courses/statistical_thinking/part1/Statistical_thinking_part_1.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Introduction to Graphical Exploratory Data Analysis\n",
    "\n",
    "> \"Exploratory data analysis can never be the whole story, but nothing else can serve as the foundation stone.\" -- John Tukey\n",
    "\n",
    "In this chapter we will explore the basics of graphical exploratory data analysis techniques. As they a picture can speak thousand words. Let's jump right into it.\n",
    "\n",
    "## Plotting a histogram\n",
    "\n",
    "One of the basic methods of graphical exploratory data analysis is to plot histogram on the data we are trying to analyze. This gives us a quick feel of the data. For our initial practice, we will use a classic dataset collected by botanist Edward Anderson and made famous by Ronald Fisher, one of the most prolific statisticians in the history. Anderson carefully measured anatomical properties of the samples of three different species of iris, *Iris Sentosa*, *Iris versicolor* and *Iris virginica*. This full dataset is available as part of `Scikit-Learn`. Let's get the data. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Target names: ['setosa' 'versicolor' 'virginica']\n",
      "Feature names: ['sepal length (cm)', 'sepal width (cm)', 'petal length (cm)', 'petal width (cm)']\n",
      "Dataset size: (150, 4)\n"
     ]
    }
   ],
   "source": [
    "from sklearn.datasets import load_iris\n",
    "\n",
    "iris = load_iris()\n",
    "print(F\"Target names: {iris.target_names}\")\n",
    "print(F\"Feature names: {iris.feature_names}\")\n",
    "print(F\"Dataset size: {iris.data.shape}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will plot distribution of the versicolor petal length as a histogram. Let's first get that into a variable."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "petal_length = iris.data[:, 2]\n",
    "versicolor_petal_length_indices = np.where(iris.target == 1)\n",
    "versicolor_petal_length = petal_length[versicolor_petal_length_indices]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import seaborn as sns\n",
    "import matplotlib.pyplot as plt  \n",
    "sns.set()\n",
    "\n",
    "_ = plt.hist(versicolor_petal_length)\n",
    "_ = plt.xlabel(\"Petal length (cm)\")\n",
    "_ = plt.ylabel(\"count\")\n",
    "_ = plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The histogram has 10 bins by default, and the count tells us how many of the values from the dataset lies in that bin. For instance, for length 5.0 cm, we have count of 4. Let's change the number of bins.\n",
    "\n",
    "***Square root rule of choosing bins***: The \"square root rule\" is the commonly used rule of thumb for choosing the number of bins: choose the number of bins to be square root of the number of samples.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYAAAAEJCAYAAACdePCvAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4yLjEsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy+j8jraAAAZPklEQVR4nO3de3TMd/7H8dfkTkWRnbit2i56aHXV1q6dVhtU1S0Nkd11WerasqvBsayocqriWpvlqHOsY+1upRQV1Ia6rl1iQ1HW6m5d6p4TkxDEJYmZ+f3h11lhJZMw30l8no9zek6+M/l8P+9+8pl5zff7ne+HzePxeAQAME5QoAsAAAQGAQAAhiIAAMBQBAAAGIoAAABDEQAAYCgCAAAMFRLoAsrq0qVrcrvLfutCVFQ15ebm+6GiRwdjVDLGp3SMUckCMT5BQTbVrPnY/3yu0gWA2+0pVwB82xYlY4xKxviUjjEqWUUaH04BAYChCAAAMBQBAACGIgAAwFAEAAAYigAAAEMRAABgqEp3HwCAiquwyCW7PTLQZfjkZsEtXb1yI9BlBBQBAOChCQsNVuyYtYEuwyefzYnT1UAXEWCcAgIAQxEAAGAoAgAADEUAAIChCAAAMBQBAACGIgAAwFAEAAAYyq8BkJ+fr27duuns2bPFHl+6dKn69evnz64BAKXwWwAcPHhQvXv31smTJ4s9fuzYMf3+97/3V7cAAB/5LQBWrFihyZMnKzo62vtYYWGhJk2apMTERH91CwDwkd/WAkpOTr7nsTlz5qhnz5767ne/669uAQA+smwxuF27dikrK0tJSUnKzMws936ioqqVu21lWaUwkBijkjE+j5ZA/D0r0hyyLADWr1+vo0ePKi4uTtevX1dOTo5GjRql3/3ud2XaT25uvtxuT5n7t9sj5XSavvZfyRijkjE+patIb26+sPrvGYg5FBRku+8HZ8sCYPr06d6fMzMzNX/+/DK/+QMAHh7uAwAAQ/n9CGDbtm33PNa6dWu1bt3a310DAErAEQAAGIoAAABDEQAAYCgCAAAMRQAAgKEIAAAwFAEAAIYiAADAUAQAABiKAAAAQxEAAGAoAgAADEUAAIChCAAAMBQBAACGIgAAwFAEAAAYigAAAEMRAABgKL8GQH5+vrp166azZ89Kkj755BN169ZNsbGxSkpKUmFhoT+7BwCUwG8BcPDgQfXu3VsnT56UJH3zzTdavHixli9frnXr1sntduvjjz/2V/cAgFL4LQBWrFihyZMnKzo6WpIUFhamyZMnq1q1arLZbHrqqad0/vx5f3UPAChFiL92nJycXGy7fv36ql+/viTp4sWLSk1N1fTp0/3VPQCgFH4LgPvJzs7WkCFD1LNnT7Vu3brM7aOiqpW7b7s9stxtTcEYlYzxebQE4u9ZkeaQpQFw/PhxDRkyRP369dOgQYPKtY/c3Hy53Z4yt7PbI+V0Xi1Xn6ZgjErG+JSuIr25+cLqv2cg5lBQkO2+H5wtC4D8/HwNHjxYo0aNUvfu3a3qFgBwH5bdB7Bq1Srl5ORoyZIliouLU1xcnObOnWtV9wCAu/j9CGDbtm2SpAEDBmjAgAH+7g4A4CPuBAYAQxEAAGAoAgAADEUAAIChCAAAMBQBAACGIgAAwFAEAAAYyvLF4IBAi6xeRRHh5Zv6Vq91c7Pglq5euWFpnzAHAQDjRISHKHbM2kCX4ZPP5sSJ5efgL5wCAgBDEQAAYCgCAAAMRQAAgKEIAAAwFAEAAIYiAADAUAQAABiKAAAAQ/k1APLz89WtWzedPXtWkpSRkaHY2Fh17NhRKSkp/uwaAFAKvwXAwYMH1bt3b508eVKSdPPmTU2YMEELFixQenq6Dh8+rB07dvirewBAKfwWACtWrNDkyZMVHR0tSTp06JAaNmyoBg0aKCQkRLGxsdq4caO/ugcAlMJvi8ElJycX275w4YLsdrt3Ozo6WtnZ2WXeb1RUtXLXZPVKjpURY1SxFBa5+Jv4USDGtiL9PS1bDdTtdstms3m3PR5PsW1f5ebmy+32lLmd3R4pp5N1FUtiyhhVpBdgacJCgyvNyqXS7dVLKxOr53sgXmNBQbb7fnC27FtAderUkdPp9G47nU7v6SEAgPUsC4AWLVrom2++0alTp+RyubR+/Xq9/PLLVnUPALiLZaeAwsPDNWPGDL399tsqKChQTEyMOnXqZFX3AIC7+D0Atm3b5v3Z4XBo3bp1/u4SAOAD7gQGAEMRAABgKAIAAAxFAACAoQgAADAUAQAAhiIAAMBQBAAAGIoAAABDEQAAYCgCAAAM5VMA/K9/uOXYsWMPvRgAgHVKDIC8vDzl5eVp6NChunz5snc7JydHI0aMsKpGAIAflLga6JgxY7Rr1y5JUuvWrf/bKCREr732mn8rAwD4VYkBsHjxYklSUlKSpk+fbklBAABr+PTvAUyfPl3nzp3T5cuX5fH899/jfeaZZ/xWGADAv3wKgHnz5mnx4sWKioryPmaz2bR161a/FQYA8C+fAmDNmjXatGmTateu7e96AAAW8elroHXr1n2ob/5r165V165d1bVrV82cOfOh7RcA4DufjgAcDodmzZqlV155RREREd7Hy3MN4MaNG0pOTtbGjRtVvXp19e7dWxkZGXrhhRfKvC8AQPn5FACrV6+WJG3cuNH7WHmvAbhcLrndbt24cUNVq1bVrVu3FB4eXub9AAAejE8BsG3btofWYbVq1TRy5Eh17txZVapU0Y9+9CP98Ic/fGj7BwD4xqcAWLJkyf98fODAgWXu8N///rc+/fRTbd++XZGRkfr1r3+txYsXa8iQIT61j4qqVuY+v2W3R5a7rSkYI5gkEPO9Ir3GfAqAr7/+2vtzYWGh9u7dK4fDUa4Od+7cKYfD4f1KaXx8vD7++GOfAyA3N19ut6f0X7yL3R4pp/NqmduZxJQxqkgvQASW1fM9EK+xoCDbfT84+3wj2J2ys7P1zjvvlKuYpk2bavbs2bp+/bqqVKmibdu26dlnny3XvgAA5edTANytdu3aOnfuXLk6bNOmjY4cOaL4+HiFhobq2Wef1ZtvvlmufQEAyq/M1wA8Ho8OHz5c7K7gsnrzzTd50weAACvzNQDp9o1h48aN80tBAABrlOkawLlz53Tr1i01bNjQr0UBAPzPpwA4deqUfvnLX+rChQtyu92qWbOmFi5cqEaNGvm7PgCAn/i0FtCUKVM0ZMgQ7d27V/v27dPw4cP13nvv+bs2AIAf+RQAubm56tGjh3e7Z8+eunTpkt+KAgD4n08B4HK5lJeX592+ePGi3woCAFjDp2sAv/jFL/Tzn/9cnTt3ls1mU3p6ut544w1/1wYA8COfjgBiYmIkSUVFRTp+/Liys7P16quv+rUwAIB/+XQEMH78ePXt21f9+/dXQUGBli1bpgkTJmjRokX+rg8A4Cc+HQFcunRJ/fv3lySFh4drwIABcjqdfi0MAOBfPl8Ezs7O9m7n5OTI4yn7ipwAgIrDp1NAAwYMUPfu3fXSSy/JZrMpIyODpSAAoJLzKQASEhLUvHlz/eMf/1BwcLAGDx6sp556yt+1AQD8yOfloJs2baqmTZv6sxYAgIV8ugYAAHj0EAAAYCgCAAAMRQAAgKEIAAAwVEACYNu2bYqPj1fnzp01derUQJQAAMazPADOnDmjyZMna8GCBVq3bp2OHDmiHTt2WF0GABjP5/sAHpbNmzerS5cuqlOnjiQpJSVF4eHhVpcBAMazPABOnTql0NBQDRs2TFlZWWrbtq1GjRrlc/uoqGrl7ttujyx3W1MwRjBFYZErIPO9vH0WFrkUFhr8UGuxPABcLpe++OILffTRR6pataqGDx+utLQ0xcfH+9Q+NzdfbnfZF6Kz2yPldF4tczuTmDJGhBwkKSw0WLFj1ga6DJ99NieuXK/PoCDbfT84W34N4Dvf+Y4cDodq1aqliIgIdejQQYcOHbK6DAAwnuUB0K5dO+3cuVNXrlyRy+XS3//+dz3zzDNWlwEAxrP8FFCLFi00ZMgQ9enTR0VFRXrxxRfVs2dPq8sAAONZHgDS7eWlExISAtE1AOD/cScwABiKAAAAQxEAAGAoAgAADEUAAIChCAAAMBQBAACGIgAAwFAEAAAYigAAAEMRAABgKAIAAAxFAACAoQgAADAUAQAAhiIAAMBQBAAAGIoAAABDBTQAZs6cqfHjxweyBAAwVsACYPfu3UpLSwtU9wBgvIAEQF5enlJSUjRs2LBAdA8AUIACYNKkSRo9erSqV68eiO4BAJJCrO5w5cqVqlu3rhwOh1avXl3m9lFR1crdt90eWe62VissciksNNjyfsszRoGqFTDNw34PszwA0tPT5XQ6FRcXp8uXL+v69euaNm2aJkyY4FP73Nx8ud2eMvdrt0fK6bxa5naBYrdHKnbM2kCX4ZPP5sRVurEFKqPyvM6Cgmz3/eBseQAsWbLE+/Pq1au1Z88en9/8AQAPD/cBAIChLD8CuFN8fLzi4+MDWQIAGIsjAAAwFAEAAIYiAADAUAQAABiKAAAAQxEAAGAoAgAADEUAAIChCAAAMFRA7wTGo6GwyMUCa0AlRADggYWFBlealUul26uXAuAUEAAYiwAAAEMRAABgKAIAAAxFAACAoQgAADAUAQAAhiIAAMBQAbkRbP78+dqwYYMkKSYmRuPGjQtEGQBgNMuPADIyMrRz506lpaVpzZo1+te//qXNmzdbXQYAGM/yIwC73a7x48crLCxMktSoUSOdP3/e6jIAwHiWB0CTJk28P588eVIbNmzQsmXLrC4DAIwXsMXgjh49qrfeekvjxo3T9773PZ/bRUVVK3efrFgJoDJ72O9hAQmAffv2KTExURMmTFDXrl3L1DY3N19ut6fMfdrtkXI6r5a5XaAQVgDuVp73sKAg230/OFseAFlZWfrVr36llJQUORwOq7sHAPw/ywNg8eLFKigo0IwZM7yP9erVS71797a6FAAwmuUBMHHiRE2cONHqbgEAd+FOYAAwFAEAAIYiAADAUAQAABiKAAAAQxEAAGAoAgAADEUAAIChArYYnNUKi1ysrwMAdzAmAMJCgxU7Zm2gy/DZZ3PiAl0CgEccp4AAwFAEAAAYigAAAEMRAABgKAIAAAxFAACAoQgAADAUAQAAhiIAAMBQAQmAzz77TF26dFHHjh2VmpoaiBIAwHiWLwWRnZ2tlJQUrV69WmFhYerVq5dat26txo0bW10KABjN8gDIyMjQT37yE9WoUUOS9Nprr2njxo0aMWKET+2Dgmzl7ju6ZpVytw2EylRvZapVqlz1VqZapcpVb2WqVSrf+19JbWwej8fzIAWV1cKFC3X9+nWNHj1akrRy5UodOnRI77//vpVlAIDxLL8G4Ha7ZbP9N5E8Hk+xbQCANSwPgDp16sjpdHq3nU6noqOjrS4DAIxneQC88MIL2r17ty5evKgbN25o06ZNevnll60uAwCMZ/lF4Nq1a2v06NHq37+/ioqKlJCQoB/84AdWlwEAxrP8IjAAoGLgTmAAMBQBAACGIgAAwFAEAAAYyvJvAfnb3Llz9fnnn8tmsykhIUEDBw4s9vxXX32ld955R9euXVOrVq303nvvKSTkkRuG+yptfObPn69PP/1U1atXlyT97Gc/U9++fQNRasDNnDlTly5d0owZM4o9bvoc+tb9xoc5JPXr108XL170zospU6aoRYsW3ucrzBzyPEIyMzM9vXr18hQVFXlu3Ljhadeunef48ePFfqdr166eAwcOeDwejycpKcmTmpoaiFIDwpfxeeuttzz79+8PUIUVR0ZGhqd169ae3/zmN/c8Z/Ic+lZJ42P6HHK73Z42bdp4ioqK7vs7FWUOPVKngH784x/rz3/+s0JCQpSbmyuXy6WqVat6nz937pxu3ryp5557TpIUHx+vjRs3Bqpcy5U2PpJ0+PBhLVy4ULGxsZoyZYoKCgoCVG3g5OXlKSUlRcOGDbvnOdPnkFTy+EjMoRMnTkiSBg0apNdff11Lly4t9nxFmkOPVABIUmhoqObNm6euXbvK4XCodu3a3ucuXLggu93u3bbb7crOzg5EmQFT0vhcu3ZNzZo109ixY5WWlqYrV65owYIFAaw2MCZNmqTRo0d7T2HciTlU8vgwh6QrV67I4XDoww8/1B//+EctX75cu3bt8j5fkebQIxcAkpSYmKjdu3crKytLK1as8D7OQnS33W98HnvsMS1atEiNGjVSSEiIBg0apB07dgSwUuutXLlSdevWlcPh+J/Pmz6HShsf5pDUsmVLzZo1S5GRkapVq5YSEhKKjUFFmkOP1JWr48ePq7CwUM2aNVOVKlXUsWNH/ec///E+f/dCdDk5OUYtRFfa+Jw/f14ZGRlKSEiQdHtimnZxMz09XU6nU3Fxcbp8+bKuX7+uadOmacKECZKYQ6WND3NI+uKLL1RUVOQNybvHoCLNoUfqCODs2bOaOHGiCgsLVVhYqK1bt+r555/3Pl+/fn2Fh4dr3759kqS1a9catRBdaeMTERGh2bNn68yZM/J4PEpNTdWrr74awIqtt2TJEq1fv15r165VYmKi2rdv731zk5hDpY0Pc0i6evWqZs2apYKCAuXn5ystLa3YGFSkOfRIBUBMTIzatm2r7t27q2fPnmrZsqW6du2qoUOH6p///Kck6YMPPtD06dPVqVMnXb9+Xf379w9w1dYpbXxq1aqlKVOmaPjw4erUqZM8Hs89XxM1FXOoZMyh/2rXrp1iYmK8r7NvX2sVcQ6xGBwAGOqROgIAAPiOAAAAQxEAAGAoAgAADEUAAIChCABUOGfPnlWzZs0UFxfn/e/111/XqlWrSm176NAhTZo0qdTfGz9+vBYvXuzz4w/D/PnztWXLljL3k5+fr8GDB+vmzZsPXMOWLVv04YcfPvB+8Ggw6xY9VBoRERFau3atdzs7O1vdunVT8+bN1bRp0/u2O3bsWIVdmyczM1ONGzcuc7sPPvhAP/3pTxUREfHANXTo0EGpqan66quv1KxZswfeHyo3AgCVQu3atdWwYUOdPHlSTZs21cqVK7Vs2TK53W7VqFFD7777rqpWrap58+bp6tWrSkpKUnJysqZNm6aDBw/q2rVr8ng8mjp1arG7n0ty/PhxJScnKy8vTy6XS/369VNCQoIyMzOVkpKiBg0a6OjRo7p165bee+89Pf/887p48aKSkpJ0+vRp1ahRQ3a7XU2aNFGtWrV0+PBhzZo1S8HBwZKkAwcOqFevXsrJyVGTJk00Z86ce1ZnzcrK0vbt2zVx4kRJtxdbmzp1qvbv36/g4GB16NBBo0ePVlJSkiIiIvT1118rNzdX7du3V40aNbR9+3Y5nU5NnTrVuzRBQkKC5s+fz5EAOAWEyuHAgQM6ffq0WrRooT179mjNmjVKTU3VmjVrNGTIEI0YMUJ169ZVYmKiWrVqpenTp+vgwYO6cOGCPvnkE6Wnp6tHjx5atGiRT/3dunVLiYmJGjNmjFavXq2lS5fqD3/4g7788ktJt081DRo0SGvWrFF8fLxSUlIkSVOnTlXjxo21YcMGzZ07V/v375ck9e3bV82bN9e4ceO8ywJkZ2dryZIl+vzzz5Wdna1NmzbdU8fWrVvlcDi8a8nMmzdPBQUFSk9P15o1a7R//37t2bNHknTkyBH96U9/8tZatWpVLV++XP379y/2//3iiy/qb3/720M5pYTKjSMAVEg3b95UXFycJMnlcqlmzZqaPXu26tatq48++kinTp1Sr169vL9/5coV5eXlFdtHy5Yt9fjjj2v58uU6c+aMMjMz9dhjj/nU/8mTJ3X69Oli69zcvHlTR44cUaNGjVSvXj3vKZSnn35aaWlpkqQdO3Z4f46OjlanTp3u20eHDh1UpUoVSVKTJk108eLFe37nxIkTeuKJJ7zbGRkZSkpKUnBwsIKDg71rzaelpaldu3YKDQ2V3W5X1apV9dJLL0mSnnjiiWJjU6NGDYWHh+vcuXNq1KiRT+OBRxMBgArp7msAd3K73YqLi9PYsWO92xcuXNDjjz9e7Pf++te/Kjk5WQMHDtQrr7yi73//+1q3bp1P/btcLkVGRharIScnR5GRkfryyy+LnY+32Wz6dkWVkJAQ3bm6SlDQ/Q+y71wh8s593Mlms8ntdhdrc+fSwVlZWd5awsLC7rv/u30bIDAbp4BQ6bRp00Z/+ctfdOHCBUnSsmXL9MYbb0i6/cZ269YtSdKuXbvUrl079enTR82bN9eWLVvkcrl86uPJJ58sFkJZWVnq1q2bDh8+XGK7mJgY77eVLl26pC1btnjfsO+szVdPPvmkzpw54912OBxKS0uT2+1WYWGhEhMTtXfv3jLt8+rVqyosLFS9evXK1A6PHgIAlU6bNm00dOhQDRo0SLGxsVq/fr3mz58vm82m5557TmfOnNGIESPUq1cv7dmzR7GxserRo4caNGigs2fPFvtEfT9hYWFasGCBVq1apdjYWA0aNEgjR44s9QJyUlKSTpw4odjYWCUmJqpevXreT+jt27fXb3/7W+8pIl906NBBmZmZ3uAaMWKEQkNDFRcXp+7duysmJkYdO3b0eX+StHPnTrVt2/aeIwaYh9VAgYcoNTVVTz/9tFq2bKnCwkL16dNHb7/9tmJiYsq9z3fffVcOh0NdunR5KDX2799fEyZMKPHrtDAD1wCAh6hx48Z6//335Xa7VVRUpE6dOj3Qm78kjR07ViNHjlT79u0f+F6AzZs3q1WrVrz5QxJHAABgLK4BAIChCAAAMBQBAACGIgAAwFAEAAAYigAAAEP9H1EmACumd60uAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n_data = len(versicolor_petal_length)\n",
    "n_bins = int(np.sqrt(n_data))\n",
    "\n",
    "_ = plt.hist(versicolor_petal_length, bins=n_bins)\n",
    "_ = plt.xlabel(\"Petal length (cm)\")\n",
    "_ = plt.ylabel(\"count\")\n",
    "_ = plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot this in varying number of bins.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 576x576 with 4 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def plot_histogram(n_bins):\n",
    "    _ = plt.hist(versicolor_petal_length, bins=n_bins)\n",
    "    _ = plt.xlabel(\"Petal length (cm)\")\n",
    "    _ = plt.ylabel(\"count\")\n",
    "    \n",
    "plt.figure(figsize=(8, 8))\n",
    "plt.subplot(221)\n",
    "plot_histogram(5)\n",
    "plt.subplot(222)\n",
    "plot_histogram(10)\n",
    "plt.subplot(223)\n",
    "plot_histogram(25)\n",
    "plt.subplot(224)\n",
    "plot_histogram(50)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plotting all of the data: Bee Swarm plots.\n",
    "\n",
    "The problem with histogram plots is that \n",
    "* Same data can look entirely different depending on how bins are chosen. Choice of bins is arbitrary. This leads to *binning bias*. In other words same data can be interpreted in entirely different way depending on the choice of bin.\n",
    "* We are not plotting all of the data. We are putting data into bins, so we always lose some information, how much, this agains depends on choice of bins which is arbitrary.\n",
    "\n",
    "To alleviate these problems, we can use ***Bee Swarm plots***. We plot all the data in Bee Swarm plots, so no information is lost and also there is no binning bias.\n",
    "\n",
    "The quantitative information is laid out on the y-axis. On the other hand, x-axis contains the various features we are trying to compare. Note that the spread of data along x-axis has no meaning. Let's draw one to understand this better. First let's convert the data into a nice `pandas DataFrame`.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>sepal length (cm)</th>\n",
       "      <th>sepal width (cm)</th>\n",
       "      <th>petal length (cm)</th>\n",
       "      <th>petal width (cm)</th>\n",
       "      <th>species_num</th>\n",
       "      <th>species</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>5.1</td>\n",
       "      <td>3.5</td>\n",
       "      <td>1.4</td>\n",
       "      <td>0.2</td>\n",
       "      <td>0</td>\n",
       "      <td>setosa</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>4.9</td>\n",
       "      <td>3.0</td>\n",
       "      <td>1.4</td>\n",
       "      <td>0.2</td>\n",
       "      <td>0</td>\n",
       "      <td>setosa</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>4.7</td>\n",
       "      <td>3.2</td>\n",
       "      <td>1.3</td>\n",
       "      <td>0.2</td>\n",
       "      <td>0</td>\n",
       "      <td>setosa</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>4.6</td>\n",
       "      <td>3.1</td>\n",
       "      <td>1.5</td>\n",
       "      <td>0.2</td>\n",
       "      <td>0</td>\n",
       "      <td>setosa</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>5.0</td>\n",
       "      <td>3.6</td>\n",
       "      <td>1.4</td>\n",
       "      <td>0.2</td>\n",
       "      <td>0</td>\n",
       "      <td>setosa</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   sepal length (cm)  sepal width (cm)  petal length (cm)  petal width (cm)  \\\n",
       "0                5.1               3.5                1.4               0.2   \n",
       "1                4.9               3.0                1.4               0.2   \n",
       "2                4.7               3.2                1.3               0.2   \n",
       "3                4.6               3.1                1.5               0.2   \n",
       "4                5.0               3.6                1.4               0.2   \n",
       "\n",
       "   species_num species  \n",
       "0            0  setosa  \n",
       "1            0  setosa  \n",
       "2            0  setosa  \n",
       "3            0  setosa  \n",
       "4            0  setosa  "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import pandas as pd\n",
    "\n",
    "iris_df = pd.DataFrame(data=iris.data, index=None, columns=iris.feature_names)\n",
    "# also add the target values, since Scikit-Learn dataset has it in target \n",
    "iris_df['species_num'] = iris.target\n",
    "\n",
    "# Add another column species which will contain the categorical name for plotting\n",
    "iris_df['species'] = iris_df['species_num']\n",
    "iris_df['species'].replace([0, 1, 2], iris.target_names, inplace=True)\n",
    "\n",
    "iris_df.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, let's plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "_ = sns.swarmplot(x='species', y='petal length (cm)', data=iris_df)\n",
    "_ = plt.xlabel('Species')\n",
    "_ = plt.ylabel('Petal Length (cm)')\n",
    "_ = plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nice, we can see every data point in the plot and we can see the variations. Clearly setosa has smaller petal width, then versicolor and then finally virginica. Note that the spread of the values along x-axis has no meaning. As you can guess, if there are lot of data points, this plot can become overwhelmingly confusing and will obfuscate the data. We can solve this problem by ***Empirical Cumulative Distribution Functions (ECDF)***.\n",
    "\n",
    "## Empirical Cumulative Distribution Functions (ECDF)\n",
    "\n",
    "ECDF is an estimator of the Cumulative Distribution Function. The ECDF allows you to plot a feature of your data from least to greatest and see whole feature as it is distributed across the dataset. The x-value of an ECDF is the qunatity you are trying to measure, and y-value on the other hand is the fraction of data points that have a smaller value than the corresponding x-value. We will understand it better with an example. Let's draw the plot first and then make sense of it. Note that this visualization utilizes all the data points too. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "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": [
    "Let's draw ECDF plot for versicolor petal length."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x_vers, y_vers = ecdf(versicolor_petal_length)\n",
    "_ = plt.plot(x_vers, y_vers, marker='.', linestyle='none')\n",
    "_ = plt.xlabel('Versicolor Petal Length')\n",
    "_ = plt.ylabel('ECDF')\n",
    "_ = plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, on x-axis we have plotted the quantity we wanted to measure. On y-axis we have put ECDF, which moves from $\\frac{1}{n}$ to $n$ where $n$ is the size of the data. Any point on the plot represents what percentage of data points is less than what is on the x-axis. So for example, for x-axis 5.0, ~99% of the data is less than 5.0. Similarly, for point 3.6, almost 10% of data is less than 3.6. We can plot many features together to compare them. Let's try this for petal length of all the members of iris family."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array(['setosa', 'versicolor', 'virginica'], dtype='<U10')"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "iris.target_names"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "setosa_petal_length = petal_length[np.where(iris.target == 0)]\n",
    "virginica_petal_length = petal_length[np.where(iris.target == 2)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x_set, y_set = ecdf(setosa_petal_length)\n",
    "x_vir, y_vir = ecdf(virginica_petal_length)\n",
    "_ = plt.plot(x_set, y_set, marker='.', linestyle='none')\n",
    "_ = plt.plot(x_vers, y_vers, marker='.', linestyle='none')\n",
    "_ = plt.plot(x_vir, y_vir, marker='.', linestyle='none')\n",
    "_ = plt.xlabel('Petal Length')\n",
    "_ = plt.ylabel('ECDF')\n",
    "_ = plt.legend(('setosa', 'versicolor', 'virginica'), loc='lower right')\n",
    "_ = plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As you can see, this is similar to Bee-Swarm plot to compare the feature but also it gives better detail for larger dataset too. We can clearly see Setosa is much shorter, also with less absolute variability in petal length than versicolor and virginica."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Introduction to Summary Statistics\n",
    "\n",
    "We have seen that the graphical plots (histogram, Bee-Swarm plot and the ECDF) provide effective summaries of the data, however sometimes we want to summarize the data even more succinctly in one or two numbers, then we use the summary statistics. Note that they are not subsitute of the EDA we have been learning, but they sometimes are useful to speak about data in as few points as possible to give a little glimpse of the data.\n",
    "\n",
    "## Mean\n",
    "\n",
    "Mean is sum of all the data points, divided by the number of points in dataset.\n",
    "\n",
    "$$\n",
    "    mean = \\bar{x} = \\frac{1}{n}\\sum_{i=1}^{n}x_{i}\n",
    "$$\n",
    "\n",
    "Mean is easy to calculate but it is susceptible to outliers. Outliers are datapoints whose value is far greater or less than most of the rest of the data.\n",
    "\n",
    "Imagine a dataset where we have numbers like \\[1, 2, 0, 2, 1000\\]. In this case 1000 is an outlier. The mean of the dataset is 201, which doesn't effectively describe the dataset.\n",
    "\n",
    "In `NumPy` mean is calculated by:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.26"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(versicolor_petal_length)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Median\n",
    "\n",
    "Median is the middle value of the sorted data. So we sort the data and choose the middle value. Since the median relies on it's position in the dataset but not on the values, it is immune to extreme values of the data set, in other words outliers have no impact on the median."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.35"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.median(versicolor_petal_length)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Percentiles, outliers and Box Plots.\n",
    "\n",
    "The median is the special name for 50% percentile, In other words, 50% of the data are less than the median. Similarly 25th percentile is the value of datapoint that is greater than 25% of the data and so on. Percentile are useful summary statistics. Let's compute 25, 50 and 75 percentile of versicolor petal length"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([4.  , 4.35, 4.6 ])"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.percentile(versicolor_petal_length, [25, 50, 75])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's understand how percentiles relate to ECDF. We will plot percentiles of the *Iris versicolor* petal length along with ECDF. Remember that ECDF are between 0 and 1, so we have to scale our percentiles too between 0 and 1. First we will plot the ECDF and overlay the percentiles 2.5, 25, 50, 75, and 97.5 on that plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# ECDF plot\n",
    "_ = plt.plot(x_vers, y_vers, marker='.', linestyle='none')\n",
    "_ = plt.xlabel('Versicolor Petal Length (cm)')\n",
    "_ = plt.ylabel('ECDF')\n",
    "\n",
    "# Calculate various percentiles\n",
    "percentiles = np.array([2.5, 25, 50, 75, 97.5])\n",
    "vers_percentiles = np.percentile(versicolor_petal_length, percentiles)\n",
    "_ = plt.plot(vers_percentiles, percentiles/100, marker='D', color='red', linestyle='none')\n",
    "_ = plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can plot the percentiles graphically, this is where quantitative EDA meets graphical EDA. We use box plots using percentile to display some of the saliant features of the dataset. In the plot below, \n",
    "\n",
    "* Center of the box is the median, which is the 50% percentile of the data\n",
    "* The edges of the boxes are $25^{th}$ and $75^{th}$ percentile (bottom and top respectively).\n",
    "* This means that total height of the box is the middle 50% of the data and is called *Inter-Quartile Range* or *IQR*.\n",
    "* The whiskers around the box extends to 1.5 times the IQR or to the extent of the data, whichever is more extreme.\n",
    "* Finally, all points outside of the whiskers are plotted individually, and marked as outliers.\n",
    "\n",
    "When number of data points are too large, and Bee Swarm plots are too cluttered, The box plots are the great alternative. Plotting a Box plot is almost same as Bee Swarm plots."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "_ = sns.boxplot(x='species', y='petal length (cm)', data=iris_df)\n",
    "_ = plt.xlabel('Species')\n",
    "_ = plt.ylabel('Petal Length (cm)')\n",
    "_ = plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One thing to remember that outlier often doesn't mean they are erroneous data points, sometimes they are valid points, and it depends on context. Imagine a case when you see a certain sensor value which shows temperature of a unit. If all datapoints from other sensor show sensible value but this one sensor show an outlier value, it might be possible that this sensor is faulty and it's values should be removed from dataset. However imagine we are looking at election data and looking at the vote share of counties in election, we may see some counties in a state entirely vote for either republicans or democrates and they might show a different value than rest of the state. As there is 0 evidence of election fraud in United States, this means that we just have valid outliers. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Variance and Standard Deviation\n",
    "\n",
    "Variance is how we measure the spread or in other words, variability of the data. Clearly *setosa* has less spread than *virginica*. Formally it is the *mean squared distance of the data from their mean*. \n",
    "\n",
    "$$\n",
    "\\text{variance} = \\sigma_{x}^{2} = \\frac{1}{n}\\sum^{n}_{i=1}(x_{i} - \\bar{x})^2\n",
    "$$\n",
    "\n",
    "In other words, we calculate the distance of each data point from the mean, square it and then calculate the mean over all data points."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.21640000000000004"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.var(versicolor_petal_length)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As the calculation of the variance involves squared quantities, it does not have the same unit as what we want to measure, for example, the variance we got for *versicolor petal length* is in square distance unit, rather than in distance unit. Therefore, we usually take square root of the variance which is called ***Standard Deviation***. Standard devision is a reasonable metric for typical spread of the data.\n",
    "$$\n",
    "\\text{standard deviation} = \\sigma = \\sqrt{\\frac{1}{N} \\sum_{i=1}^N (x_i - \\mu)^2}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.4651881339845203"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.std(versicolor_petal_length)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Scatter Plots.\n",
    "\n",
    "So far we have drawn swarm plots and ECDF plots which compared the petal length of various species. What if, we want to compare properties of a single species. That is when scatter plot come in handy.  As an example let's draw a scatter plot between *versicolor* petal length and it's width."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# let's first get the versicolor petal width in a numpy array\n",
    "petal_width = iris.data[:, 3]\n",
    "versicolor_petal_width = petal_width[np.where(iris.target == 1)]\n",
    "\n",
    "# let's plot the scatter plot\n",
    "_ = plt.plot(versicolor_petal_length, versicolor_petal_width, marker='.', linestyle='none')\n",
    "_ = plt.xlabel('Versicolor Petal Length')\n",
    "_ = plt.ylabel('Versicolor Petal Width')\n",
    "_ = plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Covariance\n",
    "\n",
    "A measure of how two quantities vary together. The covariance is essentially mean of product of individiual distances of the qunatities from their mean.\n",
    "\n",
    "$$\n",
    " \\text{covariance} = S_{xy} = \\frac{1}{n}\\sum^{n}_{i=1}(x_{i} - \\bar{x})(y_{i} - \\bar{y})\n",
    "$$\n",
    "\n",
    "Variables are positively correlated when they grow together, anti-corelated other wise. Value of 0 means there is no correlation. We can see in the plot above as the length of versicolor petal length gets bigger, the width too gets bigger, that means versicolor petal length and petal width are positively correlated. Let's see it by calculating it using `numpy`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.22081633, 0.07310204],\n",
       "       [0.07310204, 0.03910612]])"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.cov(versicolor_petal_length, versicolor_petal_width)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What we see above is a covariance matrix. It is a square matrix, given by $C_{i, j} = \\sigma(x_{i}, x_{j})$ So diagonal values is covariance of variable with itself, which is nothing but variance.  and other values represent the various pair of covariances. Also remember that covariance matrix is symmetric since $\\sigma(x_{i}, x_{j}) = \\sigma(x_{j}, x_{i})$ as you see in above matrix. values at \\[0,1\\] and \\[1,0\\] are same (which is 0.07310204) representing positive covariance of versicolor petal length and width.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Pearson Correlation Coefficient\n",
    "\n",
    "A general, more applicable measure to calculate how two variables depend on each other, we want it to be dimensionless, that is to not have any units.  Therefore, we divide the covariance with the standard deviation of the x and y.\n",
    "\n",
    "$$\n",
    "\\rho = \\frac{\\text{cov}(X,Y)}{\\sigma_x \\sigma_y}\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\rho = \\frac{\\text{variability due to codependence}}{\\text{independent variability}}\n",
    "$$\n",
    "\n",
    "As it is dimensionless, it varies from -1 (for complete anticorrelation) to 1 (for complete correlation). Value of 0 means that there is no correlation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.7866680885228169"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "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: r\n",
    "pearson_r(versicolor_petal_length, versicolor_petal_width)\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Probablistic Logic and Statistical Inference\n",
    "\n",
    "Imagine you measure the petal length of 50 versicolor flowers petal length. You calculate the mean of those petal length. Using this mean, what can you say about the mean petal length of all the versicolors flowers on the planet. It is likely that if you measure petal length of another 50 versicolor flowers' petal length, you will get similar but quantatively different set of measurements. Can we make a guess of what will be mean measurements of next 50 flowers. With what we have studied so far, we don't have anything in our tool box to explain the how certain we can be about those measurements.\n",
    "For this, we need *probabilistic reasoning*. It allows to describe uncertainty in quantitative terms. Thinking in the probabilistic terms, you might not be able to tell exact mean of the next 50 petal lengths you measure, but you could say  \n",
    "it will be probable to be close to what we got in last 50 measurements than it is to be much greater. This experiments can be repeated over and over again, and bound the mean within a range. \n",
    "\n",
    "Given a set of data, you can describe probabilistically what you might expect if those data were acquired again and again and again. This is the science of *Statistical Inference*, where we go from measured data to probabilistic conclusion about what we might expect if we collected the same data again. It also helps us draw general and actionable conclusion from relatively few data points or observations.\n",
    "\n",
    "\n",
    "\n",
    "## Generating random numbers using the `np.random` module.\n",
    "\n",
    "We will use hacker statistics to generate random values for simulated experiments e.g. to simulate coin flips. For now, let's generate a 100,000 random numbers using `np.random.random` between 0 and 1, and then plot a histogram of results. If the numbers are truly random, all the bars in histogram should be roughly equal height.\n",
    "\n",
    "`np.random.seed` allows us to reproduce the same sequence of numbers. That is why the random module is called pseudo-random."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYUAAAD7CAYAAACFfIhNAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4yLjEsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy+j8jraAAAWu0lEQVR4nO3ccUzV1/3/8deVS1G/+I3F3iuEEZKtjWTYyTKWjdlcZtNxr8K18WozhY4tW2Olm23cRseAQFxnZju+0CwbbmbNlhH3B3UVKLm9uM1JZmhSS9YZLU2bVKlFvVyUVUBhcO/n98fi+RXpeuVy4VZ5PhKD99zP5Z434H3ey/Vem2VZlgAAkLQk0RsAAHxyEAUAgEEUAAAGUQAAGEQBAGAQBQCAQRQAAIY90RuYq+HhMUUisb3UYtWqVF2+PBrnHX1yLbZ5JWZeLJj51i1ZYtPdd//Pfz3/to9CJGLFHIUbl19MFtu8EjMvFswcH/z6CABgEAUAgEEUAAAGUQAAGEQBAGAQBQCAcUtRGB0dVUlJid5//31JUk9Pj7xer4qKitTU1GSO6+vrk8/nk9vtVk1NjaampiRJFy5cUFlZmTwejyoqKjQ2NiZJunr1qnbu3KmNGzeqrKxMoVAo3vMBAGYhahT++c9/aseOHTp37pwkaXx8XNXV1Wpubpbf79fp06fV3d0tSaqsrFRdXZ26urpkWZZaW1slSXv37lVpaakCgYDWrl2r5uZmSdLzzz+v/Px8vfLKK3rkkUe0b9++eRoTWDz+PRmWw7EiIX9W/O+yRI+POYr64rXW1lbV19fr6aefliSdOnVK2dnZysrKkiR5vV4FAgHde++9Gh8fV15eniTJ5/PpF7/4hR555BGdPHlSv/rVr8z6o48+qsrKSh0/flyHDh2SJJWUlOgnP/mJJicnlZycPC/DAgtpxf8u09KUxLw+1PuD9oRc78v/97BGEnC9N0K40MYnpjRy9fqCX+98ivoTe/O998HBQTkcDnPa6XQqGAzOWHc4HAoGgxoeHlZqaqrsdvu09Zs/l91uV2pqqq5cuaLVq1fPfbJbsNh+kBI1r3Rn/uOJZmmKPSE3zi//38MLfp2JdldyUsK+1omI4Hya9d2YSCQim81mTluWJZvN9l/Xb3z8sJtPf/gyS5bM7rnvVatSZ3X8zRLxg/Sn/SUJu3FO5D3IpQmaOVFf68UokXc8EiWR887Hdc86Cunp6dOeEA6FQnI6nTPWh4aG5HQ6lZaWppGREYXDYSUlJZnjpf88yhgaGlJ6erqmpqY0NjamlStXzmo/ly+Pxvz+H4n6ZibyXk2i/HsyrLuSkxb8eicmw0pJwPUuVovxZzsUSsxjBYdjRUzXvWSJ7WPvTM86CuvWrdPZs2fV39+vT33qU+rs7NTWrVuVmZmplJQU9fb26gtf+ILa29vlcrmUnJys/Px8+f1+eb1etbW1yeVySZIKCwvV1tamXbt2ye/3Kz8/n+cT7lCJvLFI5KMj4HYz6yikpKRo//792r17tyYmJlRYWCiPxyNJamhoUG1trUZHR5Wbm6vy8nJJUn19vaqqqnTgwAFlZGSosbFRkvTUU0+pqqpKxcXFWrFihRoaGuI4GgDMr0T+uuzfk+F5+by3HIVjx46ZvxcUFKijo2PGMTk5OTp8+PCM9czMTLW0tMxYX7lypX7961/f6hYA4BMlUY+Apfl7JMormgEABlEAABhEAQBgEAUAgEEUAAAGUQAAGEQBAGAQBQCAQRQAAAZRAAAYRAEAYBAFAIBBFAAABlEAABhEAQBgEAUAgEEUAAAGUQAAGEQBAGAQBQCAQRQAAAZRAAAYRAEAYBAFAIBBFAAABlEAABhEAQBgEAUAgEEUAAAGUQAAGEQBAGAQBQCAMacotLe3q7i4WMXFxXr22WclST09PfJ6vSoqKlJTU5M5tq+vTz6fT263WzU1NZqampIkXbhwQWVlZfJ4PKqoqNDY2NhctgQAmIOYo3D9+nXt27dPLS0tam9v1+uvv65jx46purpazc3N8vv9On36tLq7uyVJlZWVqqurU1dXlyzLUmtrqyRp7969Ki0tVSAQ0Nq1a9Xc3ByfyQAAsxZzFMLhsCKRiK5fv66pqSlNTU0pNTVV2dnZysrKkt1ul9frVSAQ0MDAgMbHx5WXlydJ8vl8CgQCmpyc1MmTJ+V2u6etAwASwx7rBVNTU/XUU09p48aNWrZsmb74xS9qcHBQDofDHON0OhUMBmesOxwOBYNBDQ8PKzU1VXa7fdo6ACAxYo7CW2+9pT/96U/629/+phUrVuiHP/yhzp07J5vNZo6xLEs2m02RSOQj1298/LCbT0ezalVqrCMAwG3N4VgR988ZcxROnDihgoICrVq1StJ/fvXzwgsvKCkpyRwTCoXkdDqVnp6uUChk1oeGhuR0OpWWlqaRkRGFw2ElJSWZ42fj8uVRRSJWTDPMxxcUABZKKDQy68ssWWL72DvTMT+nkJOTo56eHl27dk2WZenYsWNat26dzp49q/7+foXDYXV2dsrlcikzM1MpKSnq7e2V9J//teRyuZScnKz8/Hz5/X5JUltbm1wuV6xbAgDMUcyPFB544AG9+eab8vl8Sk5O1v3336/du3dr/fr12r17tyYmJlRYWCiPxyNJamhoUG1trUZHR5Wbm6vy8nJJUn19vaqqqnTgwAFlZGSosbExPpMBAGYt5ihI0s6dO7Vz585pawUFBero6JhxbE5Ojg4fPjxjPTMzUy0tLXPZBgAgTnhFMwDAIAoAAIMoAAAMogAAMIgCAMAgCgAAgygAAAyiAAAwiAIAwCAKAACDKAAADKIAADCIAgDAIAoAAIMoAAAMogAAMIgCAMAgCgAAgygAAAyiAAAwiAIAwCAKAACDKAAADKIAADCIAgDAIAoAAIMoAAAMogAAMIgCAMAgCgAAgygAAAyiAAAw5hSFY8eOyefzaePGjfrpT38qSerp6ZHX61VRUZGamprMsX19ffL5fHK73aqpqdHU1JQk6cKFCyorK5PH41FFRYXGxsbmsiUAwBzEHIXz58+rvr5ezc3N6ujo0Jtvvqnu7m5VV1erublZfr9fp0+fVnd3tySpsrJSdXV16urqkmVZam1tlSTt3btXpaWlCgQCWrt2rZqbm+MzGQBg1mKOwp///Gdt2rRJ6enpSk5OVlNTk5YtW6bs7GxlZWXJbrfL6/UqEAhoYGBA4+PjysvLkyT5fD4FAgFNTk7q5MmTcrvd09YBAIlhj/WC/f39Sk5O1q5du3Tx4kV99atf1X333SeHw2GOcTqdCgaDGhwcnLbucDgUDAY1PDys1NRU2e32aeuzsWpVaqwjAMBtzeFYEffPGXMUwuGwXn/9dbW0tGj58uWqqKjQ0qVLZbPZzDGWZclmsykSiXzk+o2PH3bz6WguXx5VJGLFNMN8fEEBYKGEQiOzvsySJbaPvTMdcxTuueceFRQUKC0tTZL00EMPKRAIKCkpyRwTCoXkdDqVnp6uUChk1oeGhuR0OpWWlqaRkRGFw2ElJSWZ4wEAiRHzcwobNmzQiRMndPXqVYXDYf3973+Xx+PR2bNn1d/fr3A4rM7OTrlcLmVmZiolJUW9vb2SpPb2drlcLiUnJys/P19+v1+S1NbWJpfLFZ/JAACzFvMjhXXr1umxxx5TaWmpJicntX79eu3YsUOf/vSntXv3bk1MTKiwsFAej0eS1NDQoNraWo2Ojio3N1fl5eWSpPr6elVVVenAgQPKyMhQY2NjfCYDAMxazFGQpG3btmnbtm3T1goKCtTR0THj2JycHB0+fHjGemZmplpaWuayDQBAnPCKZgCAQRQAAAZRAAAYRAEAYBAFAIBBFAAABlEAABhEAQBgEAUAgEEUAAAGUQAAGEQBAGAQBQCAQRQAAAZRAAAYRAEAYBAFAIBBFAAABlEAABhEAQBgEAUAgEEUAAAGUQAAGEQBAGAQBQCAQRQAAAZRAAAYRAEAYBAFAIBBFAAABlEAABhEAQBgxCUKzz77rKqqqiRJPT098nq9KioqUlNTkzmmr69PPp9PbrdbNTU1mpqakiRduHBBZWVl8ng8qqio0NjYWDy2BACIwZyj8Oqrr+rIkSOSpPHxcVVXV6u5uVl+v1+nT59Wd3e3JKmyslJ1dXXq6uqSZVlqbW2VJO3du1elpaUKBAJau3atmpub57olAECM5hSFf/3rX2pqatKuXbskSadOnVJ2draysrJkt9vl9XoVCAQ0MDCg8fFx5eXlSZJ8Pp8CgYAmJyd18uRJud3uaesAgMSwz+XCdXV12rNnjy5evChJGhwclMPhMOc7nU4Fg8EZ6w6HQ8FgUMPDw0pNTZXdbp+2PhurVqXOZQQAuG05HCvi/jljjsKLL76ojIwMFRQU6KWXXpIkRSIR2Ww2c4xlWbLZbP91/cbHD7v5dDSXL48qErFimmE+vqAAsFBCoZFZX2bJEtvH3pmOOQp+v1+hUEgPP/ywPvjgA127dk0DAwNKSkoyx4RCITmdTqWnpysUCpn1oaEhOZ1OpaWlaWRkROFwWElJSeZ4AEBixPycwu9+9zt1dnaqvb1dTz75pB588EH99re/1dmzZ9Xf369wOKzOzk65XC5lZmYqJSVFvb29kqT29na5XC4lJycrPz9ffr9fktTW1iaXyxWfyQAAszan5xRulpKSov3792v37t2amJhQYWGhPB6PJKmhoUG1tbUaHR1Vbm6uysvLJUn19fWqqqrSgQMHlJGRocbGxnhuCQAwC3GJgs/nk8/nkyQVFBSoo6NjxjE5OTk6fPjwjPXMzEy1tLTEYxsAgDniFc0AAIMoAAAMogAAMIgCAMAgCgAAgygAAAyiAAAwiAIAwCAKAACDKAAADKIAADCIAgDAIAoAAIMoAAAMogAAMIgCAMAgCgAAgygAAAyiAAAwiAIAwCAKAACDKAAADKIAADCIAgDAIAoAAIMoAAAMogAAMIgCAMAgCgAAgygAAAyiAAAwiAIAwJhTFH75y1+quLhYxcXFeu655yRJPT098nq9KioqUlNTkzm2r69PPp9PbrdbNTU1mpqakiRduHBBZWVl8ng8qqio0NjY2Fy2BACYg5ij0NPToxMnTujIkSNqa2vTmTNn1NnZqerqajU3N8vv9+v06dPq7u6WJFVWVqqurk5dXV2yLEutra2SpL1796q0tFSBQEBr165Vc3NzfCYDAMxazFFwOByqqqrSXXfdpeTkZH3mM5/RuXPnlJ2draysLNntdnm9XgUCAQ0MDGh8fFx5eXmSJJ/Pp0AgoMnJSZ08eVJut3vaOgAgMeyxXvC+++4zfz937pxeeeUVPfroo3I4HGbd6XQqGAxqcHBw2rrD4VAwGNTw8LBSU1Nlt9unrc/GqlWpsY4AALc1h2NF3D9nzFG44Z133tHjjz+up59+WklJSTp37pw5z7Is2Ww2RSIR2Wy2Ges3Pn7YzaejuXx5VJGIFdPe5+MLCgALJRQamfVlliyxfeyd6Tk90dzb26tvfetb+sEPfqAtW7YoPT1doVDInB8KheR0OmesDw0Nyel0Ki0tTSMjIwqHw9OOBwAkRsxRuHjxor773e+qoaFBxcXFkqR169bp7Nmz6u/vVzgcVmdnp1wulzIzM5WSkqLe3l5JUnt7u1wul5KTk5Wfny+/3y9Jamtrk8vlisNYAIBYxPzroxdeeEETExPav3+/Wdu+fbv279+v3bt3a2JiQoWFhfJ4PJKkhoYG1dbWanR0VLm5uSovL5ck1dfXq6qqSgcOHFBGRoYaGxvnOBIAIFYxR6G2tla1tbUfeV5HR8eMtZycHB0+fHjGemZmplpaWmLdBgAgjnhFMwDAIAoAAIMoAAAMogAAMIgCAMAgCgAAgygAAAyiAAAwiAIAwCAKAACDKAAADKIAADCIAgDAIAoAAIMoAAAMogAAMIgCAMAgCgAAgygAAAyiAAAwiAIAwCAKAACDKAAADKIAADCIAgDAIAoAAIMoAAAMogAAMIgCAMAgCgAAgygAAAyiAAAwPhFRePnll7Vp0yYVFRXp0KFDid4OACxa9kRvIBgMqqmpSS+99JLuuusubd++XV/60pd07733JnprALDoJDwKPT09+vKXv6yVK1dKktxutwKBgL73ve/d0uWXLLHN6fqddy+b0+W53k/+dTPz4rjuxXa9Umy3f9EuY7Msy4p1Q/Hwm9/8RteuXdOePXskSS+++KJOnTqlZ555JpHbAoBFKeHPKUQiEdls/79clmVNOw0AWDgJj0J6erpCoZA5HQqF5HQ6E7gjAFi8Eh6Fr3zlK3r11Vd15coVXb9+XUePHpXL5Ur0tgBgUUr4E82rV6/Wnj17VF5ersnJSW3btk2f+9znEr0tAFiUEv5EMwDgkyPhvz4CAHxyEAUAgEEUAAAGUQAAGHd8FKK92V5fX598Pp/cbrdqamo0NTWVgF3GV7SZ//KXv+jhhx/W5s2b9cQTT+iDDz5IwC7j61bfVPH48eN68MEHF3Bn8yfazO+++66+8Y1vaPPmzfrOd76zKL7PZ86c0datW7V582Y9/vjjunr1agJ2GV+jo6MqKSnR+++/P+O8ebn9su5gly5dsjZs2GANDw9bY2Njltfrtd55551pxxQXF1v/+Mc/LMuyrB//+MfWoUOHErHVuIk288jIiLV+/Xrr0qVLlmVZ1vPPP28988wzidpuXNzK99myLCsUClkej8fasGFDAnYZX9FmjkQiVlFRkdXd3W1ZlmX9/Oc/t5577rlEbTcubuX7vGPHDuv48eOWZVnWz372M6uxsTERW42bN954wyopKbFyc3Ot8+fPzzh/Pm6/7uhHCh9+s73ly5ebN9u7YWBgQOPj48rLy5Mk+Xy+aeffjqLNPDk5qfr6eq1evVqStGbNGl28eDFR242LaDPfUFtbe8tvtPhJF23mM2fOaPny5eaFoLt27VJZWVmithsXt/J9jkQiGhsbkyRdv35dS5cuTcRW46a1tVX19fUf+S4P83X7dUdHYXBwUA6Hw5x2Op0KBoP/9XyHwzHt/NtRtJnvvvtufe1rX5MkjY+P6+DBg3rooYcWfJ/xFG1mSfrDH/6gz372s1q3bt1Cb29eRJv5vffe0z333KPq6mpt2bJF9fX1Wr58eSK2Gje38n2uqqpSbW2tHnjgAfX09Gj79u0Lvc242rdvn/Lz8z/yvPm6/bqjoxDtzfbuxDfju9WZRkZGtHPnTuXk5GjLli0LucW4izbz22+/raNHj+qJJ55IxPbmRbSZp6am9Nprr2nHjh06cuSIsrKytH///kRsNW6izTw+Pq6amhr9/ve/14kTJ1RaWqof/ehHidjqgpiv2687OgrR3mzv5vOHhoZu+zfju5U3GBwcHFRpaanWrFmjffv2LfQW4y7azIFAQKFQSFu3btXOnTvN/LezaDM7HA5lZ2fr/vvvlySVlJTo1KlTC77PeIo289tvv62UlBTzNjlf//rX9dprry34PhfKfN1+3dFRiPZme5mZmUpJSVFvb68kqb29/bZ/M75oM4fDYe3atUsbN25UTU3Nbf/ISIo+85NPPqmuri61t7fr4MGDcjqd+uMf/5jAHc9dtJk///nP68qVK3rrrbckSceOHVNubm6ithsX0WbOzs7WpUuX9O6770qS/vrXv5oo3onm7fZrzk9Vf8J1dHRYxcXFVlFRkXXw4EHLsizrscces06dOmVZlmX19fVZW7dutdxut/X973/fmpiYSOR24+LjZj569Ki1Zs0aa/PmzeZPdXV1gnc8d9G+zzecP3/+jvjfR5YVfeY33njD2rp1q7Vp0ybr29/+tjU0NJTI7cZFtJmPHz9ueb1eq6SkxPrmN79pvffee4ncbtxs2LDB/O+j+b794g3xAADGHf3rIwDA7BAFAIBBFAAABlEAABhEAQBgEAUAgEEUAAAGUQAAGP8PRFYXpxhSCukAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# seed the random generator\n",
    "np.random.seed(42)\n",
    "\n",
    "# Intialize a 100,000 random numbers\n",
    "random_numbers = np.empty(100000)\n",
    "\n",
    "# Generate random numbers\n",
    "for i in range(100000):\n",
    "    random_numbers[i] = np.random.random()\n",
    "\n",
    "# plot the histogram\n",
    "_ = plt.hist(random_numbers)\n",
    "_ = plt.show()\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Random Variable\n",
    "\n",
    "A random variable is a real-valued variable whose value is determined by an underlying random experiment.\n",
    "Let's consider an example: I toss a coin 5 times. This is a random experiment, and the sample space of the experiment will be :\n",
    "\n",
    "$$\n",
    "\\text{S} = \\{\\textit{HHHHH}, \\textit{HHHHT}, ....., \\textit{TTTTT}\\}\n",
    "$$\n",
    "\n",
    "The sample space will have $2^{5} = 32$ elements. Suppose in this experiment, we are interested in number of heads. This can be done by defining a random variable ***X*** whose value is the number of heads observed. The value of ***X*** will be 0, 1, 2, 3, 4 or 5, depending on the outcome of the experiment.\n",
    "\n",
    "To put it in other words based on the above example, a random variable is a real-valued function that assigns a numerical value to each possible outcome of the random experiment. For instance, the random variable ***X*** defined above assigns following values to the outcomes.\n",
    "\n",
    "| Outcome |     X     |\n",
    "| :------------: | :----------: |\n",
    "| HHHHH| 5 |\n",
    "| HHHHT| 4 |\n",
    "| ..   | ..|\n",
    "| TTTTH| 1 |\n",
    "| TTTTT| 5 |\n",
    "\n",
    "Thus,  a random variable X is a function from the sample space to the real numbers.\n",
    "$$\n",
    "    X: S \\rightarrow \\mathbb{R}\n",
    "$$\n",
    "\n",
    "Random variables are usually denoted by capital letters such as ***X***, ***Y*** and  ***Z***. Since the *Random Variables* are actually functions, it's range can be defined. The range of random variable ***X*** shown by Range(***X***) or $\\textit{R_{x}}$ is the set of all possible values of ***X***. In the above example, $Range(X) = R_{x} = \\{0, 1, 2, 3, 4, 5\\}$\n",
    "\n",
    "### Discrete Random Variable\n",
    "\n",
    "Discrete Random Variable are random variables whose range is countable set. The above example is clearly a discrete random variable. However if we define ***Y*** as a random variable accounting for amount of rain in Seattle today. This is not countable, hence ***Y*** is not a discrete random variable, it is in fact a continous random variable which we will discuss later.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "## Bernoulli Trial\n",
    "\n",
    "An experiments which has two possible outcomes : Success (True) and Failure (False). 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). Let's simulate a Bernoulli trial using `np.random.random`\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "def perform_bernoulli_trials(n, p):\n",
    "    \"\"\"Performs n Bernoulli trials with success probability p and return the number of successes\"\"\"\n",
    "    #number of successes\n",
    "    n_success = 0\n",
    "    \n",
    "    # Perform trials\n",
    "    for i in range(n):\n",
    "        random_number = np.random.random()\n",
    "        if random_number < p:\n",
    "            n_success += 1\n",
    "    \n",
    "    return n_success"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "69793"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Unfair coin with probability of head p = 0.7. Total heads in 100000 trials\n",
    "\n",
    "perform_bernoulli_trials(100000, 0.7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "49898"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Fair coin (p = 0.5), total heads in 100000 trials\n",
    "\n",
    "perform_bernoulli_trials(100000, 0.5)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Probability Mass Function (PMF)\n",
    "\n",
    "PMF is defined as the set of probabilities of all discrete outcomes. Consider this simple scenario. A person rolling a die once. The outcomes are discrete, as only certain values can be the outcome. So possible outcomes are 1, 2, 3, 4, 5, and 6. You can't roll a 5.3. Assuming the die is fair, each result has the same $\\frac{1}{6}$ probability. In other words it is uniform probability. Therefore, PMF associated with this story is a Discrete Uniform PMF. \n",
    "\n",
    "The PMF is a property of a discrete probability distribution. **Distribution is nothing but a mathematical description of outcomes.**. We can match a story to mathematical description of outcomes as we just did with discrete uniform PMF. \n",
    "\n",
    "\n",
    "### Discrete Uniform Distribution Story\n",
    "The outcome of a rolling of a single fair die is:\n",
    "* Discrete\n",
    "* Uniformly Distributed.\n",
    "\n",
    "### Binomial Distribution: the story\n",
    "* The number of *r* successes in *n* Bernoulli trials with probability *p* of success is *Binomially distributed*.\n",
    "* The number of *r* of heads in 4 coin flips with probability 0.5 of heads is *Binomially distributed*.\n",
    "\n",
    "\n",
    "### Poisson Process\n",
    "\n",
    "* In a Poisson process, the timing of the next event is completely independent of when the previous event happened\n",
    "* Many real life process follow this. For example:\n",
    "    * Natural Births in a given hospital.\n",
    "    * Hits on a website during an hour.\n",
    "    * Meteor strikes\n",
    "    * Molecular collisions \n",
    "* The number of arrivals of a Poisson process in a given amount of time is *Poisson Distributed*.\n",
    "* The number *r* arrivals of a Poisson process in a given time interval with average rate of *x* arrivals per interval is Poisson distributed.\n",
    "* The number *r* hits on a website in one hour with an average hit of rate 6 hits per hour is Poisson Distributed.\n",
    "* Poisson distribution may look like Binomial distribution because it is indeed limit of the Binomial Distribution for low probability of success and large number of trials.\n",
    "* It is for rare events."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Poisson::  mean:  9.957  std:  3.1397374094022577\n"
     ]
    }
   ],
   "source": [
    "samples_poisson = np.random.poisson(10, 10000) # mean 10, samples 10000\n",
    "print(\"Poisson:: \", \"mean: \", np.mean(samples_poisson), \" std: \", np.std(samples_poisson))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Make a list of the n and p values to consider for the Binomial distribution. Choose n = \\[20, 100, 1000\\] and p = \\[0.5, 0.1, 0.01\\] so that np is always 10."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "n = 20 Binomial:: mean:  10.0118  std:  2.243136366786469\n",
      "n = 100 Binomial:: mean:  9.98  std:  2.995463236295849\n",
      "n = 1000 Binomial:: mean:  9.9606  std:  3.162569784210303\n"
     ]
    }
   ],
   "source": [
    "n, p = ([20, 100, 1000], [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], 10000)\n",
    "\n",
    "    # Print results\n",
    "    print('n =', n[i], 'Binomial:: mean: ', np.mean(samples_binomial),\n",
    "                                 ' std: ', np.std(samples_binomial))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The mean is same for all the 3 cases. The standard deviation of the Binomial distribution gets closer and closer to that of the Poisson distribution as the probability p gets lower and lower."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}