{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercise 3.1: Exploring and sampling probability distributions\n",
    "\n",
    "<hr />"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
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       "  function now() {\n",
       "    return new Date();\n",
       "  }\n",
       "\n",
       "  const force = true;\n",
       "\n",
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       "    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",
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       "\n",
       "    if (server_id !== undefined) {\n",
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       "      const cmd_clean = \"from bokeh.io.state import curstate; print(curstate().uuid_to_server['\" + server_id + \"'].get_sessions()[0].document.roots[0]._id)\";\n",
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       "\n",
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       "      toinsert[toinsert.length - 1].firstChild.textContent = output.data[JS_MIME_TYPE];\n",
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       "\n",
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       "\n",
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       "        EXEC_MIME_TYPE\n",
       "      );\n",
       "      this.keyboard_manager.register_events(toinsert);\n",
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       "      render(props, toinsert[toinsert.length - 1]);\n",
       "      element.append(toinsert);\n",
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       "    }\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",
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       "     */\n",
       "    OutputArea.prototype.register_mime_type(EXEC_MIME_TYPE, append_mime, {\n",
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       "  }\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",
       "  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(\"fe79d49e-2de8-4438-ab3a-783756574f0b\");\n",
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       "      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",
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       "  }\n",
       "\n",
       "  function run_callbacks() {\n",
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       "      root._bokeh_onload_callbacks.forEach(function(callback) {\n",
       "        if (callback != null)\n",
       "          callback();\n",
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       "    } 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-3.2.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-3.2.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-3.2.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-3.2.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-mathjax-3.2.1.min.js\"];\n",
       "  const css_urls = [];\n",
       "\n",
       "  const inline_js = [    function(Bokeh) {\n",
       "      Bokeh.set_log_level(\"info\");\n",
       "    },\n",
       "function(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",
       "    }\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(\"fe79d49e-2de8-4438-ab3a-783756574f0b\")).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));"
      ],
      "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(\"fe79d49e-2de8-4438-ab3a-783756574f0b\");\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-3.2.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-3.2.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-3.2.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-3.2.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-mathjax-3.2.1.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(\"fe79d49e-2de8-4438-ab3a-783756574f0b\")).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",
    "import scipy.stats as st\n",
    "\n",
    "import iqplot\n",
    "\n",
    "import bokeh.io\n",
    "import bokeh.plotting\n",
    "bokeh.io.output_notebook()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n",
    "\n",
    "NumPy offers tools for drawing random number out of distributions. The `scipy.stats` module allows convenient calculation of probability density functions, cumulative distribution functions, etc., of many named probability distributions. For example, we can draw samples out of a standard Normal distribution using NumPy and can then plot a histogram of the results to get that familiar, beautifully shaped curve."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "  const render_items = [{\"docid\":\"cde6e1f5-07b0-43ed-984a-021da15e47fb\",\"roots\":{\"p1002\":\"ec08a303-f0f4-4e0e-986b-969c2a8c6825\"},\"root_ids\":[\"p1002\"]}];\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",
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   ],
   "source": [
    "# Instantiate random number generator\n",
    "rng = np.random.default_rng()\n",
    "\n",
    "# Draw 100,000 Normal samples\n",
    "samples = rng.normal(0, 1, size=100000)\n",
    "\n",
    "# Plot the histogram\n",
    "p = iqplot.histogram(samples, bins=50, rug=False, style='step', density=True)\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We could use SciPy to evaluate the PDF, which can be a pain to code up. It's formula is\n",
    "\n",
    "\\begin{align}\n",
    "f(x;\\mu, \\sigma) = \\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\,\\mathrm{e}^{-(x-\\mu)^2/2\\sigma^2}.\n",
    "\\end{align}\n",
    "\n",
    "Instead of coding this up, we can use SciPy!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
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   "source": [
    "# x-values for plotting.\n",
    "x = np.linspace(-5, 5, 200)\n",
    "\n",
    "# Compute PDF using scipy.stats\n",
    "pdf = st.norm.pdf(x, loc=0, scale=1)\n",
    "\n",
    "# Add to the plot\n",
    "p.line(x, pdf, line_color='orange', line_width=2)\n",
    "\n",
    "bokeh.io.show(p)"
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    "The usage of NumPy and SciPy for these distributions can get tricky because distributions can be parametrized in different ways. As an example, instead of parametrizing the Normal distribution with $\\sigma$, we could parametrize it with $\\tau \\equiv 1/\\sigma$. In this case, the PDF is\n",
    "\n",
    "\\begin{align}\n",
    "f(x;\\mu, \\tau) = \\frac{\\tau}{\\sqrt{2\\pi}}\\,\\mathrm{e}^{-\\tau^2(x-\\mu)^2/2}.\n",
    "\\end{align}\n",
    "\n",
    "It is important to keep everything straight. To do this, I generally use the parametrizations used by the [Stan software package](https://mc-stan.org/). To help me keep things straight, I built the [Probability Distribution Explorer](https://distribution-explorer.github.io/). I encourage you to spend some time with the Distribution Explorer to familiarize yourself with distributions and how to sample out of them with NumPy and compute PDFs, PMFs, and CDFs with `scipy.stats`.\n",
    "\n",
    "**a)** Draw 100,000 samples out of a Gamma distribution with parameters $\\alpha = 5$ and $\\beta = 2$, and plot a histogram. Then, plot the PDF of the Gamma distribution with these parameters.\n",
    "\n",
    "**b)** Draw 100,000 samples out of a Negative Bionomial distribution with parameters $\\alpha = 20$ and $\\beta = 2$ and plot a spike plot (You should use the `fraction=True` kwarg with `iqplot.spike()`). Then, plot the PMF of the Negative Binomial distribution with these parameters."
   ]
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