers\"},{\"attributes\":{},\"id\":\"1170\",\"type\":\"BasicTickFormatter\"},{\"attributes\":{\"fill_alpha\":0.3,\"fill_color\":\"#1f77b3\",\"line_alpha\":0,\"line_color\":\"#1f77b3\",\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"y\"}},\"id\":\"1180\",\"type\":\"Patch\"}],\"root_ids\":[\"1133\"]},\"title\":\"Bokeh Application\",\"version\":\"2.1.0\"}};\n",
       "  var render_items = [{\"docid\":\"81068329-588e-4327-ab2a-870f36d46c17\",\"root_ids\":[\"1133\"],\"roots\":{\"1133\":\"3bda5700-4975-4ba5-9217-3ad3c3a54404\"}}];\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",
       "    var attempts = 0;\n",
       "    var 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": "1133"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Set parameters\n",
    "mu = 10\n",
    "sigma = 1\n",
    "\n",
    "# Draw 100000 random samples\n",
    "x = rg.normal(mu, sigma, size=100000)\n",
    "\n",
    "# Plot the histogram\n",
    "p = bokeh_catplot.histogram(\n",
    "    x,\n",
    "    density=True,\n",
    "    y_axis_label='approximate PDF',\n",
    ")\n",
    "\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It looks Normal, but, again, comparing the resulting ECDF is a better way to look at this. We'll check out the ECDF with 1000 samples so as not to choke the browser. I will also make use of the theoretical CDF for the Normal distribution available from the `scipy.stats` module."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"5149c9f4-63f3-462e-ac4f-d677f3abb475\" data-root-id=\"1283\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "    \n",
       "  var docs_json = 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\",\"dtype\":\"float64\",\"order\":\"little\",\"shape\":[1000]}},\"selected\":{\"id\":\"1328\"},\"selection_policy\":{\"id\":\"1327\"}},\"id\":\"1314\",\"type\":\"ColumnDataSource\"},{\"attributes\":{},\"id\":\"1368\",\"type\":\"Selection\"},{\"attributes\":{},\"id\":\"1288\",\"type\":\"LinearScale\"},{\"attributes\":{},\"id\":\"1297\",\"type\":\"BasicTicker\"},{\"attributes\":{\"items\":[{\"id\":\"1330\"}],\"visible\":false},\"id\":\"1329\",\"type\":\"Legend\"},{\"attributes\":{\"data_source\":{\"id\":\"1331\"},\"glyph\":{\"id\":\"1332\"},\"hover_glyph\":null,\"muted_glyph\":null,\"nonselection_glyph\":{\"id\":\"1333\"},\"selection_glyph\":null,\"view\":{\"id\":\"1335\"}},\"id\":\"1334\",\"type\":\"GlyphRenderer\"}],\"root_ids\":[\"1283\"]},\"title\":\"Bokeh Application\",\"version\":\"2.1.0\"}};\n",
       "  var render_items = [{\"docid\":\"8b9337de-0f45-4df7-9d4d-e22e9116a21a\",\"root_ids\":[\"1283\"],\"roots\":{\"1283\":\"5149c9f4-63f3-462e-ac4f-d677f3abb475\"}}];\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",
       "    var attempts = 0;\n",
       "    var 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": "1283"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Compute theoretical CDF\n",
    "x_theor = np.linspace(6, 14, 400)\n",
    "y_theor = scipy.stats.norm.cdf(x_theor, mu, sigma)\n",
    "\n",
    "# Plot the ECDF of randomly generated numbers\n",
    "p = bokeh_catplot.ecdf(x[:1000])\n",
    "\n",
    "p.line(\n",
    "    x=x_theor,\n",
    "    y=y_theor,\n",
    "    line_width=2,\n",
    "    line_color='orange',\n",
    ")\n",
    "\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Yup, right on!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Selections from discrete distributions\n",
    "\n",
    "The random numbers we have generated so far from from continuous probability distributions. We can also draw random numbers from discrete distributions. We already showed how we can do this for \"coin flips,\" but we can do it for other distributions as well. Say we wanted to draw from a Binomial distribution. We can use `rg.binomial()`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Draw how many coin flips land heads in 10 files\n",
    "rg.binomial(10, 0.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are other discrete distributions we can draw from, such as Binomial, Geometric, Poisson, etc., and [the documentation](http://docs.scipy.org/doc/numpy/reference/routines.random.html) describes how to use them."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Choosing elements from an array\n",
    "\n",
    "It is often useful to randomly choose elements from an existing array. The `rg.choice()` function does this. You equivalently could do this using `rg.integers()`, where the integers represent indices in the array, *except* `rg.choice()` has a great keyword argument, `replace`, which allows random draws with or without replacement. For example, say you had 100 samples that you wanted to send to a facility for analysis, but you can only afford to send 20. If we used `rg.integers()`, we might have a problem."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1,  2,  2,  6,  7,  9, 11, 11, 11, 13, 15, 18, 21, 22, 22, 24, 31,\n",
       "       34, 37, 39])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rg = np.random.default_rng(seed=3252)\n",
    "np.sort(rg.integers(0, 52, size=20))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Samples 2, 11, and 22 we each selected twice!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 5, 38, 44, 12, 49, 10, 47, 46, 40, 23, 18,  1, 13, 21, 42, 28, 29,\n",
       "       37, 26, 32])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rg.choice(np.arange(52), size=20, replace=False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, because we chose `replace=False`, we do not get any repeats."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Generating random sequences\n",
    "\n",
    "Because it works with selecting characters as well as numbers, we can use the `rg.choice()` function to generate random DNA sequences."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'CAGGAGTCGCTGGGATAACAATTGTGACCTATGTAACTCAGCGAAGAGACTCGGGCCCGACCCACTAAAG'"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "''.join(rg.choice(list('ATGC'), replace=True, size=70))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Shuffling an array\n",
    "\n",
    "Similarly, the `np.random.permutation()` function is useful. It takes the entries in an array and shuffles them! Let's shuffle a deck of cards."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([51, 36, 30, 35,  2,  9, 19, 40, 42, 32, 21, 45, 43, 26, 16, 27,  0,\n",
       "       25, 41, 33, 24, 47, 10, 11,  5, 14, 23, 17, 29, 12, 13, 28, 48, 22,\n",
       "        8, 46,  7, 49, 44, 31,  6, 34, 50,  1, 18, 20, 15, 39,  4, 38,  3,\n",
       "       37])"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rg.permutation(np.arange(52))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## When do we need RNG?\n",
    "\n",
    "Answer: VERY OFTEN!  We will see many examples in the next lessons and in the exercises.  \n",
    "\n",
    "In many ways, probability is the language of biology. Molecular processes have energetics that are comparable to the thermal energy, which means they are always influenced by random thermal forces. The processes of the central dogma, including DNA replication, are no exceptions. This gives rise to random mutations, which are central to understanding how evolution works. If we want to understand them, it is often useful to use random number generators to model the processes.\n",
    "\n",
    "RNG also comes up A LOT in data analysis, which we will see in the lessons on hacker stats."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Computing environment"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPython 3.7.7\n",
      "IPython 7.15.0\n",
      "\n",
      "numpy 1.18.1\n",
      "scipy 1.4.1\n",
      "pandas 0.24.2\n",
      "bokeh 2.1.0\n",
      "bokeh_catplot 0.1.8\n",
      "jupyterlab 2.1.4\n"
     ]
    }
   ],
   "source": [
    "%load_ext watermark\n",
    "%watermark -v -p numpy,scipy,pandas,bokeh,bokeh_catplot,jupyterlab"
   ]
  }
 ],
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