06a Statistical Thinking in Python (Part 1).ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Graphical exploratory data analysis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Datasets\n",
    "\n",
    "    2008 election results (all states)\n",
    "    2008 election results (swing states)\n",
    "    Belmont Stakes\n",
    "    Speed of light"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plotting a histogram of iris data\n",
    "\n",
    "For the exercises in this section, you will use a classic data set collected by botanist Edward Anderson and made famous by Ronald Fisher, one of the most prolific statisticians in history. Anderson carefully measured the anatomical properties of samples of three different species of iris, Iris setosa, Iris versicolor, and Iris virginica. The full data set is available as part of scikit-learn. Here, you will work with his measurements of petal length.\n",
    "\n",
    "Plot a histogram of the petal lengths of his 50 samples of Iris versicolor using matplotlib/seaborn's default settings. Recall that to specify the default seaborn style, you can use sns.set(), where sns is the alias that seaborn is imported as.\n",
    "\n",
    "The subset of the data set containing the Iris versicolor petal lengths in units of centimeters (cm) is stored in the NumPy array versicolor_petal_length.\n",
    "\n",
    "In the video, Justin plotted the histograms by using the pandas library and indexing the DataFrame to extract the desired column. Here, however, you only need to use the provided NumPy array. Also, Justin assigned his plotting statements (except for plt.show()) to the dummy variable _. This is to prevent unnecessary output from being displayed. It is not required for your solutions to these exercises, however it is good practice to use it. Alternatively, if you are working in an interactive environment such as a Jupyter notebook, you could use a ; after your plotting statements to achieve the same effect. Justin prefers using _. Therefore, you will see it used in the solution code."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:06:08.936150Z",
     "start_time": "2019-01-22T20:06:04.051193Z"
    }
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'versicolor_petal_length' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-1-49c511140afb>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[0;32m      7\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      8\u001b[0m \u001b[1;31m# Plot histogram of versicolor petal lengths\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 9\u001b[1;33m \u001b[0m_\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mplt\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mhist\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mversicolor_petal_length\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m     10\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     11\u001b[0m \u001b[1;31m# Show histogram\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mNameError\u001b[0m: name 'versicolor_petal_length' is not defined"
     ]
    }
   ],
   "source": [
    "# Import plotting modules\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "\n",
    "# Set default Seaborn style\n",
    "sns.set()\n",
    "\n",
    "# Plot histogram of versicolor petal lengths\n",
    "_ = plt.hist(versicolor_petal_length)\n",
    "\n",
    "# Show histogram\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Axis labels!\n",
    "\n",
    "In the last exercise, you made a nice histogram of petal lengths of Iris versicolor, but you didn't label the axes! That's ok; it's not your fault since we didn't ask you to. Now, add axis labels to the plot using plt.xlabel() and plt.ylabel(). Don't forget to add units and assign both statements to _. The packages matplotlib.pyplot and seaborn are already imported with their standard aliases. This will be the case in what follows, unless specified otherwise."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:06:57.871753Z",
     "start_time": "2019-01-22T20:06:57.858763Z"
    }
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'versicolor_petal_length' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-2-55f0d88a610f>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[1;31m# Plot histogram of versicolor petal lengths\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 2\u001b[1;33m \u001b[0m_\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mplt\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mhist\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mversicolor_petal_length\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      3\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      4\u001b[0m \u001b[1;31m# Label axes\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      5\u001b[0m \u001b[0m_\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mplt\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mxlabel\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m'petal length (cm)'\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mNameError\u001b[0m: name 'versicolor_petal_length' is not defined"
     ]
    }
   ],
   "source": [
    "# Plot histogram of versicolor petal lengths\n",
    "_ = plt.hist(versicolor_petal_length)\n",
    "\n",
    "# Label axes\n",
    "_ = plt.xlabel('petal length (cm)')\n",
    "_ = plt.ylabel('count')\n",
    "\n",
    "# Show histogram\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Adjusting the number of bins in a histogram\n",
    "\n",
    "The histogram you just made had ten bins. This is the default of matplotlib. The \"square root rule\" is a commonly-used rule of thumb for choosing number of bins: choose the number of bins to be the square root of the number of samples. Plot the histogram of Iris versicolor petal lengths again, this time using the square root rule for the number of bins. You specify the number of bins using the bins keyword argument of plt.hist().\n",
    "\n",
    "The plotting utilities are already imported and the seaborn defaults already set. The variable you defined in the last exercise, versicolor_petal_length, is already in your namespace."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:07:09.700577Z",
     "start_time": "2019-01-22T20:07:09.685592Z"
    }
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'versicolor_petal_length' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-3-b4a139b95416>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[0;32m      3\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      4\u001b[0m \u001b[1;31m# Compute number of data points: n_data\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 5\u001b[1;33m \u001b[0mn_data\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mlen\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mversicolor_petal_length\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      6\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      7\u001b[0m \u001b[1;31m# Number of bins is the square root of number of data points: n_bins\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mNameError\u001b[0m: name 'versicolor_petal_length' is not defined"
     ]
    }
   ],
   "source": [
    "# Import numpy\n",
    "import numpy as np\n",
    "\n",
    "# Compute number of data points: n_data\n",
    "n_data = len(versicolor_petal_length)\n",
    "\n",
    "# Number of bins is the square root of number of data points: n_bins\n",
    "n_bins = np.sqrt(n_data)\n",
    "\n",
    "# Convert number of bins to integer: n_bins\n",
    "n_bins = int(n_bins)\n",
    "\n",
    "# Plot the histogram\n",
    "_ = plt.hist(versicolor_petal_length, bins=n_bins)\n",
    "\n",
    "# Label axes\n",
    "_ = plt.xlabel('petal length (cm)')\n",
    "_ = plt.ylabel('count')\n",
    "\n",
    "# Show histogram\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Bee swarm plot\n",
    "\n",
    "Make a bee swarm plot of the iris petal lengths. Your x-axis should contain each of the three species, and the y-axis the petal lengths. A data frame containing the data is in your namespace as df.\n",
    "\n",
    "For your reference, the code Justin used to create the bee swarm plot in the video is provided below:\n",
    "\n",
    "_ = sns.swarmplot(x='state', y='dem_share', data=df_swing)\n",
    "_ = plt.xlabel('state')\n",
    "_ = plt.ylabel('percent of vote for Obama')\n",
    "plt.show()\n",
    "\n",
    "In the IPython Shell, you can use sns.swarmplot? or help(sns.swarmplot) for more details on how to make bee swarm plots using seaborn."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:07:23.940107Z",
     "start_time": "2019-01-22T20:07:23.927141Z"
    }
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'df' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-4-a1367280264b>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[1;31m# Create bee swarm plot with Seaborn's default settings\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 2\u001b[1;33m \u001b[0m_\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0msns\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mswarmplot\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;34m'species'\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0my\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;34m'petal length (cm)'\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mdata\u001b[0m\u001b[1;33m=\u001b[0m\u001b[0mdf\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      3\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      4\u001b[0m \u001b[1;31m# Label the axes\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      5\u001b[0m \u001b[0m_\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mplt\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mxlabel\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m'species'\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mNameError\u001b[0m: name 'df' is not defined"
     ]
    }
   ],
   "source": [
    "# Create bee swarm plot with Seaborn's default settings\n",
    "_ = sns.swarmplot(x='species', y='petal length (cm)', data=df)\n",
    "\n",
    "# Label the axes\n",
    "_ = plt.xlabel('species')\n",
    "_ = plt.ylabel('petal length (cm)')\n",
    "\n",
    "# Show the plot\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Computing the ECDF\n",
    "\n",
    "In this exercise, you will write a function that takes as input a 1D array of data and then returns the x and y values of the ECDF. You will use this function over and over again throughout this course and its sequel. ECDFs are among the most important plots in statistical analysis. You can write your own function, foo(x,y) according to the following skeleton:\n",
    "\n",
    "def foo(a,b):\n",
    "    \"\"\"State what function does here\"\"\"\n",
    "    # Computation performed here\n",
    "    return x, y\n",
    "\n",
    "The function foo() above takes two arguments a and b and returns two values x and y. The function header def foo(a,b): contains the function signature foo(a,b), which consists of the function name, along with its parameters. For more on writing your own functions, see DataCamp's course Python Data Science Toolbox (Part 1) here!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:07:45.941716Z",
     "start_time": "2019-01-22T20:07:45.935733Z"
    }
   },
   "outputs": [],
   "source": [
    "def ecdf(data):\n",
    "    \"\"\"Compute ECDF for a one-dimensional array of measurements.\"\"\"\n",
    "    # Number of data points: n\n",
    "    n = len(data)\n",
    "\n",
    "    # x-data for the ECDF: x\n",
    "    x = np.sort(data)\n",
    "\n",
    "    # y-data for the ECDF: y\n",
    "    y = np.arange(1, n+1) / n\n",
    "\n",
    "    return x, y\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plotting the ECDF\n",
    "\n",
    "You will now use your ecdf() function to compute the ECDF for the petal lengths of Anderson's Iris versicolor flowers. You will then plot the ECDF. Recall that your ecdf() function returns two arrays so you will need to unpack them. An example of such unpacking is x, y = foo(data), for some function foo()."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:08:02.174825Z",
     "start_time": "2019-01-22T20:08:02.161871Z"
    }
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'versicolor_petal_length' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-6-0807a17eabce>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[1;31m# Compute ECDF for versicolor data: x_vers, y_vers\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 2\u001b[1;33m \u001b[0mx_vers\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0my_vers\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mecdf\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mversicolor_petal_length\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      3\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      4\u001b[0m \u001b[1;31m# Generate plot\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      5\u001b[0m \u001b[0m_\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mplt\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mplot\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx_vers\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0my_vers\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mmarker\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;34m'.'\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mlinestyle\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;34m'none'\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mNameError\u001b[0m: name 'versicolor_petal_length' is not defined"
     ]
    }
   ],
   "source": [
    "# Compute ECDF for versicolor data: x_vers, y_vers\n",
    "x_vers, y_vers = ecdf(versicolor_petal_length)\n",
    "\n",
    "# Generate plot\n",
    "_ = plt.plot(x_vers, y_vers, marker='.', linestyle='none')\n",
    "\n",
    "# Label the axes\n",
    "_ = plt.xlabel('petal length (cm)')\n",
    "_ = plt.ylabel('ECDF')\n",
    "\n",
    "# Display the plot\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Comparison of ECDFs\n",
    "\n",
    "ECDFs also allow you to compare two or more distributions (though plots get cluttered if you have too many). Here, you will plot ECDFs for the petal lengths of all three iris species. You already wrote a function to generate ECDFs so you can put it to good use!\n",
    "\n",
    "To overlay all three ECDFs on the same plot, you can use plt.plot() three times, once for each ECDF. Remember to include marker='.' and linestyle='none' as arguments inside plt.plot()."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:08:13.980657Z",
     "start_time": "2019-01-22T20:08:13.967693Z"
    }
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'setosa_petal_length' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-7-d889668056f2>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[1;31m# Compute ECDFs\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 2\u001b[1;33m \u001b[0mx_set\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0my_set\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mecdf\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0msetosa_petal_length\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      3\u001b[0m \u001b[0mx_vers\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0my_vers\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mecdf\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mversicolor_petal_length\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      4\u001b[0m \u001b[0mx_virg\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0my_virg\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mecdf\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mvirginica_petal_length\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      5\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mNameError\u001b[0m: name 'setosa_petal_length' is not defined"
     ]
    }
   ],
   "source": [
    "# Compute ECDFs\n",
    "x_set, y_set = ecdf(setosa_petal_length)\n",
    "x_vers, y_vers = ecdf(versicolor_petal_length)\n",
    "x_virg, y_virg = ecdf(virginica_petal_length)\n",
    "\n",
    "# Plot all ECDFs on the same plot\n",
    "_ = plt.plot(x_set, y_set, marker='.', linestyle='none')\n",
    "_ = plt.plot(x_vers, y_vers, marker='.', linestyle='none')\n",
    "_ = plt.plot(x_virg, y_virg, marker='.', linestyle='none')\n",
    "\n",
    "# Annotate the plot\n",
    "_ = plt.legend(('setosa', 'versicolor', 'virginica'), loc='lower right')\n",
    "_ = plt.xlabel('petal length (cm)')\n",
    "_ = plt.ylabel('ECDF')\n",
    "\n",
    "# Display the plot\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Quantitative exploratory data analysis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Computing means\n",
    "\n",
    "The mean of all measurements gives an indication of the typical magnitude of a measurement. It is computed using np.mean()."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:09:10.229188Z",
     "start_time": "2019-01-22T20:09:10.216222Z"
    }
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'versicolor_petal_length' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-8-3b6a97e15478>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[1;31m# Compute the mean\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 2\u001b[1;33m \u001b[0mmean_length_vers\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mmean\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mversicolor_petal_length\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      3\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      4\u001b[0m \u001b[1;31m# Print the results with some nice formatting\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      5\u001b[0m \u001b[0mprint\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m'I. versicolor:'\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mmean_length_vers\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;34m'cm'\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mNameError\u001b[0m: name 'versicolor_petal_length' is not defined"
     ]
    }
   ],
   "source": [
    "# Compute the mean\n",
    "mean_length_vers = np.mean(versicolor_petal_length)\n",
    "\n",
    "# Print the results with some nice formatting\n",
    "print('I. versicolor:', mean_length_vers, 'cm')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Computing percentiles\n",
    "\n",
    "In this exercise, you will compute the percentiles of petal length of Iris versicolor."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:09:26.945554Z",
     "start_time": "2019-01-22T20:09:26.933588Z"
    }
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'versicolor_petal_length' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-9-a6818b956d34>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[0;32m      3\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      4\u001b[0m \u001b[1;31m# Compute percentiles: ptiles_vers\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 5\u001b[1;33m \u001b[0mptiles_vers\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mpercentile\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mversicolor_petal_length\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mpercentiles\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      6\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      7\u001b[0m \u001b[1;31m# Print the result\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mNameError\u001b[0m: name 'versicolor_petal_length' is not defined"
     ]
    }
   ],
   "source": [
    "# Specify array of percentiles: percentiles\n",
    "percentiles = np.array([2.5, 25, 50, 75, 97.5])\n",
    "\n",
    "# Compute percentiles: ptiles_vers\n",
    "ptiles_vers = np.percentile(versicolor_petal_length, percentiles)\n",
    "\n",
    "# Print the result\n",
    "print(ptiles_vers)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Comparing percentiles to ECDF\n",
    "\n",
    "To see how the percentiles relate to the ECDF, you will plot the percentiles of Iris versicolor petal lengths you calculated in the last exercise on the ECDF plot you generated in chapter 1. The percentile variables from the previous exercise are available in the workspace as ptiles_vers and percentiles.\n",
    "\n",
    "Note that to ensure the Y-axis of the ECDF plot remains between 0 and 1, you will need to rescale the percentiles array accordingly - in this case, dividing it by 100."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:09:40.144796Z",
     "start_time": "2019-01-22T20:09:40.130832Z"
    }
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'x_vers' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-10-533e0fbd5873>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[1;31m# Plot the ECDF\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 2\u001b[1;33m \u001b[0m_\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mplt\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mplot\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx_vers\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0my_vers\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;34m'.'\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      3\u001b[0m \u001b[0m_\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mplt\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mxlabel\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m'petal length (cm)'\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      4\u001b[0m \u001b[0m_\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mplt\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mylabel\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m'ECDF'\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      5\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mNameError\u001b[0m: name 'x_vers' is not defined"
     ]
    }
   ],
   "source": [
    "# Plot the ECDF\n",
    "_ = plt.plot(x_vers, y_vers, '.')\n",
    "_ = plt.xlabel('petal length (cm)')\n",
    "_ = plt.ylabel('ECDF')\n",
    "\n",
    "# Overlay percentiles as red x's\n",
    "_ = plt.plot(ptiles_vers, percentiles/100, marker='D', color='red',\n",
    "         linestyle='none')\n",
    "\n",
    "# Show the plot\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Box-and-whisker plot\n",
    "\n",
    "Making a box plot for the petal lengths is unnecessary because the iris data set is not too large and the bee swarm plot works fine. However, it is always good to get some practice. Make a box plot of the iris petal lengths. You have a pandas DataFrame, df, which contains the petal length data, in your namespace. Inspect the data frame df in the IPython shell using df.head() to make sure you know what the pertinent columns are.\n",
    "\n",
    "For your reference, the code used to produce the box plot in the video is provided below:\n",
    "\n",
    "_ = sns.boxplot(x='east_west', y='dem_share', data=df_all_states)\n",
    "\n",
    "_ = plt.xlabel('region')\n",
    "\n",
    "_ = plt.ylabel('percent of vote for Obama')\n",
    "\n",
    "In the IPython Shell, you can use sns.boxplot? or help(sns.boxplot) for more details on how to make box plots using seaborn."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:09:53.733908Z",
     "start_time": "2019-01-22T20:09:53.720943Z"
    }
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'df' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-11-bf3bde1015d6>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[1;31m# Create box plot with Seaborn's default settings\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 2\u001b[1;33m \u001b[0m_\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0msns\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mboxplot\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;34m'species'\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0my\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;34m'petal length (cm)'\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mdata\u001b[0m\u001b[1;33m=\u001b[0m\u001b[0mdf\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      3\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      4\u001b[0m \u001b[1;31m# Label the axes\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      5\u001b[0m \u001b[0m_\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mplt\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mxlabel\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m'species'\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mNameError\u001b[0m: name 'df' is not defined"
     ]
    }
   ],
   "source": [
    "# Create box plot with Seaborn's default settings\n",
    "_ = sns.boxplot(x='species', y='petal length (cm)', data=df)\n",
    "\n",
    "# Label the axes\n",
    "_ = plt.xlabel('species')\n",
    "_ = plt.ylabel('petal length (cm)')\n",
    "\n",
    "# Show the plot\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Computing the variance\n",
    "\n",
    "It is important to have some understanding of what commonly-used functions are doing under the hood. Though you may already know how to compute variances, this is a beginner course that does not assume so. In this exercise, we will explicitly compute the variance of the petal length of Iris veriscolor using the equations discussed in the videos. We will then use np.var() to compute it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:10:06.630832Z",
     "start_time": "2019-01-22T20:10:06.618864Z"
    }
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'versicolor_petal_length' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-12-8dcc4df8e974>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[1;31m# Array of differences to mean: differences\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 2\u001b[1;33m \u001b[0mdifferences\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mversicolor_petal_length\u001b[0m \u001b[1;33m-\u001b[0m \u001b[0mnp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mmean\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mversicolor_petal_length\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      3\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      4\u001b[0m \u001b[1;31m# Square the differences: diff_sq\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      5\u001b[0m \u001b[0mdiff_sq\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mdifferences\u001b[0m\u001b[1;33m**\u001b[0m\u001b[1;36m2\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mNameError\u001b[0m: name 'versicolor_petal_length' is not defined"
     ]
    }
   ],
   "source": [
    "# Array of differences to mean: differences\n",
    "differences = versicolor_petal_length - np.mean(versicolor_petal_length)\n",
    "\n",
    "# Square the differences: diff_sq\n",
    "diff_sq = differences**2\n",
    "\n",
    "# Compute the mean square difference: variance_explicit\n",
    "variance_explicit = np.mean(diff_sq)\n",
    "\n",
    "# Compute the variance using NumPy: variance_np\n",
    "variance_np = np.var(versicolor_petal_length)\n",
    "\n",
    "# Print the results\n",
    "print(variance_explicit, variance_np)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The standard deviation and the variance\n",
    "\n",
    "As mentioned in the video, the standard deviation is the square root of the variance. You will see this for yourself by computing the standard deviation using np.std() and comparing it to what you get by computing the variance with np.var() and then computing the square root."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:10:17.235935Z",
     "start_time": "2019-01-22T20:10:17.221952Z"
    }
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'versicolor_petal_length' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-13-a4159a867484>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[1;31m# Compute the variance: variance\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 2\u001b[1;33m \u001b[0mvariance\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mvar\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mversicolor_petal_length\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      3\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      4\u001b[0m \u001b[1;31m# Print the square root of the variance\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      5\u001b[0m \u001b[0mprint\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mnp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0msqrt\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mvariance\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mNameError\u001b[0m: name 'versicolor_petal_length' is not defined"
     ]
    }
   ],
   "source": [
    "# Compute the variance: variance\n",
    "variance = np.var(versicolor_petal_length)\n",
    "\n",
    "# Print the square root of the variance\n",
    "print(np.sqrt(variance))\n",
    "\n",
    "# Print the standard deviation\n",
    "print(np.std(versicolor_petal_length))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Scatter plots\n",
    "\n",
    "When you made bee swarm plots, box plots, and ECDF plots in previous exercises, you compared the petal lengths of different species of iris. But what if you want to compare two properties of a single species? This is exactly what we will do in this exercise. We will make a scatter plot of the petal length and width measurements of Anderson's Iris versicolor flowers. If the flower scales (that is, it preserves its proportion as it grows), we would expect the length and width to be correlated.\n",
    "\n",
    "For your reference, the code used to produce the scatter plot in the video is provided below:\n",
    "\n",
    "_ = plt.plot(total_votes/1000, dem_share, marker='.', linestyle='none')\n",
    "\n",
    "_ = plt.xlabel('total votes (thousands)')\n",
    "\n",
    "_ = plt.ylabel('percent of vote for Obama')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:10:31.299776Z",
     "start_time": "2019-01-22T20:10:31.284817Z"
    }
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'versicolor_petal_length' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-14-d835f88ad111>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[1;31m# Make a scatter plot\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 2\u001b[1;33m _ = plt.plot(versicolor_petal_length, versicolor_petal_width,\n\u001b[0m\u001b[0;32m      3\u001b[0m              marker='.', linestyle='none')\n\u001b[0;32m      4\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      5\u001b[0m \u001b[1;31m# Label the axes\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mNameError\u001b[0m: name 'versicolor_petal_length' is not defined"
     ]
    }
   ],
   "source": [
    "# Make a scatter plot\n",
    "_ = plt.plot(versicolor_petal_length, versicolor_petal_width,\n",
    "             marker='.', linestyle='none')\n",
    "\n",
    "# Label the axes\n",
    "_ = plt.xlabel('petal length (cm)')\n",
    "_ = plt.ylabel('petal width (cm)')\n",
    "\n",
    "# Show the result\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Computing the covariance\n",
    "\n",
    "The covariance may be computed using the Numpy function np.cov(). For example, we have two sets of data x and y, np.cov(x, y) returns a 2D array where entries [0,1] and [1,0] are the covariances. Entry [0,0] is the variance of the data in x, and entry [1,1] is the variance of the data in y. This 2D output array is called the covariance matrix, since it organizes the self- and covariance.\n",
    "\n",
    "To remind you how the I. versicolor petal length and width are related, we include the scatter plot you generated in a previous exercise."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:10:47.119973Z",
     "start_time": "2019-01-22T20:10:47.106985Z"
    }
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'versicolor_petal_length' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-15-d0c52408115e>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[1;31m# Compute the covariance matrix: covariance_matrix\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 2\u001b[1;33m \u001b[0mcovariance_matrix\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mcov\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mversicolor_petal_length\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mversicolor_petal_width\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      3\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      4\u001b[0m \u001b[1;31m# Print covariance matrix\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      5\u001b[0m \u001b[0mprint\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mcovariance_matrix\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mNameError\u001b[0m: name 'versicolor_petal_length' is not defined"
     ]
    }
   ],
   "source": [
    "# Compute the covariance matrix: covariance_matrix\n",
    "covariance_matrix = np.cov(versicolor_petal_length, versicolor_petal_width)\n",
    "\n",
    "# Print covariance matrix\n",
    "print(covariance_matrix)\n",
    "\n",
    "# Extract covariance of length and width of petals: petal_cov\n",
    "petal_cov = covariance_matrix[0,1]\n",
    "\n",
    "# Print the length/width covariance\n",
    "print(petal_cov)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Computing the Pearson correlation coefficient\n",
    "\n",
    "As mentioned in the video, the Pearson correlation coefficient, also called the Pearson r, is often easier to interpret than the covariance. It is computed using the np.corrcoef() function. Like np.cov(), it takes two arrays as arguments and returns a 2D array. Entries [0,0] and [1,1] are necessarily equal to 1 (can you think about why?), and the value we are after is entry [0,1].\n",
    "\n",
    "In this exercise, you will write a function, pearson_r(x, y) that takes in two arrays and returns the Pearson correlation coefficient. You will then use this function to compute it for the petal lengths and widths of I. versicolor.\n",
    "\n",
    "Again, we include the scatter plot you generated in a previous exercise to remind you how the petal width and length are related."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:11:00.839649Z",
     "start_time": "2019-01-22T20:11:00.826682Z"
    }
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'versicolor_petal_width' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-16-6c2899cb68ae>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[0;32m      8\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      9\u001b[0m \u001b[1;31m# Compute Pearson correlation coefficient for I. versicolor\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m---> 10\u001b[1;33m \u001b[0mr\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mpearson_r\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mversicolor_petal_width\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mversicolor_petal_length\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m     11\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     12\u001b[0m \u001b[1;31m# Print the result\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mNameError\u001b[0m: name 'versicolor_petal_width' is not defined"
     ]
    }
   ],
   "source": [
    "def pearson_r(x, y):\n",
    "    \"\"\"Compute Pearson correlation coefficient between two arrays.\"\"\"\n",
    "    # Compute correlation matrix: corr_mat\n",
    "    corr_mat = np.corrcoef(x, y)\n",
    "\n",
    "    # Return entry [0,1]\n",
    "    return corr_mat[0,1]\n",
    "\n",
    "# Compute Pearson correlation coefficient for I. versicolor\n",
    "r = pearson_r(versicolor_petal_width, versicolor_petal_length)\n",
    "\n",
    "# Print the result\n",
    "print(r)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Thinking probabilistically-- Discrete variables"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:11:25.831147Z",
     "start_time": "2019-01-22T20:11:25.822169Z"
    }
   },
   "source": [
    "## Generating random numbers using the np.random module\n",
    "\n",
    "We will be hammering the np.random module for the rest of this course and its sequel. Actually, you will probably call functions from this module more than any other while wearing your hacker statistician hat. Let's start by taking its simplest function, np.random.random() for a test spin. The function returns a random number between zero and one. Call np.random.random() a few times in the IPython shell. You should see numbers jumping around between zero and one.\n",
    "\n",
    "In this exercise, we'll generate lots of random numbers between zero and one, and then plot a histogram of the results. If the numbers are truly random, all bars in the histogram should be of (close to) equal height.\n",
    "\n",
    "You may have noticed that, in the video, Justin generated 4 random numbers by passing the keyword argument size=4 to np.random.random(). Such an approach is more efficient than a for loop: in this exercise, however, you will write a for loop to experience hacker statistics as the practice of repeating an experiment over and over again."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:11:36.896870Z",
     "start_time": "2019-01-22T20:11:36.610638Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Seed the random number generator\n",
    "np.random.seed(42)\n",
    "\n",
    "# Initialize random numbers: random_numbers\n",
    "random_numbers = np.empty(100000)\n",
    "\n",
    "# Generate random numbers by looping over range(100000)\n",
    "for i in range(100000):\n",
    "    random_numbers[i] = np.random.random()\n",
    "\n",
    "# Plot a histogram\n",
    "_ = plt.hist(random_numbers)\n",
    "\n",
    "# Show the plot\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The np.random module and Bernoulli trials\n",
    "\n",
    "You can think of a Bernoulli trial as a flip of a possibly biased coin. Specifically, each coin flip has a probability \\(p\\) of landing heads (success) and probability \\(1-p\\) of landing tails (failure). In this exercise, you will write a function to perform n Bernoulli trials, perform_bernoulli_trials(n, p), which returns the number of successes out of n Bernoulli trials, each of which has probability p of success. To perform each Bernoulli trial, use the np.random.random() function, which returns a random number between zero and one."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:11:52.194755Z",
     "start_time": "2019-01-22T20:11:52.189772Z"
    }
   },
   "outputs": [],
   "source": [
    "def perform_bernoulli_trials(n, p):\n",
    "    \"\"\"Perform n Bernoulli trials with success probability p\n",
    "    and return number of successes.\"\"\"\n",
    "    # Initialize number of successes: n_success\n",
    "    n_success = 0\n",
    "\n",
    "    # Perform trials\n",
    "    for i in range(n):\n",
    "        # Choose random number between zero and one: random_number\n",
    "        random_number = np.random.random()\n",
    "\n",
    "        # If less than p, it's a success  so add one to n_success\n",
    "        if random_number < p:\n",
    "            n_success += 1\n",
    "\n",
    "    return n_success"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## How many defaults might we expect?\n",
    "\n",
    "Let's say a bank made 100 mortgage loans. It is possible that anywhere between 0 and 100 of the loans will be defaulted upon. You would like to know the probability of getting a given number of defaults, given that the probability of a default is p = 0.05. To investigate this, you will do a simulation. You will perform 100 Bernoulli trials using the perform_bernoulli_trials() function you wrote in the previous exercise and record how many defaults we get. Here, a success is a default. (Remember that the word \"success\" just means that the Bernoulli trial evaluates to True, i.e., did the loan recipient default?) You will do this for another 100 Bernoulli trials. And again and again until we have tried it 1000 times. Then, you will plot a histogram describing the probability of the number of defaults."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:12:09.664813Z",
     "start_time": "2019-01-22T20:12:09.386556Z"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Users\\Admin\\Anaconda3\\lib\\site-packages\\matplotlib\\axes\\_axes.py:6521: MatplotlibDeprecationWarning: \n",
      "The 'normed' kwarg was deprecated in Matplotlib 2.1 and will be removed in 3.1. Use 'density' instead.\n",
      "  alternative=\"'density'\", removal=\"3.1\")\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Seed random number generator\n",
    "np.random.seed(42)\n",
    "\n",
    "# Initialize the number of defaults: n_defaults\n",
    "n_defaults = np.empty(1000)\n",
    "\n",
    "# Compute the number of defaults\n",
    "for i in range(1000):\n",
    "    n_defaults[i] = perform_bernoulli_trials(100, 0.05)\n",
    "\n",
    "# Plot the histogram with default number of bins; label your axes\n",
    "_ = plt.hist(n_defaults, normed=True)\n",
    "_ = plt.xlabel('number of defaults out of 100 loans')\n",
    "_ = plt.ylabel('probability')\n",
    "\n",
    "# Show the plot\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Will the bank fail?\n",
    "\n",
    "Plot the number of defaults you got from the previous exercise, in your namespace as n_defaults, as a CDF. The ecdf() function you wrote in the first chapter is available.\n",
    "\n",
    "If interest rates are such that the bank will lose money if 10 or more of its loans are defaulted upon, what is the probability that the bank will lose money?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:12:22.359735Z",
     "start_time": "2019-01-22T20:12:22.198192Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Probability of losing money = 0.022\n"
     ]
    }
   ],
   "source": [
    "# Compute ECDF: x, y\n",
    "x, y = ecdf(n_defaults)\n",
    "\n",
    "# Plot the CDF with labeled axes\n",
    "_ = plt.plot(x, y, marker='.', linestyle='none')\n",
    "_ = plt.xlabel('number of defaults out of 100')\n",
    "_ = plt.ylabel('CDF')\n",
    "\n",
    "# Show the plot\n",
    "plt.show()\n",
    "\n",
    "# Compute the number of 100-loan simulations with 10 or more defaults: n_lose_money\n",
    "n_lose_money = np.sum(n_defaults >= 10)\n",
    "\n",
    "# Compute and print probability of losing money\n",
    "print('Probability of losing money =', n_lose_money / len(n_defaults))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sampling out of the Binomial distribution\n",
    "\n",
    "Compute the probability mass function for the number of defaults we would expect for 100 loans as in the last section, but instead of simulating all of the Bernoulli trials, perform the sampling using np.random.binomial(). This is identical to the calculation you did in the last set of exercises using your custom-written perform_bernoulli_trials() function, but far more computationally efficient. Given this extra efficiency, we will take 10,000 samples instead of 1000. After taking the samples, plot the CDF as last time. This CDF that you are plotting is that of the Binomial distribution.\n",
    "\n",
    "Note: For this exercise and all going forward, the random number generator is pre-seeded for you (with np.random.seed(42)) to save you typing that each time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:12:43.657709Z",
     "start_time": "2019-01-22T20:12:43.466222Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Take 10,000 samples out of the binomial distribution: n_defaults\n",
    "n_defaults = np.random.binomial(n=100, p=0.05, size=10000)\n",
    "\n",
    "# Compute CDF: x, y\n",
    "x, y = ecdf(n_defaults)\n",
    "\n",
    "# Plot the CDF with axis labels\n",
    "_ = plt.plot(x, y, marker='.', linestyle='none')\n",
    "_ = plt.xlabel('number of defaults out of 100 loans')\n",
    "_ = plt.ylabel('CDF')\n",
    "\n",
    "# Show the plot\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plotting the Binomial PMF\n",
    "\n",
    "As mentioned in the video, plotting a nice looking PMF requires a bit of matplotlib trickery that we will not go into here. Instead, we will plot the PMF of the Binomial distribution as a histogram with skills you have already learned. The trick is setting up the edges of the bins to pass to plt.hist() via the bins keyword argument. We want the bins centered on the integers. So, the edges of the bins should be -0.5, 0.5, 1.5, 2.5, ... up to max(n_defaults) + 1.5. You can generate an array like this using np.arange() and then subtracting 0.5 from the array.\n",
    "\n",
    "You have already sampled out of the Binomial distribution during your exercises on loan defaults, and the resulting samples are in the NumPy array n_defaults."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:12:57.952585Z",
     "start_time": "2019-01-22T20:12:57.745140Z"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Users\\Admin\\Anaconda3\\lib\\site-packages\\matplotlib\\axes\\_axes.py:6521: MatplotlibDeprecationWarning: \n",
      "The 'normed' kwarg was deprecated in Matplotlib 2.1 and will be removed in 3.1. Use 'density' instead.\n",
      "  alternative=\"'density'\", removal=\"3.1\")\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Compute bin edges: bins\n",
    "bins = np.arange(0, max(n_defaults) + 1.5) - 0.5\n",
    "\n",
    "# Generate histogram\n",
    "_ = plt.hist(n_defaults, normed=True, bins=bins)\n",
    "\n",
    "# Label axes\n",
    "_ = plt.xlabel('number of defaults out of 100 loans')\n",
    "_ = plt.ylabel('PMF')\n",
    "\n",
    "# Show the plot\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Relationship between Binomial and Poisson distributions\n",
    "\n",
    "You just heard that the Poisson distribution is a limit of the Binomial distribution for rare events. This makes sense if you think about the stories. Say we do a Bernoulli trial every minute for an hour, each with a success probability of 0.1. We would do 60 trials, and the number of successes is Binomially distributed, and we would expect to get about 6 successes. This is just like the Poisson story we discussed in the video, where we get on average 6 hits on a website per hour. So, the Poisson distribution with arrival rate equal to \\(np\\) approximates a Binomial distribution for \\(n\\) Bernoulli trials with probability \\(p\\) of success (with \\(n\\) large and \\(p\\) small). Importantly, the Poisson distribution is often simpler to work with because it has only one parameter instead of two for the Binomial distribution.\n",
    "\n",
    "Let's explore these two distributions computationally. You will compute the mean and standard deviation of samples from a Poisson distribution with an arrival rate of 10. Then, you will compute the mean and standard deviation of samples from a Binomial distribution with parameters \\(n\\) and \\(p\\) such that \\(np = 10\\)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:13:13.349204Z",
     "start_time": "2019-01-22T20:13:13.331254Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Poisson:      10.0145 3.1713545607516043\n",
      "n = 20 Binom: 10.0592 2.23523944131272\n",
      "n = 100 Binom: 10.0441 2.9942536949964675\n",
      "n = 1000 Binom: 10.0129 3.139639085946026\n"
     ]
    }
   ],
   "source": [
    "# Draw 10,000 samples out of Poisson distribution: samples_poisson\n",
    "samples_poisson = np.random.poisson(10, size=10000)\n",
    "\n",
    "# Print the mean and standard deviation\n",
    "print('Poisson:     ', np.mean(samples_poisson),\n",
    "                       np.std(samples_poisson))\n",
    "\n",
    "# Specify values of n and p to consider for Binomial: n, p\n",
    "n = [20, 100, 1000]\n",
    "p = [0.5, 0.1, 0.01]\n",
    "\n",
    "# Draw 10,000 samples for each n,p pair: samples_binomial\n",
    "for i in range(3):\n",
    "    samples_binomial = np.random.binomial(n[i], p[i], size=10000)\n",
    "\n",
    "    # Print results\n",
    "    print('n =', n[i], 'Binom:', np.mean(samples_binomial),\n",
    "                                 np.std(samples_binomial))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Was 2015 anomalous?\n",
    "\n",
    "1990 and 2015 featured the most no-hitters of any season of baseball (there were seven). Given that there are on average 251/115 no-hitters per season, what is the probability of having seven or more in a season?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:13:28.599016Z",
     "start_time": "2019-01-22T20:13:28.591058Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Probability of seven or more no-hitters: 0.0072\n"
     ]
    }
   ],
   "source": [
    "# Draw 10,000 samples out of Poisson distribution: n_nohitters\n",
    "n_nohitters = np.random.poisson(251/115, size=10000)\n",
    "\n",
    "# Compute number of samples that are seven or greater: n_large\n",
    "n_large = np.sum(n_nohitters >= 7)\n",
    "\n",
    "# Compute probability of getting seven or more: p_large\n",
    "p_large = n_large / 10000\n",
    "\n",
    "# Print the result\n",
    "print('Probability of seven or more no-hitters:', p_large)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Thinking probabilistically-- Continuous variables"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Normal PDF\n",
    "\n",
    "In this exercise, you will explore the Normal PDF and also learn a way to plot a PDF of a known distribution using hacker statistics. Specifically, you will plot a Normal PDF for various values of the variance."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:14:31.299455Z",
     "start_time": "2019-01-22T20:14:31.095998Z"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Users\\Admin\\Anaconda3\\lib\\site-packages\\matplotlib\\axes\\_axes.py:6521: MatplotlibDeprecationWarning: \n",
      "The 'normed' kwarg was deprecated in Matplotlib 2.1 and will be removed in 3.1. Use 'density' instead.\n",
      "  alternative=\"'density'\", removal=\"3.1\")\n",
      "C:\\Users\\Admin\\Anaconda3\\lib\\site-packages\\matplotlib\\axes\\_axes.py:6521: MatplotlibDeprecationWarning: \n",
      "The 'normed' kwarg was deprecated in Matplotlib 2.1 and will be removed in 3.1. Use 'density' instead.\n",
      "  alternative=\"'density'\", removal=\"3.1\")\n",
      "C:\\Users\\Admin\\Anaconda3\\lib\\site-packages\\matplotlib\\axes\\_axes.py:6521: MatplotlibDeprecationWarning: \n",
      "The 'normed' kwarg was deprecated in Matplotlib 2.1 and will be removed in 3.1. Use 'density' instead.\n",
      "  alternative=\"'density'\", removal=\"3.1\")\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Draw 100000 samples from Normal distribution with stds of interest: samples_std1, samples_std3, samples_std10\n",
    "samples_std1 = np.random.normal(20, 1, size=100000)\n",
    "samples_std3 = np.random.normal(20, 3, size=100000)\n",
    "samples_std10 = np.random.normal(20, 10, size=100000)\n",
    "\n",
    "# Make histograms\n",
    "_ = plt.hist(samples_std1, bins=100, normed=True, histtype='step')\n",
    "_ = plt.hist(samples_std3, bins=100, normed=True, histtype='step')\n",
    "_ = plt.hist(samples_std10, bins=100, normed=True, histtype='step')\n",
    "\n",
    "# Make a legend, set limits and show plot\n",
    "_ = plt.legend(('std = 1', 'std = 3', 'std = 10'))\n",
    "plt.ylim(-0.01, 0.42)\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Normal CDF\n",
    "\n",
    "Now that you have a feel for how the Normal PDF looks, let's consider its CDF. Using the samples you generated in the last exercise (in your namespace as samples_std1, samples_std3, and samples_std10), generate and plot the CDFs."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:15:27.911625Z",
     "start_time": "2019-01-22T20:15:27.161654Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Generate CDFs\n",
    "x_std1, y_std1 = ecdf(samples_std1)\n",
    "x_std3, y_std3 = ecdf(samples_std3)\n",
    "x_std10, y_std10 = ecdf(samples_std10)\n",
    "\n",
    "# Plot CDFs\n",
    "_ = plt.plot(x_std1, y_std1, marker='.', linestyle='none')\n",
    "_ = plt.plot(x_std3, y_std3, marker='.', linestyle='none')\n",
    "_ = plt.plot(x_std10, y_std10, marker='.', linestyle='none')\n",
    "\n",
    "# Make a legend and show the plot\n",
    "_ = plt.legend(('std = 1', 'std = 3', 'std = 10'), loc='lower right')\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Are the Belmont Stakes results Normally distributed?\n",
    "\n",
    "Since 1926, the Belmont Stakes is a 1.5 mile-long race of 3-year old thoroughbred horses. Secretariat ran the fastest Belmont Stakes in history in 1973. While that was the fastest year, 1970 was the slowest because of unusually wet and sloppy conditions. With these two outliers removed from the data set, compute the mean and standard deviation of the Belmont winners' times. Sample out of a Normal distribution with this mean and standard deviation using the np.random.normal() function and plot a CDF. Overlay the ECDF from the winning Belmont times. Are these close to Normally distributed?\n",
    "\n",
    "Note: Justin scraped the data concerning the Belmont Stakes from the Belmont Wikipedia page."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:15:43.005112Z",
     "start_time": "2019-01-22T20:15:42.989128Z"
    }
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'belmont_no_outliers' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-28-7db0344ecc56>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[1;31m# Compute mean and standard deviation: mu, sigma\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 2\u001b[1;33m \u001b[0mmu\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mmean\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mbelmont_no_outliers\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      3\u001b[0m \u001b[0msigma\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mstd\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mbelmont_no_outliers\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      4\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      5\u001b[0m \u001b[1;31m# Sample out of a normal distribution with this mu and sigma: samples\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mNameError\u001b[0m: name 'belmont_no_outliers' is not defined"
     ]
    }
   ],
   "source": [
    "# Compute mean and standard deviation: mu, sigma\n",
    "mu = np.mean(belmont_no_outliers)\n",
    "sigma = np.std(belmont_no_outliers)\n",
    "\n",
    "# Sample out of a normal distribution with this mu and sigma: samples\n",
    "samples = np.random.normal(mu, sigma, size=10000)\n",
    "\n",
    "# Get the CDF of the samples and of the data\n",
    "x_theor, y_theor = ecdf(samples)\n",
    "x, y = ecdf(belmont_no_outliers)\n",
    "\n",
    "# Plot the CDFs and show the plot\n",
    "_ = plt.plot(x_theor, y_theor)\n",
    "_ = plt.plot(x, y, marker='.', linestyle='none')\n",
    "_ = plt.xlabel('Belmont winning time (sec.)')\n",
    "_ = plt.ylabel('CDF')\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## What are the chances of a horse matching or beating Secretariat's record?\n",
    "\n",
    "Assume that the Belmont winners' times are Normally distributed (with the 1970 and 1973 years removed), what is the probability that the winner of a given Belmont Stakes will run it as fast or faster than Secretariat?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:15:56.658070Z",
     "start_time": "2019-01-22T20:15:56.645117Z"
    }
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'mu' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-29-337d9f3b75ac>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[1;31m# Take a million samples out of the Normal distribution: samples\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 2\u001b[1;33m \u001b[0msamples\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mrandom\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mnormal\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mmu\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0msigma\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0msize\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;36m1000000\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      3\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      4\u001b[0m \u001b[1;31m# Compute the fraction that are faster than 144 seconds: prob\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      5\u001b[0m \u001b[0mprob\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0msum\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0msamples\u001b[0m \u001b[1;33m<=\u001b[0m \u001b[1;36m144\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;33m/\u001b[0m \u001b[0mlen\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0msamples\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mNameError\u001b[0m: name 'mu' is not defined"
     ]
    }
   ],
   "source": [
    "# Take a million samples out of the Normal distribution: samples\n",
    "samples = np.random.normal(mu, sigma, size=1000000)\n",
    "\n",
    "# Compute the fraction that are faster than 144 seconds: prob\n",
    "prob = np.sum(samples <= 144) / len(samples)\n",
    "\n",
    "# Print the result\n",
    "print('Probability of besting Secretariat:', prob)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## If you have a story, you can simulate it!\n",
    "\n",
    "Sometimes, the story describing our probability distribution does not have a named distribution to go along with it. In these cases, fear not! You can always simulate it. We'll do that in this and the next exercise.\n",
    "\n",
    "In earlier exercises, we looked at the rare event of no-hitters in Major League Baseball. Hitting the cycle is another rare baseball event. When a batter hits the cycle, he gets all four kinds of hits, a single, double, triple, and home run, in a single game. Like no-hitters, this can be modeled as a Poisson process, so the time between hits of the cycle are also Exponentially distributed.\n",
    "\n",
    "How long must we wait to see both a no-hitter and then a batter hit the cycle? The idea is that we have to wait some time for the no-hitter, and then after the no-hitter, we have to wait for hitting the cycle. Stated another way, what is the total waiting time for the arrival of two different Poisson processes? The total waiting time is the time waited for the no-hitter, plus the time waited for the hitting the cycle.\n",
    "\n",
    "Now, you will write a function to sample out of the distribution described by this story."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:16:13.609984Z",
     "start_time": "2019-01-22T20:16:13.605972Z"
    }
   },
   "outputs": [],
   "source": [
    "def successive_poisson(tau1, tau2, size=1):\n",
    "    \"\"\"Compute time for arrival of 2 successive Poisson processes.\"\"\"\n",
    "    # Draw samples out of first exponential distribution: t1\n",
    "    t1 = np.random.exponential(tau1, size=size)\n",
    "\n",
    "    # Draw samples out of second exponential distribution: t2\n",
    "    t2 = np.random.exponential(tau2, size=size)\n",
    "\n",
    "    return t1 + t2\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Distribution of no-hitters and cycles\n",
    "\n",
    "Now, you'll use your sampling function to compute the waiting time to observe a no-hitter and hitting of the cycle. The mean waiting time for a no-hitter is 764 games, and the mean waiting time for hitting the cycle is 715 games."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-22T20:16:27.770247Z",
     "start_time": "2019-01-22T20:16:27.586737Z"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Users\\Admin\\Anaconda3\\lib\\site-packages\\matplotlib\\axes\\_axes.py:6521: MatplotlibDeprecationWarning: \n",
      "The 'normed' kwarg was deprecated in Matplotlib 2.1 and will be removed in 3.1. Use 'density' instead.\n",
      "  alternative=\"'density'\", removal=\"3.1\")\n"
     ]
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Draw samples of waiting times\n",
    "waiting_times = successive_poisson(764, 715, size=100000)\n",
    "\n",
    "# Make the histogram\n",
    "_ = plt.hist(waiting_times, bins=100, histtype='step',\n",
    "             normed=True)\n",
    "\n",
    "# Label axes\n",
    "_ = plt.xlabel('total waiting time (games)')\n",
    "_ = plt.ylabel('PDF')\n",
    "\n",
    "# Show the plot\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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