Wasserman_Chapter_4.ipynb

{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "name": "Wasserman_Chapter_4.ipynb",
      "provenance": [],
      "collapsed_sections": [],
      "include_colab_link": true
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/sajad13901/Statistics_Wasserman/blob/main/Wasserman_Chapter_4.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 279
        },
        "id": "cG2CIjrgiD8_",
        "outputId": "eeb8be04-8726-4193-e82e-0372097eac70"
      },
      "source": [
        "# Chapter 4, problem 4, part b\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "def calc_eps(n, alpha):\n",
        "    return np.sqrt(np.log(2/alpha)/(2*n))\n",
        "\n",
        "def toss(n, probHead):\n",
        "  cnt = 0\n",
        "  for i in range(n):\n",
        "    cnt += np.random.choice(a = [1, 0], p = [probHead, 1-probHead]) \n",
        "  return cnt/n\n",
        "\n",
        "N = 5000\n",
        "# vec_n = np.linspace(1, N+1, 51)\n",
        "vec_n = np.array([500, 1000, 1500, 2000, 2500, 3000, 3500, 4000])\n",
        "sim_runs = 100\n",
        "vec_coverage = []\n",
        "for n in vec_n:\n",
        "    eps = calc_eps(n, 0.05)\n",
        "    cnt = 0\n",
        "    for i in range(sim_runs):\n",
        "        phat = toss(int(n), .4)\n",
        "        if (0.4 > phat - eps) and (0.4 < phat + eps):\n",
        "            cnt += 1\n",
        "    vec_coverage.append(cnt/sim_runs)\n",
        "\n",
        "plt.plot(vec_n, vec_coverage, 'o')\n",
        "plt.xlabel(\"n\")\n",
        "plt.ylabel(\"coverage frac\")\n",
        "plt.show()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 279
        },
        "id": "cdJwoiqrM-Wo",
        "outputId": "b1120a97-e33b-4c37-ecc4-e9678de24c4d"
      },
      "source": [
        "# Chapter 4, problem 4, part c\n",
        "vec_n = np.array([1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000, 10000])\n",
        "len_interval = []\n",
        "for n in vec_n:\n",
        "    len_interval.append(2*calc_eps(n, 0.05))\n",
        "\n",
        "plt.plot(vec_n, len_interval, 'o')\n",
        "plt.xlabel(\"n\")\n",
        "plt.ylabel(\"interval length\")\n",
        "plt.show()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 279
        },
        "id": "JDRMvDve7vjE",
        "outputId": "932d27e0-3c64-413f-b799-9c4b0d2e18f1"
      },
      "source": [
        "# Chapter 4, problem 6\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "import math\n",
        "\n",
        "N = 10000\n",
        "vec_x = np.random.normal(0, 1, N)\n",
        "vec_t = np.linspace(0.5, 4, 100)\n",
        "\n",
        "y = abs(vec_x)\n",
        "expected_x1 = np.mean(y)\n",
        "expected_x2 = np.mean(pow(y, 2))\n",
        "expected_x3 = np.mean(pow(y, 3))\n",
        "markov1 = expected_x1/pow(vec_t, 1)\n",
        "markov2 = expected_x2/pow(vec_t, 2)\n",
        "markov3 = expected_x3/pow(vec_t, 3)\n",
        "mill = np.sqrt(2/math.pi) * np.exp(-0.5*vec_t**2) / vec_t\n",
        "\n",
        "# empirical p(|x| > t)\n",
        "vec_prob = []\n",
        "for t in vec_t:\n",
        "    cnt = 0\n",
        "    for x in vec_x:\n",
        "        if abs(x)>t:\n",
        "            cnt += 1\n",
        "    vec_prob.append(cnt/len(vec_x))\n",
        "\n",
        "plt.figure()\n",
        "plt.plot(vec_t, vec_prob, label='p( |x| > t )')\n",
        "plt.plot(vec_t, markov1, label='markov1')\n",
        "plt.plot(vec_t, markov2, label='markov2')\n",
        "plt.plot(vec_t, mill, label='mill')\n",
        "plt.xlabel(\"t\")\n",
        "plt.legend()\n",
        "plt.ylabel(\"Probability\")\n",
        "plt.show()\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 279
        },
        "id": "mIZv29hfqUYg",
        "outputId": "8d3f2fa3-5fe9-408f-a1bb-ead4da9af13e"
      },
      "source": [
        "# Chapter 4, problem 7\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "import math\n",
        "\n",
        "def calc_mill(t, n):\n",
        "    return np.sqrt(2/math.pi) * np.exp(-n * t**2) / (t * np.sqrt(n))\n",
        "\n",
        "def calc_cheb(t, n):\n",
        "    return 1 / (n * t**2)\n",
        "\n",
        "vec_t = np.linspace(0.3, 3, 100)\n",
        "\n",
        "n = 10\n",
        "mill = calc_mill(vec_t, n)\n",
        "cheb = calc_cheb(vec_t, n)\n",
        "\n",
        "plt.figure()\n",
        "plt.plot(vec_t, mill, label='Mill Bound')\n",
        "plt.plot(vec_t, cheb, label='Chebyshev Bound')\n",
        "plt.xlabel(\"t\")\n",
        "plt.legend()\n",
        "plt.ylabel(\"Probability\")\n",
        "plt.show()\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    }
  ]
}