{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "name": "Chapter 2 exercises.ipynb",
      "provenance": [],
      "collapsed_sections": [
        "mitu2snKRzzg"
      ]
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "code",
      "metadata": {
        "id": "wQEDjCdS-iAo"
      },
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "YyO3mta3-xp3"
      },
      "source": [
        "# Section 2.2\n",
        "\n",
        "## Exercise 2.1"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "O3-sg2Hs-y7b",
        "outputId": "53bdb873-3616-429f-eaa1-b926c3dc4b49"
      },
      "source": [
        "xi = 0.005\n",
        "sigma = 0.05\n",
        "\n",
        "m1 = 100\n",
        "m2 = 1000\n",
        "m3 = 10000\n",
        "eps = np.random.normal(size=m3)\n",
        "R1 = np.exp(xi+sigma*eps[:m1])-1\n",
        "R2 = np.exp(xi+sigma*eps[:m2])-1\n",
        "R3 = np.exp(xi+sigma*eps)-1\n",
        "\n",
        "# theoretical values\n",
        "print(np.exp(xi + 0.5 * sigma**2)-1)\n",
        "print(np.exp(xi + 0.5 * sigma**2)*np.sqrt(np.exp(sigma**2) - 1))\n",
        "\n",
        "# simulated values\n",
        "print(np.mean(R1), np.mean(R2), np.mean(R3))\n",
        "print(np.std(R1), np.std(R2), np.std(R3))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "0.006269572003761992\n",
            "0.05034494090854318\n",
            "0.004844023735673695 0.0038784473576364046 0.005831923224707375\n",
            "0.04823652245532138 0.04932223919014895 0.05025515035212742\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "XKLfSQHGAUWs"
      },
      "source": [
        "## Exercise 2.2"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 269
        },
        "id": "YebNp3gNAa2M",
        "outputId": "33251605-bb48-47b3-da19-6957575792f6"
      },
      "source": [
        "from scipy.stats import norm\n",
        "xi = 0.005\n",
        "sigma = 0.05\n",
        "n = 1000\n",
        "u = (np.arange(n) + 0.5)/n\n",
        "\n",
        "q1 = xi+sigma*norm.ppf(u)\n",
        "q2 = np.exp(xi+sigma*norm.ppf(u))-1\n",
        "\n",
        "plt.plot(q1, q2)\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": 265
        },
        "id": "ahSrin3oBoM_",
        "outputId": "2e514ea6-115e-4664-f52d-57a04a1e8827"
      },
      "source": [
        "sigma = 0.25\n",
        "q1 = xi+sigma*norm.ppf(u)\n",
        "q2 = np.exp(xi+sigma*norm.ppf(u))-1\n",
        "\n",
        "plt.plot(q1, q2)\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": "markdown",
      "metadata": {
        "id": "KnPV7_07CGBf"
      },
      "source": [
        "## Exercise 2.4"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "id": "qccQuuYrCHtd",
        "outputId": "bf424d49-58fe-47fc-ac4f-33fd5bfeffcc"
      },
      "source": [
        "xi = 0.005\n",
        "sigma = 0.05\n",
        "m = 10000\n",
        "\n",
        "X_star = xi+sigma*norm.rvs(size=m)\n",
        "Y_star = np.exp(xi+sigma*norm.rvs(size=m))-1\n",
        "\n",
        "plt.plot(sorted(X_star), sorted(Y_star))\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": {
        "id": "rx4XP6PHC8BN",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "outputId": "5049eb65-70b2-407a-e7cb-cbda675c584b"
      },
      "source": [
        "sigma = 0.25\n",
        "X_star = xi+sigma*norm.rvs(size=m)\n",
        "Y_star = np.exp(xi+sigma*norm.rvs(size=m))-1\n",
        "\n",
        "plt.plot(sorted(X_star), sorted(Y_star))\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": "markdown",
      "metadata": {
        "id": "4NAwTuhXDTR8"
      },
      "source": [
        "## Exercise 2.5"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "id": "ldKEhkhLDUvx",
        "outputId": "943a1913-6af7-4dc1-c262-5807ff5bacbc"
      },
      "source": [
        "from scipy.stats import kde\n",
        "\n",
        "xi = 0.005\n",
        "sigma = 0.05\n",
        "\n",
        "m = 1000000\n",
        "eps = norm.rvs(size=m)\n",
        "R = np.exp(xi+sigma*eps)-1\n",
        "\n",
        "density = kde.gaussian_kde(R)\n",
        "x = np.linspace(-0.2, 0.2, 200)\n",
        "y = density(x)\n",
        "plt.plot(x, y, label='full')\n",
        "\n",
        "density1 = kde.gaussian_kde(R[:100])\n",
        "y1 = density1(x)\n",
        "plt.plot(x, y1, label='restricted')\n",
        "plt.legend()\n",
        "plt.show()\n",
        "\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": 265
        },
        "id": "M3_rDA6qGq-L",
        "outputId": "a4468bea-32f1-41af-9a7e-813637efac89"
      },
      "source": [
        "bandwidth = [0.1, 0.2, 0.3, 0.4, 0.5]\n",
        "x = np.linspace(-0.2, 0.2, 200)\n",
        "\n",
        "density = kde.gaussian_kde(R)\n",
        "y = density(x)\n",
        "plt.plot(x, y, label='default')\n",
        "\n",
        "density = [kde.gaussian_kde(R[:100], bw_method=bw) for bw in bandwidth]\n",
        "for i, bw in enumerate(bandwidth):\n",
        "  y = density[i](x)\n",
        "  plt.plot(x, y, label=bw)\n",
        "plt.legend()\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": "markdown",
      "metadata": {
        "id": "BHWuqBtsPIC7"
      },
      "source": [
        "## Exercise 2.6"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "id": "Lg7km-KaPJ2p",
        "outputId": "6cbfb982-3f1a-48b3-dba3-949fbe9b900b"
      },
      "source": [
        "xi = 0.005\n",
        "sigma = 0.05\n",
        "yps = 0.01\n",
        "\n",
        "m = 1000\n",
        "mb = 1000\n",
        "\n",
        "eps = np.random.normal(size=(m,mb))\n",
        "R = np.exp(xi+sigma*eps)-1\n",
        "q = np.sort(R)[:, int(yps*m)]\n",
        "\n",
        "x = np.linspace(-0.13, -0.08, 200)\n",
        "density = kde.gaussian_kde(q)\n",
        "y = density(x)\n",
        "plt.plot(x, y)\n",
        "plt.axvline(x=np.exp(xi+sigma*norm.ppf(yps))-1, color='red')\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": "markdown",
      "metadata": {
        "id": "mitu2snKRzzg"
      },
      "source": [
        "## Exercise 2.9"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "y1ySXNTiR23M",
        "outputId": "ebcd07ce-f06d-4925-cd58-5e35cb980e5e"
      },
      "source": [
        "xi = 0.005\n",
        "sigma = 0.05\n",
        "\n",
        "n1 = 100\n",
        "n2 = 10000\n",
        "n3 = 1000000\n",
        "\n",
        "eps = np.random.normal(size=n3)\n",
        "X1 = np.exp(xi+sigma*eps[:n1])\n",
        "X2 = np.exp(xi+sigma*eps[:n2])\n",
        "X3 = np.exp(xi+sigma*eps)\n",
        "\n",
        "print(np.exp(4*sigma**2)+2*np.exp(3*sigma**2)+3*np.exp(2*sigma**2)-6)\n",
        "print(np.mean(np.power(X1-np.mean(X1), 4))/(np.var(X1))**2-3)\n",
        "print(np.mean(np.power(X2-np.mean(X2), 4))/(np.var(X2))**2-3)\n",
        "print(np.mean(np.power(X3-np.mean(X3), 4))/(np.var(X3))**2-3)\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "0.04014412055143879\n",
            "0.4050980178603498\n",
            "0.15274479338225477\n",
            "0.034515389707492794\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "WCYsr8jjTAgF",
        "outputId": "9d1ebcc2-9091-4b68-f8cb-cbc763ccb3e5"
      },
      "source": [
        "xi = 0\n",
        "sigma = 1\n",
        "X1 = np.exp(xi+sigma*eps[:n1])\n",
        "X2 = np.exp(xi+sigma*eps[:n2])\n",
        "X3 = np.exp(xi+sigma*eps)\n",
        "print(np.exp(4*sigma**2)+2*np.exp(3*sigma**2)+3*np.exp(2*sigma**2)-6)\n",
        "print(np.mean(np.power(X1-np.mean(X1), 4))/(np.var(X1))**2-3)\n",
        "print(np.mean(np.power(X2-np.mean(X2), 4))/(np.var(X2))**2-3)\n",
        "print(np.mean(np.power(X3-np.mean(X3), 4))/(np.var(X3))**2-3)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "110.93639217631153\n",
            "36.257846370880145\n",
            "56.26955108355349\n",
            "114.53336944761007\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ttm0ZU8lBg7x"
      },
      "source": [
        "# Section 2.3\n",
        "\n",
        "## Exercise 2.10"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "Pl8-4I84Bi0n",
        "outputId": "f55b3d53-1bdd-4de9-da32-f96d17b17947"
      },
      "source": [
        "beta = 1\n",
        "m1 = 100\n",
        "m2 = 1000\n",
        "m3 = 10000\n",
        "m4 = 100000\n",
        "\n",
        "U = np.random.random(size=m4)\n",
        "X1 = beta*np.tan(np.pi*(U[:m1]-0.5))\n",
        "X2 = beta*np.tan(np.pi*(U[:m2]-0.5))\n",
        "X3 = beta*np.tan(np.pi*(U[:m3]-0.5))\n",
        "X4 = beta*np.tan(np.pi*(U[:m4]-0.5))\n",
        "\n",
        "print(np.mean(X1), np.mean(X2), np.mean(X3), np.mean(X4))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "-3.9353839543538722 1.5278362004845483 1.7625191710819024 2.6776842290726646\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "JPzM3DJuCgg_"
      },
      "source": [
        "## Exercise 2.12"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "sO-8DBIRCh2d",
        "outputId": "0adc3227-41f1-452f-867d-4b672313605e"
      },
      "source": [
        "from scipy.stats import norm\n",
        "\n",
        "xi = 0\n",
        "sigma = 1\n",
        "m = 10000\n",
        "a = np.arange(1,4)\n",
        "\n",
        "p = 1-norm.cdf(a, loc=xi, scale=sigma)\n",
        "U = np.random.random(size=m)\n",
        "X = [norm.ppf(1-p[i]+p[i]*U) for i in range(3)]\n",
        "\n",
        "print(np.mean(X, axis=1))\n",
        "print(np.std(X, axis=1))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[1.5263215 2.3742273 3.2839256]\n",
            "[0.44825041 0.33921669 0.26625842]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "eJiIEdTeFM_R"
      },
      "source": [
        "# Section 2.4\n",
        "## Exercise 2.18"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 299
        },
        "id": "98G5cn2HFPQu",
        "outputId": "5ba8e37f-6cca-4383-d3ad-31569f013a6c"
      },
      "source": [
        "from scipy.stats import kde\n",
        "\n",
        "xi = 1\n",
        "sigma = 0.25\n",
        "m = 10000\n",
        "\n",
        "eps = np.random.normal(size=m)\n",
        "X = xi*np.exp(-sigma**2/2+sigma*eps)\n",
        "print('mean:', np.mean(X))\n",
        "\n",
        "density = kde.gaussian_kde(X)\n",
        "x = np.linspace(-0.5, 2.5, 200)\n",
        "y = density(x)\n",
        "plt.plot(x, y)\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "mean: 0.9997845278011434\n"
          ]
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f7d91778e90>]"
            ]
          },
          "metadata": {},
          "execution_count": 2
        },
        {
          "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": 299
        },
        "id": "4cnzVi2MFsJk",
        "outputId": "02cf14dc-88e6-4270-f6c8-779c43ffb5ad"
      },
      "source": [
        "sigma = 0.5\n",
        "X = xi*np.exp(-sigma**2/2+sigma*eps)\n",
        "print('mean:', np.mean(X))\n",
        "\n",
        "density = kde.gaussian_kde(X)\n",
        "x = np.linspace(-0.5, 2.5, 200)\n",
        "y = density(x)\n",
        "plt.plot(x, y)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "mean: 1.0002338662862473\n"
          ]
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f7d91272590>]"
            ]
          },
          "metadata": {},
          "execution_count": 3
        },
        {
          "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": 299
        },
        "id": "k5m9Mx5KF0Oy",
        "outputId": "dc828bed-aeb6-4929-e1aa-a8f446ae6bac"
      },
      "source": [
        "sigma = 1\n",
        "X = xi*np.exp(-sigma**2/2+sigma*eps)\n",
        "print('mean:', np.mean(X))\n",
        "\n",
        "density = kde.gaussian_kde(X)\n",
        "x = np.linspace(-0.5, 2.5, 200)\n",
        "y = density(x)\n",
        "plt.plot(x, y)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "mean: 1.0021311799462342\n"
          ]
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f7d911f6dd0>]"
            ]
          },
          "metadata": {},
          "execution_count": 4
        },
        {
          "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": "markdown",
      "metadata": {
        "id": "pYV0SqMeGf_m"
      },
      "source": [
        "## Exercise 2.19"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 282
        },
        "id": "L9SLXP9LGhLw",
        "outputId": "3555d6b8-8392-49a7-c380-327a84aacc7d"
      },
      "source": [
        "from scipy.stats import gamma, norm\n",
        "\n",
        "\n",
        "xi = 0.005\n",
        "xi_sigma = 0.08\n",
        "alpha = 10\n",
        "m = 10000\n",
        "\n",
        "eps = norm.rvs(size=m)\n",
        "Z1 = 1/gamma.rvs(alpha, size=m, scale=1/alpha)\n",
        "R1 = np.exp(xi+xi_sigma*np.sqrt(Z1)*eps)-1\n",
        "\n",
        "density = kde.gaussian_kde(R1)\n",
        "x = np.linspace(-0.4, 0.4, 200)\n",
        "y = density(x)\n",
        "plt.plot(x, y)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f7d91179a10>]"
            ]
          },
          "metadata": {},
          "execution_count": 5
        },
        {
          "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": "markdown",
      "metadata": {
        "id": "O-2H9qdwIWEr"
      },
      "source": [
        "## Exercise 2.20"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "id": "HXN7E-NSIXWP",
        "outputId": "fbc5ecb8-e6d6-4665-b512-b6d9c09de648"
      },
      "source": [
        "\n",
        "xi = 0.005\n",
        "xi_sigma = 0.08\n",
        "alpha = 10\n",
        "tau = np.sqrt(np.log(1+1/alpha))\n",
        "m = 10000\n",
        "\n",
        "eps = np.random.normal(size=m)\n",
        "eta = np.random.normal(size=m)\n",
        "Z2 = np.exp(0.5*tau**2+tau*eta)\n",
        "R2 = np.exp(xi+xi_sigma*np.sqrt(Z2)*eps)-1\n",
        "\n",
        "\n",
        "density = kde.gaussian_kde(R1)\n",
        "x = np.linspace(-0.4, 0.4, 200)\n",
        "y = density(x)\n",
        "plt.plot(x, y)\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": 282
        },
        "id": "NbIMwd7hI91I",
        "outputId": "b31fd741-2a42-4a6a-ec78-162622e6f201"
      },
      "source": [
        "plt.plot(sorted(R1), sorted(R2))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f7d91001450>]"
            ]
          },
          "metadata": {},
          "execution_count": 7
        },
        {
          "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": "markdown",
      "metadata": {
        "id": "GHAbm-6zJFch"
      },
      "source": [
        "## Exercise 2.21"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "njRoeSdJJG26",
        "outputId": "9d6a4742-33cb-4c3c-a4b4-a7d643914858"
      },
      "source": [
        "xi = 0.05\n",
        "sigma = 0.25\n",
        "rho = [0, 0.6, 0.9]\n",
        "\n",
        "m = 100000\n",
        "eta1 = np.random.normal(size=m)\n",
        "eta2 = np.random.normal(size=m)\n",
        "eps1 = eta1 \n",
        "eps2 = np.array([rho[i]*eta1+np.sqrt(1-rho[i]**2)*eta2 for i in range(3)])\n",
        "\n",
        "R1 = np.exp(xi+sigma*eps1)-1\n",
        "R2 = np.exp(xi+sigma*eps2)-1\n",
        "\n",
        "\n",
        "Rcal = 0.5*(R1+R2)\n",
        "\n",
        "print(np.mean(Rcal, axis=1))\n",
        "print(np.std(Rcal, axis=1))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[0.08429065 0.08406965 0.08393683]\n",
            "[0.19428213 0.24498444 0.26741124]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "1ilI9O-XJvuu"
      },
      "source": [
        "## Exercise 2.22"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "sNK1InoJJxW1",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "8c552b8f-adb0-4220-def8-290e29dbd479"
      },
      "source": [
        "from scipy.stats import gamma\n",
        "\n",
        "xi = 0.05\n",
        "xi_sigma = 0.25\n",
        "alpha = 5\n",
        "rho = [0,0.6,0.9]\n",
        "\n",
        "m = 100000\n",
        "eta1 = np.random.normal(size=m)\n",
        "eta2 = np.random.normal(size=m)\n",
        "eps1 = eta1 \n",
        "eps2 = [rho[i]*eta1+np.sqrt(1-rho[i]**2)*eta2 for i in range(3)]\n",
        "G = gamma.rvs(alpha, size=m, scale=1/alpha)\n",
        "sigma = xi_sigma/np.sqrt(G)\n",
        "\n",
        "R1 = np.exp(xi+sigma*eps1)-1\n",
        "R2 = np.exp(xi+sigma*eps2)-1\n",
        "\n",
        "Rcal1 = 0.5*(R1+R2[0])\n",
        "Rcal2 = 0.5*(R1+R2[1])\n",
        "Rcal3 = 0.5*(R1+R2[2])\n",
        "\n",
        "print(np.mean(Rcal1), np.mean(Rcal2), np.mean(Rcal3))\n",
        "print(np.std(Rcal1), np.std(Rcal2), np.std(Rcal3))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "0.09390779657744129 0.09352385520375457 0.09305158243658057\n",
            "0.22684786493275338 0.28476640919565566 0.3100275071809336\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "h8wCulRMlK8g"
      },
      "source": [
        "#Exercise 2.23"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "_swbmcyUlQAc",
        "outputId": "a3b38d17-d163-4ce4-cc03-78834663030f"
      },
      "source": [
        "xi = 0.05\n",
        "sigma = 0.25\n",
        "rho = [0, 0.6, 0.9]\n",
        "J = 5\n",
        "m = 10000\n",
        "\n",
        "eps = np.random.normal(size=(m,J+1))\n",
        "eps1 = np.sqrt(rho[0])*eps[:,0:1] + np.sqrt(1-rho[0])*eps[:,1:J+1]\n",
        "R1 = np.exp(xi+sigma*eps1)-1\n",
        "Rcal1 = np.mean(R1,axis=1)\n",
        "\n",
        "eps2 = np.sqrt(rho[1])*eps[:,0:1] + np.sqrt(1-rho[1])*eps[:,1:J+1]\n",
        "R2 = np.exp(xi+sigma*eps2)-1\n",
        "Rcal2 = np.mean(R2,axis=1)\n",
        "\n",
        "eps3 = np.sqrt(rho[2])*eps[:,0:1] + np.sqrt(1-rho[2])*eps[:,1:J+1]\n",
        "R3 = np.exp(xi+sigma*eps3)-1\n",
        "Rcal3 = np.mean(R3,axis=1)\n",
        "\n",
        "print('mean:', np.mean(Rcal1), np.mean(Rcal2), np.mean(Rcal3))\n",
        "print('std:', np.std(Rcal1), np.std(Rcal2), np.std(Rcal3))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "mean: 0.08393365441539767 0.0857597081818457 0.08631239455811435\n",
            "std: 0.1233840252685101 0.22378196824982471 0.26092466732258685\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "CXsmmQ6xWj2x"
      },
      "source": [
        "# Exercise 2.24"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "vsEjltFRWoaN",
        "outputId": "486fdeb8-a9c5-4316-a571-71830b6d060d"
      },
      "source": [
        "xi = 0.05\n",
        "sigma = 0.25\n",
        "rho0 = 0.6\n",
        "tau = [0, 0.5, 1]\n",
        "m = 10000\n",
        "\n",
        "eps = np.random.normal(size=(m,2))\n",
        "eps = np.concatenate((eps, np.tile(eps[:, 1:2], (1,2))), axis=1)\n",
        "eta = np.random.normal(size=m)\n",
        "Z1 = np.exp(tau[0]*eta)\n",
        "rho1 = ((1-rho0)*Z1-(1-rho0))/((1+rho0)*Z1+1-rho0)\n",
        "eps[:,1] = rho1*eps[:,0]+np.sqrt(1-np.power(rho1,2))*eps[:,1]\n",
        "\n",
        "Z2 = np.exp(tau[1]*eta)\n",
        "rho2 = ((1-rho0)*Z2-(1-rho0))/((1+rho0)*Z2+1-rho0)\n",
        "eps[:,2] = rho2*eps[:,0]+np.sqrt(1-np.power(rho2, 2))*eps[:,1]\n",
        "\n",
        "Z3 = np.exp(tau[2]*eta)\n",
        "rho3 = ((1-rho0)*Z3-(1-rho0))/((1+rho0)*Z3+1-rho0)\n",
        "eps[:,3] = rho3*eps[:,0]+np.sqrt(1-np.power(rho3, 2))*eps[:,1]\n",
        "R = np.exp(xi+sigma*eps)-1\n",
        "Rcal1 = 0.5*(R[:,0]+R[:,1])\n",
        "Rcal2 = 0.5*(R[:,0]+R[:,2])\n",
        "Rcal3 = 0.5*(R[:,0]+R[:,3])\n",
        "\n",
        "print('mean:', np.mean(Rcal1), np.mean(Rcal2), np.mean(Rcal3))\n",
        "print('std:', np.std(Rcal1), np.std(Rcal2), np.std(Rcal3))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "mean: 0.08307080659155507 0.08292876463400954 0.08280850003316316\n",
            "std: 0.19027604161888145 0.18896769054942056 0.18580731170495382\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "cEugHdYNdezO"
      },
      "source": [
        "# Section 2.5"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "HRbnENLjdinG"
      },
      "source": [
        "# Exercise 2.25\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "9P0weFV9dlOP",
        "outputId": "29550e39-09c1-453b-a25d-bf883e37d14e"
      },
      "source": [
        "xi = 10\n",
        "alpha = 4\n",
        "m = 10000\n",
        "\n",
        "G = gamma.rvs(alpha, size=m, scale=1/alpha)\n",
        "X = xi*G\n",
        "\n",
        "print(xi, np.mean(X))\n",
        "print(xi/np.sqrt(alpha), np.std(X))\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "10 10.008506725126363\n",
            "5.0 4.999747325688828\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "CPqKRrDO4YHN"
      },
      "source": [
        "# Exercise 2.26"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ROyNI0p34aRw"
      },
      "source": [
        "from scipy.stats import gamma\n",
        "import numpy as np\n",
        "\n",
        "\n",
        "xi = 10\n",
        "alpha = 4\n",
        "a = [5, 15, 25]\n",
        "m = 100000"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "WVkRGBHW5Pa6"
      },
      "source": [
        "X = xi * gamma.rvs(alpha, size=m, scale=1/alpha)\n",
        "X0 = X[X > a[0]] - a[0]\n",
        "X1 = X[X > a[1]] - a[1]\n",
        "X2 = X[X > a[2]] - a[2]"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "JUppHFRC5SBo",
        "outputId": "8b34d2e6-6cf0-4c98-c180-880f3a999dac"
      },
      "source": [
        "print('acceptance:', float(len(X0))/m, float(len(X1))/m, float(len(X2))/m)\n",
        "print(np.mean(X0), np.mean(X1), np.mean(X2))\n",
        "print(np.std(X0), np.std(X1), np.std(X2))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "acceptance: 0.85419 0.15101 0.00998\n",
            "6.053707339618842 3.8347448665674766 3.295440375692355\n",
            "4.600480479487391 3.5798071372136056 3.1745111059329694\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "osEHNJvhAiCY",
        "outputId": "fe5dc9f5-4e95-4e87-e83a-647addd86085"
      },
      "source": [
        "alpha = 1\n",
        "X = xi * gamma.rvs(alpha, size=m, scale=1/alpha)\n",
        "X0 = X[X > a[0]] - a[0]\n",
        "X1 = X[X > a[1]] - a[1]\n",
        "X2 = X[X > a[2]] - a[2]\n",
        "print('acceptance:', float(len(X0))/m, float(len(X1))/m, float(len(X2))/m)\n",
        "print(np.mean(X0), np.mean(X1), np.mean(X2))\n",
        "print(np.std(X0), np.std(X1), np.std(X2))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "acceptance: 0.60425 0.22089 0.08195\n",
            "9.97992965780108 10.06589515193078 10.097337388478948\n",
            "10.04117768547202 10.085813645953275 10.034361258967547\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "I3Qt3MwxAqNE"
      },
      "source": [
        "# Exercise 2.27"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "E_MhJhjGAsPx"
      },
      "source": [
        "xi = 10\n",
        "alpha = 4\n",
        "a = 5\n",
        "b = [10, 15, 20]\n",
        "m = 100000"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "CpE-n4siAz_x"
      },
      "source": [
        "X = xi * gamma.rvs(alpha, size=m, scale=1/alpha)\n",
        "X0 = X[(X > a) & (X < b[0])]\n",
        "X1 = X[(X > a) & (X < b[1])]\n",
        "X2 = X[(X > a) & (X < b[2])]"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "-yskBB80BDbA",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "21039ae1-7544-4073-ff75-58ca1ea29803"
      },
      "source": [
        "print('acceptance:',float(len(X0))/m, float(len(X1))/m, float(len(X2))/m)\n",
        "print(np.mean(X0), np.mean(X1), np.mean(X2))\n",
        "print(np.std(X0), np.std(X1), np.std(X2))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "acceptance: 0.42423 0.70545 0.8138\n",
            "7.517401934779921 9.3753631499259 10.39768192094822\n",
            "1.408962673875513 2.6813192927326255 3.6468861031394026\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "I8NEt14jBP-H"
      },
      "source": [
        "# Exercise 2.29"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "0sYa21HdBSVg"
      },
      "source": [
        "alpha = 2\n",
        "beta = 1\n",
        "m = 100000"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "HnSFcZVQBW8a"
      },
      "source": [
        "U = np.random.uniform(size=m)\n",
        "X = beta * np.power(-np.log(U), 1/alpha)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ABxScFaQBnfF"
      },
      "source": [
        "X_sorted = sorted(X)\n",
        "med = X_sorted[int(0.5 * m)]\n",
        "qd = X_sorted[int(0.75 * m)] - X_sorted[int(0.25 * m)]"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "1HRDM0GjB_WU",
        "outputId": "8f806eab-c72f-4639-b802-6a9c614dea0c"
      },
      "source": [
        "print(beta * np.power(np.log(2), 1/alpha), med)\n",
        "print(beta * (np.power(np.log(4), 1/alpha)-np.power(np.log(4)-np.log(3), 1/alpha)), qd)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "0.8325546111576977 0.8325262435228936\n",
            "0.6410500012128232 0.6405696043821102\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "AweQzLeFCm9g"
      },
      "source": [
        "# Exercise 2.30"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "xkFXIcgLCl0i"
      },
      "source": [
        "from scipy.stats import weibull_min, kde\n",
        "\n",
        "beta = 1\n",
        "alpha = [2, 3.6, 5]\n",
        "m = 10000\n"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 282
        },
        "id": "Ew7e1VHbEfHp",
        "outputId": "a071f712-2ced-4669-c7a4-de42ef5f80ca"
      },
      "source": [
        "X0 = weibull_min.rvs(alpha[0], scale=beta, size=m)\n",
        "density0 = kde.gaussian_kde(X0)\n",
        "x = np.linspace(0.0, 3.0, 200)\n",
        "y0 = density0(x)\n",
        "plt.plot(x, y0)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f7d9114c310>]"
            ]
          },
          "metadata": {},
          "execution_count": 25
        },
        {
          "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": 286
        },
        "id": "o_Qn1NvYFDKl",
        "outputId": "55a1a88e-ca70-4704-8ae0-23caf7ee97cb"
      },
      "source": [
        "X1 = weibull_min.rvs(alpha[1], scale=beta, size=m)\n",
        "density1 = kde.gaussian_kde(X1)\n",
        "x = np.linspace(0.0, 3.0, 200)\n",
        "y1 = density1(x)\n",
        "plt.plot(x, y1)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f7d90c88310>]"
            ]
          },
          "metadata": {},
          "execution_count": 26
        },
        {
          "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": 282
        },
        "id": "rKzESWt3Ffoq",
        "outputId": "1413179e-42b4-4a18-b1e8-654f86fff1ed"
      },
      "source": [
        "X2 = weibull_min.rvs(alpha[2], scale=beta, size=m)\n",
        "density2 = kde.gaussian_kde(X2)\n",
        "x = np.linspace(0.0, 3.0, 200)\n",
        "y2 = density2(x)\n",
        "plt.plot(x, y2)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7fdb28830650>]"
            ]
          },
          "metadata": {},
          "execution_count": 42
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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XZk9DByvnFKfE+DajWVJRpFEsRSYo1jP6rwN3Xmb9XcCyyOMh4MsAZlZGeI7ZLcBm4DNmlvxDJqaZUMjZU9/JuprUa7YZtqSyiOauPjovDARdikjKiSno3f1ZoO0ym9wLfNPDtgMzzGwO8DbgKXdvc/d24Cku/wNDpkBdSzddfYOsm5fCQa9pBUUmLF5t9HOB+qjXDZFlYy1/AzN7yMxqzay2ubk5TmUJhJttANandNCHBzdT843IlUuai7Hu/rC7b3L3TRUVqTGyYqrYXd/B9Lzsi2fFqWh+2XRys00XZEUmIF5BfwqYF/W6JrJsrOWSQLsaOrl6binZKXajVLSc7CwWzipUF0uRCYhX0G8F3hfpfXMd0OnuZ4AngDvMbGbkIuwdkWWSIP2DIQ6cPpfSzTbDwj1vFPQiVyonlo3M7FHgZqDczBoI96TJBXD3rwDbgN8BjgI9wB9F1rWZ2V8DOyNv9Tl3v9xFXYmzg43n6B8KpfSF2GFLKgv5xYGzDAyFyM1OmlZHkaQXU9C7+wPjrHfgo2OsewR45MpLk3jY3dAJwNqa0oArmbwlFUUMhpyTrT0srUzd6w0iiabTojS3u76D8qI85s6YFnQpkzYc7rogK3JlFPRpbnd9B+tqZqTMjFKXs1iDm4lMiII+jZ3vG+Ro8/m0aJ8HKMrPYXZJgYJe5Aop6NPYaw2duKdH+/ywJZWFumlK5Aop6NPY8B2xa1N4jJuRllQUcazpPOHr/yISCwV9GtvT0Mm8smmUFeYFXUrcLK0s4nzfIE1dfUGXIpIyFPRpbFd9R1qdzYOmFRSZCAV9mmo938epjgusS6P2edC0giIToaBPU3siN0ql8hj0o6kqyacoP0cXZEWugII+Te1u6CDL4Kq56XVGb2YsqSjUTVMiV0BBn6b2NHSytLKIwvyYRrlIKUsqNH+syJVQ0Kchd2dPQ/pdiB22rKqYxnO9mlZQJEYK+jR0quMCLef70+5C7LAVs8MXZA+f7Qq4EpHUoKBPQ3sujliZnmf0K2aXAHCoUUEvEgsFfRra3dBBbraxck5x0KVMierSAorzcxT0IjFS0KehPfWdrJpTQn5OdtClTAkzY/nsYg6p6UYkJgr6NBMKOa+d6ky7/vMjLa8q5lBjl8a8EYmBgj7N1LWc53zfYFqNWDmalbOL6bwwoDFvRGIQU9Cb2Z1mdsjMjprZJ0dZ/w9mtivyOGxmHVHrhqLWbY1n8fJGr5xsB2DDgpkBVzK1VswOX384qHZ6kXGNezeNmWUDXwJuBxqAnWa21d33D2/j7v89avv/ClwT9RYX3H19/EqWy9l5op2ywjwWlxcGXcqUWlEVDvrDjV3ctLwi4GpEklssZ/SbgaPuXufu/cBjwL2X2f4B4NF4FCdX7pWT7WyYPzMtpg68nJmFeVQW5+uMXiQGsQT9XKA+6nVDZNkbmNkCYBHwq6jFBWZWa2bbzewdY32ImT0U2a62ubk5hrJkpJbzfRxv6ebahendbDNsxexi3TQlEoN4X4y9H/iBuw9FLVvg7puA9wD/aGZLRtvR3R92903uvqmiQr+KT0TtiXD7/KZMCfqqcNAPhdTzRuRyYgn6U8C8qNc1kWWjuZ8RzTbufiryZx3wDJe230scvXKyjbycrLQbsXIsy2cX0zcY4vW2nqBLEUlqsQT9TmCZmS0yszzCYf6G3jNmthKYCbwUtWymmeVHnpcDbwb2j9xX4uPlE+2sqylN2xulRloZ6XlzqPFcwJWIJLdxg97dB4GPAU8AB4Dvufs+M/ucmd0Tten9wGN+6R0sq4BaM9sNPA38bXRvHYmfc70DvNbQwfWLZwVdSsIsqyzGDA41ashikcuJabByd98GbBux7NMjXn92lP1eBK6eRH0So53H2wg5XLckc4J+Wl42C8qmc+iszuhFLkd3xqaJF4+1kpeTxYb5mXEhdtjwUAgiMjYFfZp46VgrG+fPpCA3M9rnh62cU8Lxlm4u9A+Nv7FIhlLQp4H27n4ONJ7j+gxqthm2ek4JIUcjWYpchoI+DbxU14o7vCkDg35NdXgSkn2nOwOuRCR5KejTwDOHmigpyGH9vPQemng0NTOnUVKQw/7TuiArMhYFfYpzd359uJkbllWQk515X6eZsbq6hH0KepExZV4ypJmDjV2cPdeX0SM4rp5TysHGcxoKQWQMCvoU98yh8ABwN63I3KBfU11C70CI4y26cUpkNAr6FPfMoSZWzi6mqqQg6FICs/riBVk134iMRkGfwtq6+9l5oo3bV1cFXUqgllYWkZeTpaAXGYOCPoX94sBZQg53rJ4ddCmBys3OYkVVsXreiIxBQZ/CntzXyNwZ07hqbknQpQRu9ZwS9p3u5NIx9UQEFPQpq7tvkGePtHD76qq0nzYwFmvmltDeM0Djud6gSxFJOgr6FPXs4Wb6B0O8bU1mN9sMWz0nckH2lJpvREZS0KeoJ/Y1MnN6bsbMDzueVXNKMFPPG5HRKOhTUP9giF8ebOKtq6oy8m7Y0RTm57BoViH7z2jMG5GRlBIpaHtdK129g9yhZptLrNJQCCKjiinozexOMztkZkfN7JOjrH+/mTWb2a7I44NR6x40syORx4PxLD5TPbGvkel52dywrDzoUpLKVdWlNLRfoKOnP+hSRJLKuEFvZtnAl4C7gNXAA2a2epRNv+vu6yOPr0X2LQM+A2wBNgOfMTM1Kk/CUMh5cv9ZblpekXGTjIxnbU0pAHsa1HwjEi2WM/rNwFF3r3P3fuAx4N4Y3/9twFPu3ubu7cBTwJ0TK1UAdhxvpbmrj99dWx10KUnnqrnDQd8RcCUiySWWoJ8L1Ee9bogsG+k+M9tjZj8ws3lXuC9m9pCZ1ZpZbXNzcwxlZaaf7jnD9Lxsbl1ZGXQpSad0Wi6Lygt1Ri8yQrwuxv4nsNDd1xI+a//Glb6Buz/s7pvcfVNFReaOxHg5A0Mhfr63kdtWVTEtT802o1lbU6qgFxkhlqA/BcyLel0TWXaRu7e6e1/k5deAjbHuK7F78Vgrbd393L12TtClJK21NTNoPNdLk+6QFbkolqDfCSwzs0VmlgfcD2yN3sDMopPnHuBA5PkTwB1mNjNyEfaOyDKZgJ/uPk1xfk5Gjz0/nnWRC7K7dVYvctG4Qe/ug8DHCAf0AeB77r7PzD5nZvdENvu4me0zs93Ax4H3R/ZtA/6a8A+LncDnIsvkCvUNDvHEvkZuX1NFfo6abcayprqULNMFWZFoObFs5O7bgG0jln066vmngE+Nse8jwCOTqFGA5w63cK53kLvXqbfN5UzLy2Z5VbHO6EWi6M7YFPHTPaeZMT2XtyzVTVLjWVtTymsNHRqyWCRCQZ8CuvsGeXL/We66aja5GttmXGtrZtDeM0BD+4WgSxFJCkqNFLDttTP09A9x34aaoEtJCetqZgCwW+30IoCCPiV8/5UGFpcXsnGBRo+IxYrZxeRlZ6k/vUiEgj7JnWzt5uXjbdy3sUYzScUoLyeLVdUl7K7XGb0IKOiT3nd31pNl8HsbRh05QsawrqaUvac6GQrpgqyIgj6J9Q4M8djOem5fXcWc0mlBl5NS1tXMoLt/iMNnu4IuRSRwCvok9rM9Z2jr7ufB6xcGXUrK2byoDICdJ3R/noiCPkm5O9986QRLK4u4fsmsoMtJOTUzpzGntIAdxxX0Igr6JLXzRDu7Gzp58E0LdRF2AsyMzYvK2Hm8TTdOScZT0Cepf/n1MWYV5vH7G9V3fqKuXVhGU1cfJ1t7gi5FJFAK+iR05GwXvzzYxPuuX6jpAidhS6Sd/mW100uGU9AnoYefrWNabjbvu35B0KWktKWVRZQV5rG9rjXoUkQCpaBPMo2dvfx41ynefe08ZhbmBV1OSjMz3rRkFs8faVE7vWQ0BX2S+bcXjjMUcj7wlkVBl5IWblxeQVNXHwcb1Z9eMpeCPomc6x3g2zte5+1rq5lXNj3octLCjcvCs3E9e1gTzkvmUtAnkW+9dJLzfYP86Y2Lgy4lbcwuLWBFVTHPHlHQS+aKKejN7E4zO2RmR83sk6Os/x9mtt/M9pjZL81sQdS6ITPbFXlsHbmvhPUODPFvLxznpuUVXDW3NOhy0sqNy8vZebydnv7BoEsRCcS4QW9m2cCXgLuA1cADZrZ6xGavApvcfS3wA+DvotZdcPf1kcc9yKi+X1tPy/l+PnzzkqBLSTu3rKikfyjEs4dbgi5FJBCxnNFvBo66e5279wOPAfdGb+DuT7v78F0p2wHd5XMFBodC/MuzdWyYP+Ni32+Jn82LypgxPZcn9jUGXYpIIGIJ+rlAfdTrhsiysXwAeDzqdYGZ1ZrZdjN7xwRqTHs/3XOGhvYLfPjmpRruYArkZGdx28oqfnngLANDoaDLEUm4uF6MNbP3ApuAv49avMDdNwHvAf7RzEZtmzCzhyI/EGqbmzPnwlko5Hz5mWMsryritpWVQZeTtu68ajbnegd185RkpFiC/hQwL+p1TWTZJczsrcD/BO5x977h5e5+KvJnHfAMcM1oH+LuD7v7JnffVFFREfMBpLqfvXaGQ2e7+MjNS8nK0tn8VLlhWTnT87J5fK+abyTzxBL0O4FlZrbIzPKA+4FLes+Y2TXAvxAO+aao5TPNLD/yvBx4M7A/XsWnuv7BEJ9/8hArZxdz97rqoMtJawW52dy2qorHXztD/6CabySzjBv07j4IfAx4AjgAfM/d95nZ58xsuBfN3wNFwPdHdKNcBdSa2W7gaeBv3V1BH/Hoy69zsrWHv7hzJdk6m59y71hfTXvPgG6ekoyTE8tG7r4N2DZi2aejnr91jP1eBK6eTIHpqulcL59/8hBvWjKLm1dkTlNVkG5cXsHM6bn8eNcp3rq6KuhyRBJGd8YG5H/95376BkP873derZ42CZKbncXd66p5av9ZunoHgi5HJGEU9AH4xf6z/Oy1M3z81qUsKi8MupyM8s5r5tI3GGLr7tNBlyKSMAr6BOvuG+TTP9nL8qoiHrpRd8Em2vp5M1g5u5jv7HhdQxdLxlDQJ9jf/fwgpzt7+Zvfu5q8HP31J5qZ8Qdb5rPv9Dn2NHQGXY5IQihpEujpg01846WTvP9NC9m4QEMdBOXea+YyLTebb+84GXQpIgmhoE+Q5q4+PvGD3ayoKuaTd60MupyMVlKQyzs3zOXHu07T3NU3/g4iKU5BnwChkPPn399NV+8gX3jgGk34nQQ++JZFDAyF+OZLJ4IuRWTKKegT4JEXjvPrw8381dtXsWJ2cdDlCLC4oojbV1Xxre0nNU69pD0F/RR77kgzf/P4QW5fXcV7r1sw/g6SMH960xI6egb4xotqq5f0pqCfQgcbz/GRb/+GZZVF/MO71+vGqCSzccFMbl1ZyZefOUpnj26gkvSloJ8iR5u6+IOv7qAwL4evPbiJovyYRpuQBPvE21bQ1TfIPz9zNOhSRKaMgn4KHG/p5j1f3YGZ8e0/2ULNzOlBlyRjWDWnhPs21PCvzx/nwJlzQZcjMiUU9HFW39bDe766ncGQ850/2cKSiqKgS5Jx/OXvrKJ0Wi6f/OEehkK6W1bSj4I+jk53XOCBr26np3+If//AFpZXqYdNKigrzOOz96xhd0Mnn3/yUNDliMSdgj5Oms718p6vbqezZ4Bv/vFmVleXBF2SXIHfXTuH92yZz5efOcaPX33DBGoiKU1XCOOgqauX93xtB01dfXzrA5tZN29G0CXJFTIzPnv3Go42nefPv78bgHdcMzfgqkTiQ2f0k3Ss+Tz3fflFTrVf4JH3X6sxbFJYXk4W//rgJjYtnMmffXcXn3/ikKYdlLSgoJ8gd+d7tfW844sv0NM3xGMPXcd1i2cFXZZMUnFBLl//o828a2MNX3z6KHf/0/M8ua+RkC7SSgqzWMbkNrM7gf8HZANfc/e/HbE+H/gmsBFoBd7t7ici6z4FfAAYAj7u7k+M93mbNm3y2traKzuSBAiFnPr2Hp4/2sK3XjrJwcYutiwq4/O/v455ZepCmW6e2NfI32w7wInWHubOmMbb185h88IyNi6YyczCvKDLE7mEmb3i7ptGXTde0JtZNnAYuB1oAHYCD0RP8m1mHwHWuvuHzOx+4J3u/m4zWw08CmwGqoFfAMvdfehyn9QZfA8AAAZJSURBVJnIoA+FnO7+QVrP99N8vo/mrhGPqGUt5/sYjJzZragq5oM3LOK+DTVkaWLvtDU4FGLb3kZ++EoDLxxtufj9LyovZHF5IQvLC1kwazqzCvMpK8xjVlEeZYV5lBTkkpttuhtaEuZyQR/LxdjNwFF3r4u82WPAvcD+qG3uBT4bef4D4IsW/hd+L/CYu/cBx83saOT9XprIgYzn7V94jgsDQ+DghJtXwn+C44Qiza3uzpA7PX1DnO8fZLSfddlZRnlRHhXF+VQU5bNydjEVxfnMnTmNzQvLWFpZpP/EGSAnO4t71lVzz7pqegeG2F3fQe3Jdl5r6OREazcvHGuhd2D0dvzsLGNabjYFuVnk52STlQWGMfzPxghfBLbIi+F/TReXScaZOT2P733o+ri/byxBPxeoj3rdAGwZaxt3HzSzTmBWZPn2EfuO2pXBzB4CHgKYP39+LLW/wfKqYvqHQpf8B7LIf6CsyJPh/2hZBoX5ORTn51BUkEN5UX441CPBPnN6ns7U5RIFudlsWTyLLVHXYkIhp6W7j9bz/bR199Pa3U/b+T7O9w1yYWCI3oFQ5M+h8AlH5Kzitycgvz0hGV4R9UoyTElB7pS8b9J0r3T3h4GHIdx0M5H3+Id3r49rTSLjycoyKosLqCwuCLoUkTHF0uvmFDAv6nVNZNmo25hZDlBK+KJsLPuKiMgUiiXodwLLzGyRmeUB9wNbR2yzFXgw8vxdwK88/DvqVuB+M8s3s0XAMuDl+JQuIiKxGLfpJtLm/jHgCcLdKx9x931m9jmg1t23Av8KfCtysbWN8A8DItt9j/CF20Hgo+P1uBERkfiKqR99oiVrP3oRkWR1ue6VujNWRCTNKehFRNKcgl5EJM0p6EVE0lxSXow1s2bg5AR3Lwda4lhOkNLlWNLlOEDHkozS5ThgcseywN0rRluRlEE/GWZWO9aV51STLseSLscBOpZklC7HAVN3LGq6ERFJcwp6EZE0l45B/3DQBcRRuhxLuhwH6FiSUbocB0zRsaRdG72IiFwqHc/oRUQkioJeRCTNpWzQm9mdZnbIzI6a2SdHWZ9vZt+NrN9hZgsTX+X4YjiO95tZs5ntijw+GESd4zGzR8ysycz2jrHezOwLkePcY2YbEl1jrGI4lpvNrDPqO/l0omuMlZnNM7OnzWy/me0zs/82yjZJ/93EeBwp8b2YWYGZvWxmuyPH8r9G2Sa++eXuKfcgPFzyMWAxkAfsBlaP2OYjwFciz+8Hvht03RM8jvcDXwy61hiO5UZgA7B3jPW/AzxOeGbH64AdQdc8iWO5Gfhp0HXGeCxzgA2R58XA4VH+jSX9dxPjcaTE9xL5ey6KPM8FdgDXjdgmrvmVqmf0Fycsd/d+YHjC8mj3At+IPP8BcJsl32zesRxHSnD3ZwnPRTCWe4Fveth2YIaZzUlMdVcmhmNJGe5+xt1/E3neBRzgjfM2J/13E+NxpITI3/P5yMvcyGNkr5i45leqBv1oE5aP/NIvmbAcGJ6wPJnEchwA90V+pf6Bmc0bZX0qiPVYU8X1kV+9HzezNUEXE4vIr//XED6DjJZS381ljgNS5Hsxs2wz2wU0AU+5+5jfSTzyK1WDPpP8J7DQ3dcCT/Hbn/ISnN8QHldkHfBPwI8DrmdcZlYE/BD4M3c/F3Q9EzXOcaTM9+LuQ+6+nvA82pvN7Kqp/LxUDfrJTFieTMY9Dndvdfe+yMuvARsTVFu8pc1E8e5+bvhXb3ffBuSaWXnAZY3JzHIJh+O33f0/RtkkJb6b8Y4j1b4XAHfvAJ4G7hyxKq75lapBP5kJy5PJuMcxoq30HsJtk6loK/C+SA+P64BOdz8TdFETYWazh9tLzWwz4f9HyXYSAYR71BCe0/mAu//fMTZL+u8mluNIle/FzCrMbEbk+TTgduDgiM3iml/jTg6ejHwSE5YnkxiP4+Nmdg/hydXbCPfCSTpm9ijhXg/lZtYAfIbwRSbc/SvANsK9O44CPcAfBVPp+GI4lncBHzazQeACcH8SnkQMezPwh8BrkTZhgL8E5kNKfTexHEeqfC9zgG+YWTbhH0bfc/efTmV+aQgEEZE0l6pNNyIiEiMFvYhImlPQi4ikOQW9iEiaU9CLiKQ5Bb2ISJpT0IuIpLn/D198BiuUi1g7AAAAAElFTkSuQmCC\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 282
        },
        "id": "H50dAqWaF3Ld",
        "outputId": "2e6e07ab-6929-4c45-f0d6-d1211e952127"
      },
      "source": [
        "Y = np.random.normal(size=m)\n",
        "plt.plot(sorted(X0), sorted(Y))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f7d90b94c50>]"
            ]
          },
          "metadata": {},
          "execution_count": 27
        },
        {
          "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": "markdown",
      "metadata": {
        "id": "PKrLopc9GE8z"
      },
      "source": [
        "# Exercise 2.31"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "r8CtGhtjGHSa"
      },
      "source": [
        "alpha = 2\n",
        "beta = 1\n",
        "m = 10000"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "cDaReplOGOHn"
      },
      "source": [
        "U = np.random.uniform(size=m)\n",
        "X = beta * np.power(-np.log(U), -1/alpha)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "tAdt22XtGWAn",
        "outputId": "bc33a6c5-75ae-4b58-c29d-758620169f4e"
      },
      "source": [
        "X_sorted = sorted(X)\n",
        "med = X_sorted[int(0.5 * m)]\n",
        "qd = X_sorted[int(0.75 * m)] - X_sorted[int(0.25 * m)]\n",
        "print(beta * np.power(np.log(2), -1/alpha), med)\n",
        "print(beta * np.power(np.log(4) - np.log(3), -1/alpha)-beta * np.power(np.log(4), -1/alpha), qd)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "1.2011224087864498 1.1749774213933124\n",
            "1.0150975454553706 0.9990845378772046\n"
          ]
        }
      ]
    }
  ]
}