UltimateGuidetoMachineLearning_Statistics.ipynb

{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "provenance": []
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "source": [
        "## Probability Theory"
      ],
      "metadata": {
        "id": "fNU0mtGDMrzV"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Expected value and variance"
      ],
      "metadata": {
        "id": "sSfj6TIAM0_R"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "\n",
        "# Generate 10000 random numbers from a normal distribution\n",
        "rand_nums = np.random.normal(loc=0, scale=1, size=10000)\n",
        "\n",
        "# Calculate the expected value and variance\n",
        "expected_value = np.mean(rand_nums)\n",
        "variance = np.var(rand_nums)\n",
        "\n",
        "print(f\"Expected value: {expected_value:.2f}\")\n",
        "print(f\"Variance: {variance:.2f}\")"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "cwyFpHAgMsBG",
        "outputId": "e477a480-1c71-4765-ac7c-824b0ee4cdf1"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Expected value: -0.01\n",
            "Variance: 1.03\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Conditional probability and Bayes' theorem"
      ],
      "metadata": {
        "id": "a4H6IGa5MzPE"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Define the probabilities\n",
        "p_disease = 0.01\n",
        "p_no_disease = 1 - p_disease\n",
        "p_positive_given_disease = 0.95\n",
        "p_positive_given_no_disease = 0.05\n",
        "\n",
        "# Calculate the probability of testing positive\n",
        "p_positive = (p_positive_given_disease * p_disease) + (p_positive_given_no_disease * p_no_disease)\n",
        "\n",
        "# Calculate the probability of having the disease given a positive test\n",
        "p_disease_given_positive = (p_positive_given_disease * p_disease) / p_positive\n",
        "\n",
        "print(f\"Probability of having the disease given a positive test: {p_disease_given_positive:.2f}\")"
      ],
      "metadata": {
        "id": "YxwMApSVMvTG"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Normal Distribution"
      ],
      "metadata": {
        "id": "uqFGSApwM5OR"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "from scipy.stats import norm\n",
        "\n",
        "# Create a normal distribution with mean 0 and standard deviation 1\n",
        "mu, sigma = 0, 1\n",
        "x = np.linspace(mu - 3*sigma, mu + 3*sigma, 100)\n",
        "plt.plot(x, norm.pdf(x, mu, sigma))\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "id": "fB9DGkVeM3dY",
        "outputId": "f68cd298-9ee6-4793-871b-266dacd42453"
      },
      "execution_count": null,
      "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"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Binomial Distribution"
      ],
      "metadata": {
        "id": "PefZqRIlM-TZ"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "from scipy.stats import binom\n",
        "\n",
        "# Create a binomial distribution with n=10 and p=0.5\n",
        "n, p = 10, 0.5\n",
        "x = np.arange(binom.ppf(0.01, n, p), binom.ppf(0.99, n, p))\n",
        "plt.plot(x, binom.pmf(x, n, p), 'bo', ms=8)\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "id": "5GYM-b-HM7v_",
        "outputId": "add56c36-d568-456d-f7bb-afd79b6677bb"
      },
      "execution_count": null,
      "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"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Poisson Distribution"
      ],
      "metadata": {
        "id": "MT4m7T-pNA48"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "from scipy.stats import poisson\n",
        "\n",
        "# Create a Poisson distribution with lambda=2\n",
        "mu = 2\n",
        "x = np.arange(poisson.ppf(0.01, mu), poisson.ppf(0.99, mu))\n",
        "plt.plot(x, poisson.pmf(x, mu), 'bo', ms=8)\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "id": "h8peaaUDM_oo",
        "outputId": "bed80c81-1fad-473b-e3c2-425f29a07852"
      },
      "execution_count": null,
      "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"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Exponential Distribution"
      ],
      "metadata": {
        "id": "B9by2A4JNEo4"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "from scipy.stats import expon\n",
        "\n",
        "# Create an exponential distribution with lambda=0.5\n",
        "rate = 0.5\n",
        "x = np.linspace(0, 10, 100)\n",
        "plt.plot(x, expon.pdf(x, scale=1/rate))\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "id": "zxwqjNHINDqr",
        "outputId": "d6144335-df9c-4ca6-faf3-18d7777650f9"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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zKwr4pyvewbMb9/NfOj5dRE7CYAp9BVBpZuPNLAQsBJb0H2Bms4B76CvzhtjHTC4fO6+Cq84s4TtPb+blrQe8jiMiCeKEhe6ciwCLgKeBTcDDzrkNZnaXmc2PDvs2kA08YmavmdmS4/w6GQQz41vvP5PKkTl85oFVusG0iAyKeXXcc3V1taupqfFk24li18F25t/9MsXZaTz26XPJSU/1OpKIeMzMVjrnqgd6T2eKDmPlhZn8+Kaz2H6gjX946DV9SSoib0uFPsydO6mIr15dxR83NfCtp1/3Oo6IDGMpXgeQE7t57jg272vhnhe3U1GYxY1zyr2OJCLDkAo9AZgZX58/lfqmDr78u/WU5mdw4WQd9ikib6UplwSREgzwo5tmUTkym0/fv4rX9+nyACLyVir0BJKTnsq9Hz2bzFCQj967gt3NHV5HEpFhRIWeYMbkZ3Dfx+fQ1hXhwz9fTlNb2OtIIjJMqNAT0DtKcvnpR6qpa+rg4/etoD0c8TqSiAwDKvQENXdCIT9YOJM1dc3c9utVhCO9XkcSEY+p0BPYvGkl/Nv7pvPilkbueHA1kR6VukgyU6EnuIVzyvmXq6p4av0+vvjoWp1NKpLEdBy6D9xy/njauyL857NbSA8F+b8LphEImNexRGSIqdB9YtF7JtHR3cOPX+i7hrpKXST5qNB9wsz4x8vOwAE/UamLJCUVuo+YGV+87AwM+PEL23DO8a/XTFepiyQJFbrPvLmnHjDjR8/X0hHu4TvXzyAlqO+/RfxOhe5DZsYXLjuDjFCQbz+9mfZwDz+8aRZpKUGvo4lIHGm3zcc+8+5JfO3qKp7ZuJ9P3FdDa5fOKBXxMxW6z330vPF85/oZ/GXbQW766TIOtHZ5HUlE4kSFngSumz2WxTfPZsv+Fq77yV/YdbDd60giEgcq9CRx8TtGcf8n5tLc0c21P3mF1+qavY4kIjGmQk8is8eN4NFPnUtGKMjCxa/yh/V7vY4kIjGkQk8yk0Zm89tPn0dVSS633b+Ke17sO15dRBKfCj0JFWWn8cAn53LF9BL+/anX+cIja+ns7vE6loicJh2HnqTSU4P8cOEsJo/M4Xt/3EJtYyuLb57NqNx0r6OJyCnSHnoSCwSMOy6p5L8+NJut+1u4+ocvU7PjkNexROQUqdCFedNG89inzyUzFGTh4mXc+/IbmlcXSUAqdAFgyuhcHl90Pu+eMpK7ntjI3//Pap1ZKpJgVOhyVF5GKvd8aDZfnHcGS9ft5aof/Jn1uw97HUtEBkmFLm8RCBifvmgSD956Dp3dvVz747/wi1c0BSOSCFToMqA54wt46o4LuHByEV///UY+9ssVNLR0eh1LRN7GoArdzOaZ2WYzqzWzOwd4/0IzW2VmETO7LvYxxQsjskL89MPVfH3+VF7ddpDLvvcSf1i/z+tYInIcJyx0MwsCdwOXA1XAjWZWdcywXcBHgQdiHVC8ZWZ85NwKnrz9fEpHZPCpX6/kcw+9xuH2bq+jicgxBrOHPgeodc5td86FgQeBBf0HOOd2OOfWAr1xyCjDwKSROTx223nc/p5JPL5mD5d+70X+tGm/17FEpJ/BFHopUNdvuT667qSZ2a1mVmNmNY2NjafyK8RDoZQAn3vvGTz+mfMoyApxy3013PHgal1jXWSYGNIvRZ1zi51z1c656uLi4qHctMTQtNI8liw6nzsurmTpur1c8t0XeaSmTkfCiHhsMIW+Gyjrtzw2uk6SWCglwD9cOpmn7riAypHZ/OOja/nA4mVs3tfidTSRpDWYQl8BVJrZeDMLAQuBJfGNJYli0sgcHrr1HL557XS27G/hih/8mW88sZGWTn1pKjLUTljozrkIsAh4GtgEPOyc22Bmd5nZfAAzO9vM6oHrgXvMbEM8Q8vwEggYC+eU8/znL+KG6rHc+8obvPs7L/LQil309GoaRmSomFfzntXV1a6mpsaTbUt8ralr5q4nNrJyZxNTx+Ty5SurOGdiodexRHzBzFY656oHek9nikrMzSjL59FPncMPbpxFU1uYG3+6jI//coXm10XiTIUucWFmzJ8xhue+cBF3Xj6FFTsOcfn3X+LzD6+h7lC71/FEfElTLjIkmtrC3P18Lb9athPnHB84u4xF765kdJ7ukCRyMt5uykWFLkNq3+FOfvjcVh5aUdf3ZerZZXzqXRMZk5/hdTSRhKBCl2Gn7lA7dz9fy6Mr6zGD62aP5e8unEhFUZbX0USGNRW6DFv1Te385IVtPFJTT6S3l8unl3DbuyYyrTTP62giw5IKXYa9hiOd3PvKDu5ftpOWrghzJxTwifMn8J4pIwkEzOt4IsOGCl0SxpHObh76ax2/eOUN9hzuZHxRFjfPHcd11WPJTU/1Op6I51ToknC6e3p5av0+fvnKG6za1UxmKMj7ZpXywXeOo2pMrtfxRDyjQpeEtq7+ML96dQePr9lDONLLzLJ8bppTzpVnlpCVluJ1PJEhpUIXX2huD/ObVbt5YPlOtjW2kRUKctWZY7i+eiyzx43ATHPt4n8qdPEV5xw1O5t4pKaOJ9bupT3cQ0VhJtfMKuV9s0oZV6hDH8W/VOjiW21dEZau28tvV+/m1e0HcQ5mluVz9YwxXHVmCaNydSaq+IsKXZLCnuYOlqzZw5LX9rBx7xHM4OyKAi6fNpp500ZTkqezUSXxqdAl6dQ2tPLE2j08tW4fm/f3XeVxZlk+7506ivdWjWbSyGyPE4qcGhW6JLXahlb+sH4vz2zcz9r6wwCML8riPVNGcvGUkVRXFBBK0YVHJTGo0EWi9jR38MdN+/nTpgZe3XaQcE8v2WkpnDuxkHedUcyFlcWUFWR6HVPkuFToIgNo64rwSu0BXtjSyIubG9nd3AFARWEm500q4vxJRcydUMiIrJDHSUX+RoUucgLOObY1tvLnrQd4eesBlm0/SFu4B4B3lORyzoRC5owvYM74AgpU8OIhFbrISeru6WVtfTOvbjvIX7YdZNWuJjq7ewGYNDKb6nEjmD1uBNUVBVQUZuqkJhkyKnSR0xSO9LJudzPLth+iZschVu5s4khnBIARmanMLMtnVvkIZpTlc2ZpnqZpJG7ertB1IQyRQQilBJg9roDZ4woA6O111Da2snJnE6t3NbF6VzPPb248Or68IJPppXlMK81jemkeU8fkquQl7lToIqcgEDAmj8ph8qgcbpxTDvRd+nd9/WHW1B9mbX0za+qbeXLd3qM/U5KXTlVJLlNKcpgyOpcpo3OoKMoiNahDJiU2VOgiMZKbnsq5k4o4d1LR0XXN7WHW7z7Cpr1H2Lj3CBv2HObFLY1EevumOkPBABOKs5g8KofKkdlMGplN5ahsyguydGy8nDQVukgc5WeGOL+yiPMr/1byXZEeahtaeX1vC1saWtiyr4WVO5tYsmbP0THBgFFekMnE4izGF2UxviibiqJMxhdlMSonXXdxkgGp0EWGWFpKkKlj8pg65q33TW3rirC9sY2tDS1sa2xle2Mb2xvbeGnrAcKR3n4/H2BcYSblBVmUF2RSVpBBeUEmY0dkMnZEhq4Rn8T0Jy8yTGSlpTB9bB7Tx7616Ht7HXsOd/DGgTZ2HGxn18G+57pD7fxl2wHao8fLv2lEZipj8jMozc9gTH4GY/LTKcnrex6dl8HInDTN2/uUCl1kmAsELLr3nckFlW99zznHgdYw9U3t1Dd1RB/t7GnuYMfBNl6pPXD0BKk3mUFRdhqjc9MZlZvGyNx0RuakMTKn77k4+ijKTtM8foJRoYskMDM7WsCzykcMOOZIZzd7mjvY29zJviOd7Dvc99jf0snu5k5W7WrmUFt4wJ/Ny0ilMDtEUVYahdkhCrNDFGSlUZgVYkRWqO85M8SIrFRGZIZITw3G8+PKCajQRXwuNz2V3NGpTBl9/JtrhyO9HGjtoqGliwMtXTS2dtHY0sXB1i4OtIZpbO1ia0Mry7Z30dzRzfHOR8xIDZKfmUp+Zoj8jFTyM1PJy+h75PZ7zk1POfqck55KTnoKGalBnXF7mlToIkIoJRCdbz/xTUAiPb00d3TT1BbmYFuYprYwTe3dNLWHaW7ve93cHuZwRze1Da00d3RzuKP7LV/sDiQlYGSnp5Cd1vfIib7OSvvbc9/rIJmhFLLefA6lkBEKkhkKHn2dEQqSkRokmGRHAw2q0M1sHvB9IAj8zDn3zWPeTwN+BcwGDgIfcM7tiG1UERkOUoIBirL75tgrTzz8qM7uHo5Ey/1IZ4Qjnd0c6eimpTMSfXTT2hWhtTPCkc4IbV0RDrSG2XGwnbauvuVjvw84kbSUAJnRck8PBUlP6Sv79NQA6SlB0lODpKUGSE8NRpf7XqelBPoeR18HCUXXhaKPN8eEgn3vpQYt+hwgFAx4cmjpCQvdzILA3cClQD2wwsyWOOc29ht2C9DknJtkZguBbwEfiEdgEUlM6al9BTryNO7z2tvr6OjuoS0coa2rh/ZwhPZwD21dETrCPbSFe+gIR+jo7qE93ENHuIfO7p6jy53dvXR29/1cU1s3nZEeuqLruiJ9z2+e9HW6UgLWV+5HS95ICfYV/2cvmczVM8bEZDtv2eYgxswBap1z2wHM7EFgAdC/0BcAX4u+fhT4kZmZ8+rKXyLiS4GAHZ16ISc+24j09BLu6aWzu5euSA/hSO/Rsg9Heo8ud0X6xoUjvXRHn8PRdd391nf3OLoivUR6/racn5kal+yDKfRSoK7fcj3wzuONcc5FzOwwUAgc6D/IzG4FbgUoLy8/xcgiIvGTEgyQEgyQmYDXUhvSg0ydc4udc9XOueri4uKh3LSIiO8NptB3A2X9lsdG1w04xsxSgDz6vhwVEZEhMphCXwFUmtl4MwsBC4Elx4xZAnwk+vo64DnNn4uIDK0TzqFH58QXAU/Td9jivc65DWZ2F1DjnFsC/Bz4bzOrBQ7RV/oiIjKEBnUcunNuKbD0mHVf6fe6E7g+ttFERORk6Mo7IiI+oUIXEfEJFbqIiE+YVwejmFkjsPMUf7yIY05aSgL6zMlBnzk5nM5nHuecG/BEHs8K/XSYWY1zrtrrHENJnzk56DMnh3h9Zk25iIj4hApdRMQnErXQF3sdwAP6zMlBnzk5xOUzJ+QcuoiI/G+JuocuIiLHUKGLiPhEwhW6mc0zs81mVmtmd3qdJ97MrMzMnjezjWa2wczu8DrTUDCzoJmtNrMnvM4yFMws38weNbPXzWyTmZ3jdaZ4M7N/iP6dXm9m/2Nmp35vumHKzO41swYzW99vXYGZPWtmW6PPI2K1vYQq9H73N70cqAJuNLMqb1PFXQT4vHOuCpgLfCYJPjPAHcAmr0MMoe8Df3DOTQFm4PPPbmalwO1AtXNuGn1XcvXjVVp/Ccw7Zt2dwJ+cc5XAn6LLMZFQhU6/+5s658LAm/c39S3n3F7n3Kro6xb6/kMv9TZVfJnZWOBK4GdeZxkKZpYHXEjfZahxzoWdc82ehhoaKUBG9KY4mcAej/PEnHPuJfouKd7fAuC+6Ov7gGtitb1EK/SB7m/q63Lrz8wqgFnAco+jxNv/A74I9HqcY6iMBxqBX0SnmX5mZlleh4on59xu4DvALmAvcNg594y3qYbMKOfc3ujrfcCoWP3iRCv0pGVm2cBvgM865454nSdezOwqoME5t9LrLEMoBTgL+IlzbhbQRgz/GT4cReeNF9D3P7MxQJaZfcjbVEMveme3mB07nmiFPpj7m/qOmaXSV+b3O+ce8zpPnJ0HzDezHfRNqb3HzH7tbaS4qwfqnXNv/svrUfoK3s8uAd5wzjU657qBx4BzPc40VPabWQlA9LkhVr840Qp9MPc39RUzM/rmVjc5577rdZ54c879H+fcWOdcBX1/vs8553y95+ac2wfUmdkZ0VUXAxs9jDQUdgFzzSwz+nf8Ynz+RXA//e/B/BHg8Vj94kHdgm64ON79TT2OFW/nATcD68zstei6f4reFlD84++B+6M7KtuBj3mcJ66cc8vN7FFgFX1Hcq3Gh5cAMLP/AS4CisysHvgq8E3gYTO7hb5LiN8Qs+3p1H8REX9ItCkXERE5DhW6iIhPqNBFRHxChS4i4hMqdBERn1Chi4j4hApdRMQn/j/59vlZKo9akQAAAABJRU5ErkJggg==\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Gamma Distribution"
      ],
      "metadata": {
        "id": "S7pXoZ82NJLg"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "from scipy.stats import gamma\n",
        "\n",
        "# Create a gamma distribution with k=2 and lambda=0.5\n",
        "k, theta = 2, 1/0.5\n",
        "x = np.linspace(gamma.ppf(0.01, k), gamma.ppf(0.99, k), 100)\n",
        "plt.plot(x, gamma.pdf(x, k, scale=theta))\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "id": "7Tz6DcxWNHwq",
        "outputId": "de963b7d-b347-4193-b89c-1bf7847d55fa"
      },
      "execution_count": null,
      "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"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Descriptive Statistics"
      ],
      "metadata": {
        "id": "dq8ilduuP5jw"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Measures of central tendency and variability"
      ],
      "metadata": {
        "id": "SBVuX8NQP7Zq"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "#Mean\n",
        "import numpy as np\n",
        "data = np.array([2, 4, 6, 8, 10])\n",
        "mean = np.mean(data)\n",
        "print(\"Mean:\", mean)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "_pogj0d8NKho",
        "outputId": "20eb7aa0-4a75-4f86-d64a-9cfcdd7ac271"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Mean: 6.0\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#Median\n",
        "import numpy as np\n",
        "data = np.array([2, 4, 6, 8, 10])\n",
        "median = np.median(data)\n",
        "print(\"Median:\", median)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "s-RKAn3hP9oM",
        "outputId": "1dcfba4e-2513-437d-d785-9fb5f697a40d"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Median: 6.0\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#Mode\n",
        "import scipy.stats as stats\n",
        "data = [2, 4, 6, 8, 10, 10]\n",
        "mode = stats.mode(data)\n",
        "print(\"Mode:\", mode)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "TkgCiTRqP_bq",
        "outputId": "23d5d670-0fe3-4b30-8340-2956a08af33d"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Mode: ModeResult(mode=array([10]), count=array([2]))\n"
          ]
        },
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "<ipython-input-9-a11b655e42a6>:4: FutureWarning: Unlike other reduction functions (e.g. `skew`, `kurtosis`), the default behavior of `mode` typically preserves the axis it acts along. In SciPy 1.11.0, this behavior will change: the default value of `keepdims` will become False, the `axis` over which the statistic is taken will be eliminated, and the value None will no longer be accepted. Set `keepdims` to True or False to avoid this warning.\n",
            "  mode = stats.mode(data)\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#Range\n",
        "import numpy as np\n",
        "data = np.array([2, 4, 6, 8, 10])\n",
        "range = np.max(data) - np.min(data)\n",
        "print(\"Range:\", range)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "yCPgeGBpQBJV",
        "outputId": "155fb5be-6394-4ee3-cec4-705311878230"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Range: 8\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#Variance\n",
        "import numpy as np\n",
        "data = np.array([2, 4, 6, 8, 10])\n",
        "variance = np.var(data, ddof=1)\n",
        "print(\"Variance:\", variance)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "U84xI1mrQCyN",
        "outputId": "050fce71-4574-4887-bd0a-c87c7ec03272"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Variance: 10.0\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#Std\n",
        "import numpy as np\n",
        "data = np.array([2, 4, 6, 8, 10])\n",
        "std_dev = np.std(data, ddof=1)\n",
        "print(\"Standard Deviation:\", std_dev)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "rSKxKpKYQE2l",
        "outputId": "ea298fd7-e4ee-46c4-e1f8-04fd3a518314"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Standard Deviation: 3.1622776601683795\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Inferential Statistics"
      ],
      "metadata": {
        "id": "PlycFmx0UdYR"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Hypothesis Testing"
      ],
      "metadata": {
        "id": "WO0f7qDcUeW8"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "from scipy.stats import ttest_ind\n",
        "\n",
        "# Generate some sample data\n",
        "group1 = np.random.normal(10, 2, size=50)\n",
        "group2 = np.random.normal(12, 2, size=50)\n",
        "\n",
        "# Perform a two-sample t-test\n",
        "t, p = ttest_ind(group1, group2)\n",
        "\n",
        "print(\"t-statistic:\", t)\n",
        "print(\"p-value:\", p)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "5cACcRnlQGge",
        "outputId": "59f0e676-f2f1-41e3-8f75-bc633de423f4"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "t-statistic: -4.822809251458353\n",
            "p-value: 5.188860276395987e-06\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Confidence Intervals"
      ],
      "metadata": {
        "id": "zZRGGffeUkmM"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "from scipy.stats import t\n",
        "\n",
        "# Generate some sample data\n",
        "data = np.random.normal(0, 1, size=100)\n",
        "\n",
        "# Calculate the sample mean and standard deviation\n",
        "xbar = np.mean(data)\n",
        "s = np.std(data, ddof=1)\n",
        "\n",
        "# Calculate the t-value for a 95% confidence level with n-1 degrees of freedom\n",
        "tval = t.ppf(0.975, len(data)-1)\n",
        "\n",
        "# Calculate the confidence interval\n",
        "ci = (xbar - tval*s/np.sqrt(len(data)), xbar + tval*s/np.sqrt(len(data)))\n",
        "\n",
        "print(ci)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "mYhwlxR9Uhud",
        "outputId": "7fe8ef08-3649-4c11-ef03-cecb8eeb3974"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "(-0.1594362734433123, 0.24420582753756548)\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Power Analysis"
      ],
      "metadata": {
        "id": "zxtLFqQsUoh0"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import statsmodels.stats.power as smp\n",
        "\n",
        "# Set parameters for power analysis\n",
        "effect_size = 0.5\n",
        "alpha = 0.05\n",
        "power = 0.8\n",
        "\n",
        "# Conduct power analysis\n",
        "nobs = smp.tt_ind_solve_power(effect_size=effect_size, alpha=alpha, power=power)\n",
        "\n",
        "print(\"Sample size: \", round(nobs))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "mIlXZD_gUjMH",
        "outputId": "83d2a30e-8660-4e3f-bd76-147cbd675ba1"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Sample size:  64\n"
          ]
        },
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "/usr/local/lib/python3.8/dist-packages/scipy/stats/_continuous_distns.py:6832: RuntimeWarning: invalid value encountered in _nct_sf\n",
            "  return np.clip(_boost._nct_sf(x, df, nc), 0, 1)\n",
            "/usr/local/lib/python3.8/dist-packages/scipy/stats/_continuous_distns.py:6826: RuntimeWarning: invalid value encountered in _nct_cdf\n",
            "  return np.clip(_boost._nct_cdf(x, df, nc), 0, 1)\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Generalized Linear Models"
      ],
      "metadata": {
        "id": "_2N0cIfJy_n4"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import statsmodels.api as sm\n",
        "\n",
        "# Load data\n",
        "data = sm.datasets.get_rdataset(\"titanic\", \"vcd\").data\n",
        "\n",
        "# Fit logistic regression model\n",
        "model = sm.formula.glm(\"survived ~ age + sex\", data=data, family=sm.families.Binomial()).fit()\n",
        "\n",
        "# Print model summary\n",
        "print(model.summary())"
      ],
      "metadata": {
        "id": "2tHi_lE3UqkK"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# Poisson\n",
        "import statsmodels.api as sm\n",
        "import pandas as pd\n",
        "\n",
        "# Load data\n",
        "data = pd.read_csv(\"data.csv\")\n",
        "\n",
        "# Fit Poisson regression model\n",
        "model = sm.GLM(data[\"y\"], data[[\"x1\", \"x2\", \"x3\"]], family=sm.families.Poisson()).fit()\n",
        "\n",
        "# Print model summary\n",
        "print(model.summary())"
      ],
      "metadata": {
        "id": "v_scYqjEzChM"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "!pip install statsmodels"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "rc2WDnYB5vd1",
        "outputId": "31c39645-8d5b-4d52-dad6-411a7439b7fd"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Looking in indexes: https://pypi.org/simple, https://us-python.pkg.dev/colab-wheels/public/simple/\n",
            "Requirement already satisfied: statsmodels in /usr/local/lib/python3.8/dist-packages (0.13.5)\n",
            "Requirement already satisfied: numpy>=1.17 in /usr/local/lib/python3.8/dist-packages (from statsmodels) (1.22.1)\n",
            "Requirement already satisfied: packaging>=21.3 in /usr/local/lib/python3.8/dist-packages (from statsmodels) (23.0)\n",
            "Requirement already satisfied: patsy>=0.5.2 in /usr/local/lib/python3.8/dist-packages (from statsmodels) (0.5.3)\n",
            "Requirement already satisfied: pandas>=0.25 in /usr/local/lib/python3.8/dist-packages (from statsmodels) (1.3.5)\n",
            "Requirement already satisfied: scipy>=1.3 in /usr/local/lib/python3.8/dist-packages (from statsmodels) (1.7.3)\n",
            "Requirement already satisfied: python-dateutil>=2.7.3 in /usr/local/lib/python3.8/dist-packages (from pandas>=0.25->statsmodels) (2.8.2)\n",
            "Requirement already satisfied: pytz>=2017.3 in /usr/local/lib/python3.8/dist-packages (from pandas>=0.25->statsmodels) (2022.7.1)\n",
            "Requirement already satisfied: six in /usr/local/lib/python3.8/dist-packages (from patsy>=0.5.2->statsmodels) (1.15.0)\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "import pandas as pd\n",
        "import statsmodels.genmod.bayes_mixed_glm as smf\n",
        "\n",
        "# Load the iris dataset\n",
        "iris = pd.read_csv('https://archive.ics.uci.edu/ml/machine-learning-databases/iris/iris.data',\n",
        "                   names=['sepal_length', 'sepal_width', 'petal_length', 'petal_width', 'species'])\n",
        "\n",
        "# Fit a Bayesian linear regression model\n",
        "random = {\"a\": \"1 + petal_width\", \"b\": \"1 + petal_width*sepal_length\"}\n",
        "\n",
        "\n",
        "model = smf.BinomialBayesMixedGLM.from_formula('sepal_length ~ petal_length + C(species)', random, iris)\n",
        "\n",
        "# Print the summary of the model\n",
        "print(model.summary())\n"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 416
        },
        "id": "MdFS5e4338qx",
        "outputId": "330a8384-b6ea-47a1-cd83-ce5ecb6df226"
      },
      "execution_count": 22,
      "outputs": [
        {
          "output_type": "error",
          "ename": "ValueError",
          "evalue": "ignored",
          "traceback": [
            "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
            "\u001b[0;31mValueError\u001b[0m                                Traceback (most recent call last)",
            "\u001b[0;32m<ipython-input-22-e63fbe7cb58c>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[1;32m     10\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     11\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 12\u001b[0;31m \u001b[0mmodel\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msmf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mBinomialBayesMixedGLM\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfrom_formula\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'sepal_length ~ petal_length + C(species)'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mrandom\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0miris\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     13\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     14\u001b[0m \u001b[0;31m# Print the summary of the model\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
            "\u001b[0;32m/usr/local/lib/python3.8/dist-packages/statsmodels/genmod/bayes_mixed_glm.py\u001b[0m in \u001b[0;36mfrom_formula\u001b[0;34m(cls, formula, vc_formulas, data, vcp_p, fe_p)\u001b[0m\n\u001b[1;32m   1043\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1044\u001b[0m         \u001b[0;31m# Copy over to the intended class structure\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1045\u001b[0;31m         mod = BinomialBayesMixedGLM(\n\u001b[0m\u001b[1;32m   1046\u001b[0m             \u001b[0mx\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mendog\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1047\u001b[0m             \u001b[0mx\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mexog\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
            "\u001b[0;32m/usr/local/lib/python3.8/dist-packages/statsmodels/genmod/bayes_mixed_glm.py\u001b[0m in \u001b[0;36m__init__\u001b[0;34m(self, endog, exog, exog_vc, ident, vcp_p, fe_p, fep_names, vcp_names, vc_names)\u001b[0m\n\u001b[1;32m   1033\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mall\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0munique\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mendog\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mr_\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1034\u001b[0m             \u001b[0mmsg\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m\"endog values must be 0 and 1, and not all identical\"\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1035\u001b[0;31m             \u001b[0;32mraise\u001b[0m \u001b[0mValueError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mmsg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1036\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1037\u001b[0m     \u001b[0;34m@\u001b[0m\u001b[0mclassmethod\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
            "\u001b[0;31mValueError\u001b[0m: endog values must be 0 and 1, and not all identical"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "import pandas as pd\n",
        "import statsmodels.api as sm\n",
        "\n",
        "# Load the iris dataset\n",
        "iris = pd.read_csv('https://archive.ics.uci.edu/ml/machine-learning-databases/iris/iris.data',\n",
        "                   names=['sepal_length', 'sepal_width', 'petal_length', 'petal_width', 'species'])\n",
        "\n",
        "# Fit a mixed-effects linear regression model\n",
        "random = {'sepal_width': '1 + petal_width', 'petal_length': '1 + petal_width*sepal_length'}\n",
        "model = sm.MixedLM.from_formula('sepal_length ~ petal_length + C(species)', data=iris, re_formula='1', vc_formula=random,groups='species')\n",
        "\n",
        "# Print the summary of the model\n",
        "result = model.fit()\n",
        "result.summary()"
      ],
      "metadata": {
        "id": "0PiC20c73_m8",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 439
        },
        "outputId": "80e8233c-cf7f-412e-c750-e595334d1d2e"
      },
      "execution_count": 31,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "/usr/local/lib/python3.8/dist-packages/statsmodels/regression/mixed_linear_model.py:2237: ConvergenceWarning: The MLE may be on the boundary of the parameter space.\n",
            "  warnings.warn(msg, ConvergenceWarning)\n",
            "/usr/local/lib/python3.8/dist-packages/statsmodels/regression/mixed_linear_model.py:2261: ConvergenceWarning: The Hessian matrix at the estimated parameter values is not positive definite.\n",
            "  warnings.warn(msg, ConvergenceWarning)\n",
            "/usr/local/lib/python3.8/dist-packages/statsmodels/regression/mixed_linear_model.py:2705: RuntimeWarning: invalid value encountered in sqrt\n",
            "  sdf[0:self.k_fe, 1] = np.sqrt(np.diag(self.cov_params()[0:self.k_fe]))\n"
          ]
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<class 'statsmodels.iolib.summary2.Summary'>\n",
              "\"\"\"\n",
              "                 Mixed Linear Model Regression Results\n",
              "=======================================================================\n",
              "Model:                 MixedLM     Dependent Variable:     sepal_length\n",
              "No. Observations:      150         Method:                 REML        \n",
              "No. Groups:            3           Scale:                  0.0000      \n",
              "Min. group size:       50          Log-Likelihood:         1675.3089   \n",
              "Max. group size:       50          Converged:              Yes         \n",
              "Mean group size:       50.0                                            \n",
              "-----------------------------------------------------------------------\n",
              "                              Coef.  Std.Err.   z   P>|z| [0.025 0.975]\n",
              "-----------------------------------------------------------------------\n",
              "Intercept                      0.003                                   \n",
              "C(species)[T.Iris-versicolor] -0.028                                   \n",
              "C(species)[T.Iris-virginica]   0.009                                   \n",
              "petal_length                   0.000    0.000 0.000 1.000 -0.000  0.000\n",
              "species Var                    0.000                                   \n",
              "petal_length Var               0.022  786.985                          \n",
              "sepal_width Var                0.000                                   \n",
              "=======================================================================\n",
              "\n",
              "\"\"\""
            ],
            "text/html": [
              "<table class=\"simpletable\">\n",
              "<tr>\n",
              "       <td>Model:</td>       <td>MixedLM</td> <td>Dependent Variable:</td> <td>sepal_length</td>\n",
              "</tr>\n",
              "<tr>\n",
              "  <td>No. Observations:</td>   <td>150</td>         <td>Method:</td>           <td>REML</td>    \n",
              "</tr>\n",
              "<tr>\n",
              "     <td>No. Groups:</td>       <td>3</td>          <td>Scale:</td>           <td>0.0000</td>   \n",
              "</tr>\n",
              "<tr>\n",
              "  <td>Min. group size:</td>    <td>50</td>      <td>Log-Likelihood:</td>     <td>1675.3089</td> \n",
              "</tr>\n",
              "<tr>\n",
              "  <td>Max. group size:</td>    <td>50</td>        <td>Converged:</td>           <td>Yes</td>    \n",
              "</tr>\n",
              "<tr>\n",
              "  <td>Mean group size:</td>   <td>50.0</td>            <td></td>                 <td></td>      \n",
              "</tr>\n",
              "</table>\n",
              "<table class=\"simpletable\">\n",
              "<tr>\n",
              "                <td></td>                 <th>Coef.</th> <th>Std.Err.</th>   <th>z</th>   <th>P>|z|</th> <th>[0.025</th> <th>0.975]</th>\n",
              "</tr>\n",
              "<tr>\n",
              "  <th>Intercept</th>                      <td>0.003</td>     <td></td>       <td></td>      <td></td>       <td></td>       <td></td>   \n",
              "</tr>\n",
              "<tr>\n",
              "  <th>C(species)[T.Iris-versicolor]</th> <td>-0.028</td>     <td></td>       <td></td>      <td></td>       <td></td>       <td></td>   \n",
              "</tr>\n",
              "<tr>\n",
              "  <th>C(species)[T.Iris-virginica]</th>   <td>0.009</td>     <td></td>       <td></td>      <td></td>       <td></td>       <td></td>   \n",
              "</tr>\n",
              "<tr>\n",
              "  <th>petal_length</th>                   <td>0.000</td>   <td>0.000</td>  <td>0.000</td> <td>1.000</td> <td>-0.000</td>  <td>0.000</td>\n",
              "</tr>\n",
              "<tr>\n",
              "  <th>species Var</th>                    <td>0.000</td>     <td></td>       <td></td>      <td></td>       <td></td>       <td></td>   \n",
              "</tr>\n",
              "<tr>\n",
              "  <th>petal_length Var</th>               <td>0.022</td>  <td>786.985</td>   <td></td>      <td></td>       <td></td>       <td></td>   \n",
              "</tr>\n",
              "<tr>\n",
              "  <th>sepal_width Var</th>                <td>0.000</td>     <td></td>       <td></td>      <td></td>       <td></td>       <td></td>   \n",
              "</tr>\n",
              "</table>"
            ]
          },
          "metadata": {},
          "execution_count": 31
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Markov Chain Monte Carlo (MCMC)"
      ],
      "metadata": {
        "id": "espnB8QTkbXF"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "# Define the target distribution\n",
        "def target(x):\n",
        "    return np.exp(-0.5*(x-2)**2) + np.exp(-0.5*(x+2)**2)\n",
        "\n",
        "# Define the proposal distribution\n",
        "def proposal(x, sigma=1):\n",
        "    return np.random.normal(x, sigma)\n",
        "\n",
        "# Initialize the Markov chain\n",
        "x = 0\n",
        "samples = []\n",
        "\n",
        "# Run the algorithm\n",
        "for i in range(10000):\n",
        "    # Propose a new state\n",
        "    y = proposal(x)\n",
        "\n",
        "    # Compute the acceptance probability\n",
        "    alpha = min(1, target(y)/target(x))\n",
        "\n",
        "    # Accept or reject the proposed state\n",
        "    if np.random.rand() < alpha:\n",
        "        x = y\n",
        "\n",
        "    # Add the sample to the list of samples\n",
        "    samples.append(x)\n",
        "\n",
        "# Plot the samples\n",
        "plt.hist(samples, bins=50, density=True, label='Estimated distribution')\n",
        "plt.legend()\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "id": "fRxnzQY6dWtz",
        "outputId": "e14551cc-3fde-46be-af10-21fe444ec76a"
      },
      "execution_count": 32,
      "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"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "from scipy.stats import gaussian_kde\n",
        "\n",
        "# Estimate the posterior distribution using kernel density estimation\n",
        "kde = gaussian_kde(samples)\n",
        "\n",
        "# Define the range of the x-axis for the plot\n",
        "x_min = min(samples)\n",
        "x_max = max(samples)\n",
        "x_range = np.linspace(x_min, x_max, 1000)\n",
        "\n",
        "# Plot the posterior distribution\n",
        "plt.plot(x_range, kde(x_range), label='Posterior distribution')\n",
        "plt.legend()\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "id": "t3QpB3jHkZtR",
        "outputId": "6bc3d6c0-545f-4741-ee55-f71c7a1983f9"
      },
      "execution_count": 33,
      "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"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Compute the 95% credible interval\n",
        "lower_bound = np.percentile(samples, 2.5)\n",
        "upper_bound = np.percentile(samples, 97.5)\n",
        "print(f\"95% credible interval: [{lower_bound}, {upper_bound}]\")\n"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "FzuC4QzvkgdF",
        "outputId": "582027c1-24ab-43e4-90ad-a11a138a8d48"
      },
      "execution_count": 34,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "95% credible interval: [-3.5701802296026166, 3.762132517614992]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [],
      "metadata": {
        "id": "af1hm4BfkjI6"
      },
      "execution_count": null,
      "outputs": []
    }
  ]
}