{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Mo8QV6zESV0H"
      },
      "source": [
        "Lecture 16 - Normal distributions, central limit theorem, cdfs\n",
        "---------\n",
        "\n",
        "CS 2810, Spring 2022, 3/21/2022, Muzny\n",
        "\n",
        "We recommend downloading this notebook and then uploading it to [Google Colaboratory](https://colab.research.google.com/).\n",
        "\n",
        "We'll upload finalized html/ipynb versions of this notebook after lecture."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 22,
      "metadata": {
        "id": "X_2oMEfpSV0J"
      },
      "outputs": [],
      "source": [
        "# for the pi, e, sqrt functions\n",
        "import math\n",
        "\n",
        "# for our coin flips\n",
        "import random\n",
        "\n",
        "# for our normal distribution functions\n",
        "from scipy.stats import norm\n",
        "\n",
        "# for graphing (optional)\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "# for the linspace function (optional)\n",
        "import numpy as np"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 23,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "Z13xYZSsSV0K",
        "outputId": "338dfa2a-8014-4936-fd2f-a3e3295deb57"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "0.5\n",
            "8\n",
            "2.718281828459045\n",
            "3.141592653589793\n",
            "0.1991696681468755\n",
            "from scipy 0.19916966814687553\n"
          ]
        }
      ],
      "source": [
        "# check your calculation for the pdf value \n",
        "# of a normal distribution here\n",
        "\n",
        "x = 5.11\n",
        "mean = 5\n",
        "stddev = 2\n",
        "\n",
        "# do the math \"by hand\" first:\n",
        "print( 1 / 2) # to divide\n",
        "print( 2 ** 3) # ** is the exponent operator\n",
        "print(math.e) # the e constant\n",
        "print(math.pi) # give you the value of pi\n",
        "# carful of order of operations!\n",
        "pdf_val = (1 / (stddev * math.sqrt((2 * math.pi)))) * math.e ** (-0.5 * ((x - mean) / stddev) ** 2)\n",
        "print(pdf_val)\n",
        "\n",
        "# then, run the code:\n",
        "# verify our pdf calculations using scipy\n",
        "print(\"from scipy\", norm.pdf(x, mean, stddev))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 24,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "id": "TNF3VVWYSV0M",
        "outputId": "49a1082b-ff34-4c4a-b124-79caf0e1f39f"
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "# central limit theorem example\n",
        "def flip_coin(times):\n",
        "    \"\"\"\n",
        "    Function that generates a list of fair two-sided coin flips.\n",
        "    Parameters:\n",
        "    times (int): number of times to flip the coin\n",
        "    Return:\n",
        "    list of values from the coin flips\n",
        "    \"\"\"\n",
        "    return [random.randint(0, 1) for t in range(times)]\n",
        "\n",
        "\n",
        "# fill in numbers here\n",
        "# number of samples to take\n",
        "samples = 10000\n",
        "# flips per sample\n",
        "times = 40\n",
        "\n",
        "# accumulate a list of the average value of each sample\n",
        "averages = []\n",
        "# repeat for samples times\n",
        "for sample_num in range(samples):\n",
        "    # flip the coins\n",
        "    coin_flips = flip_coin(times)\n",
        "    # calculate the average\n",
        "    averages.append(sum(coin_flips) / len(coin_flips))\n",
        "    \n",
        "# display a histogram of the averages\n",
        "plt.hist(averages)\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# if I wanted to apply the central limit theorem to people's heights, what are samples and times?\n",
        "# samples: groups of people, chosen with replacement\n",
        "# times: asking each person in a given group (sample) how tall they are\n",
        "\n",
        "# we will do this sufficient number of times to get a good estimate of \n",
        "# our population mean (this is a mean of means)\n",
        "# law of large numbers :)"
      ],
      "metadata": {
        "id": "L2TC6M5bXp1w"
      },
      "execution_count": 25,
      "outputs": []
    },
    {
      "cell_type": "code",
      "execution_count": 26,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "GfMwiVsaSV0M",
        "outputId": "2871ab2a-fd7b-4098-8105-f1eeec651a81"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "5.11\n",
            "5\n",
            "2\n",
            "0.5\n",
            "0.15865525393145707\n",
            "0.3413447460685429\n"
          ]
        }
      ],
      "source": [
        "# using sci py to access the cdf of a distribution:\n",
        "# norm.cdf(x, mean, stddev)\n",
        "print(x)\n",
        "print(mean)\n",
        "print(stddev)\n",
        "\n",
        "# what percent is <= 5?\n",
        "print(norm.cdf(5, mean, stddev))\n",
        "\n",
        "# what percent is between 3 and 5?\n",
        "print(norm.cdf(3, mean, stddev))\n",
        "print(norm.cdf(5, mean, stddev) - norm.cdf(3, mean, stddev))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 27,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "JTJnBRlcSV0N",
        "outputId": "f33725e9-c275-45a9-ed33-e940eeb36721"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "2.4368968689107993\n",
            "0.10000027475074158\n"
          ]
        }
      ],
      "source": [
        "# using sci py to access the ppf of a distribution:\n",
        "# norm.ppf(percentage, mean, stddev)\n",
        "\n",
        "print(norm.ppf(.1, mean, stddev))\n",
        "print(norm.cdf(2.4369, mean, stddev))\n"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# ICA question 4\n",
        "# bottom 10 % of travel times\n",
        "print(norm.ppf(.1, mean, stddev))\n",
        "\n",
        "# % trips between 5 and 8 minutes\n",
        "print(norm.cdf(8, mean, stddev) - norm.cdf(5, mean, stddev))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "ipjXzm28dGb8",
        "outputId": "59f36549-25c4-47f5-ba11-88821ed0c661"
      },
      "execution_count": 28,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "2.4368968689107993\n",
            "0.4331927987311419\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 29,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "id": "VL8T-2kNSV0O",
        "outputId": "8f557c4b-82c0-41a0-ceb7-7579e9cd4954"
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "# example adapted from:\n",
        "# https://docs.scipy.org/doc/scipy-1.8.0/html-scipyorg/reference/generated/scipy.stats.norm.html\n",
        "\n",
        "# we're plotting the calculated pdf in combination with\n",
        "# the histogram generated from sampling a normally distributed \n",
        "# random variable with the same mean and standard deviation \n",
        "# that we've been working with\n",
        "fig, ax = plt.subplots(1, 1)\n",
        "xs = np.linspace(norm.ppf(0.01, mean, stddev),\n",
        "                norm.ppf(0.99, mean, stddev), 100)\n",
        "ax.plot(xs, norm.pdf(xs, mean, stddev),\n",
        "       'o', lw=5, alpha=0.6, label='norm pdf')\n",
        "r = norm.rvs(mean, stddev,size=1000)\n",
        "ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)\n",
        "ax.legend(loc='best', frameon=False)\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        ""
      ],
      "metadata": {
        "id": "sCyElfGsbaSI"
      },
      "execution_count": 29,
      "outputs": []
    }
  ],
  "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": "3.8.12"
    },
    "colab": {
      "name": "16_pdf_cdf_lec.ipynb",
      "provenance": [],
      "collapsed_sections": []
    }
  },
  "nbformat": 4,
  "nbformat_minor": 0
}