Discrete_Probability_Distributions.ipynb

{
  "cells": [
    {
      "cell_type": "markdown",
      "source": [
        "# Discrete Probability Distributions for Machine Learning"
      ],
      "metadata": {
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          "transient": {
            "deleting": false
          }
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import pandas as pd\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "import seaborn as sns\n",
        "\n",
        "import warnings\n",
        "warnings.filterwarnings(\"ignore\")\n",
        "\n",
        "import yfinance as yf\n",
        "yf.pdr_override()"
      ],
      "outputs": [],
      "execution_count": 1,
      "metadata": {
        "collapsed": true,
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          "source_hidden": false,
          "outputs_hidden": false
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        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# input\n",
        "start = '2021-01-01'\n",
        "end = '2022-01-01'\n",
        "symbol = 'AMD'\n",
        "\n",
        "# Read data \n",
        "dataset = yf.download(symbol,start,end)\n",
        "\n",
        "# View Columns\n",
        "dataset.head()"
      ],
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[*********************100%***********************]  1 of 1 completed\n"
          ]
        },
        {
          "output_type": "execute_result",
          "execution_count": 2,
          "data": {
            "text/plain": "                 Open       High        Low      Close  Adj Close    Volume\nDate                                                                       \n2021-01-04  92.110001  96.059998  90.919998  92.300003  92.300003  51802600\n2021-01-05  92.099998  93.209999  91.410004  92.769997  92.769997  34208000\n2021-01-06  91.620003  92.279999  89.459999  90.330002  90.330002  51911700\n2021-01-07  91.330002  95.510002  91.199997  95.160004  95.160004  42897200\n2021-01-08  95.980003  96.400002  93.269997  94.580002  94.580002  39816400",
            "text/html": "<div>\n<style scoped>\n    .dataframe tbody tr th:only-of-type {\n        vertical-align: middle;\n    }\n\n    .dataframe tbody tr th {\n        vertical-align: top;\n    }\n\n    .dataframe thead th {\n        text-align: right;\n    }\n</style>\n<table border=\"1\" class=\"dataframe\">\n  <thead>\n    <tr style=\"text-align: right;\">\n      <th></th>\n      <th>Open</th>\n      <th>High</th>\n      <th>Low</th>\n      <th>Close</th>\n      <th>Adj Close</th>\n      <th>Volume</th>\n    </tr>\n    <tr>\n      <th>Date</th>\n      <th></th>\n      <th></th>\n      <th></th>\n      <th></th>\n      <th></th>\n      <th></th>\n    </tr>\n  </thead>\n  <tbody>\n    <tr>\n      <th>2021-01-04</th>\n      <td>92.110001</td>\n      <td>96.059998</td>\n      <td>90.919998</td>\n      <td>92.300003</td>\n      <td>92.300003</td>\n      <td>51802600</td>\n    </tr>\n    <tr>\n      <th>2021-01-05</th>\n      <td>92.099998</td>\n      <td>93.209999</td>\n      <td>91.410004</td>\n      <td>92.769997</td>\n      <td>92.769997</td>\n      <td>34208000</td>\n    </tr>\n    <tr>\n      <th>2021-01-06</th>\n      <td>91.620003</td>\n      <td>92.279999</td>\n      <td>89.459999</td>\n      <td>90.330002</td>\n      <td>90.330002</td>\n      <td>51911700</td>\n    </tr>\n    <tr>\n      <th>2021-01-07</th>\n      <td>91.330002</td>\n      <td>95.510002</td>\n      <td>91.199997</td>\n      <td>95.160004</td>\n      <td>95.160004</td>\n      <td>42897200</td>\n    </tr>\n    <tr>\n      <th>2021-01-08</th>\n      <td>95.980003</td>\n      <td>96.400002</td>\n      <td>93.269997</td>\n      <td>94.580002</td>\n      <td>94.580002</td>\n      <td>39816400</td>\n    </tr>\n  </tbody>\n</table>\n</div>"
          },
          "metadata": {}
        }
      ],
      "execution_count": 2,
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    },
    {
      "cell_type": "code",
      "source": [
        "dataset.shape"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 3,
          "data": {
            "text/plain": "(252, 6)"
          },
          "metadata": {}
        }
      ],
      "execution_count": 3,
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        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "x = np.array(dataset['Adj Close']).squeeze()"
      ],
      "outputs": [],
      "execution_count": 4,
      "metadata": {
        "collapsed": true,
        "jupyter": {
          "source_hidden": false,
          "outputs_hidden": false
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          "shell.execute_reply": "2022-05-10T00:54:27.381Z"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import collections\n",
        "\n",
        "repeat_numbers = [item for item, count in collections.Counter(x).items() if count > 1]"
      ],
      "outputs": [],
      "execution_count": 5,
      "metadata": {
        "collapsed": true,
        "jupyter": {
          "source_hidden": false,
          "outputs_hidden": false
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        },
        "execution": {
          "iopub.status.busy": "2022-05-10T00:54:27.137Z",
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          "shell.execute_reply": "2022-05-10T00:54:27.384Z"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "repeat_numbers"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 6,
          "data": {
            "text/plain": "[79.05999755859375, 82.76000213623047, 78.55000305175781, 106.1500015258789]"
          },
          "metadata": {}
        }
      ],
      "execution_count": 6,
      "metadata": {
        "collapsed": true,
        "jupyter": {
          "source_hidden": false,
          "outputs_hidden": false
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          "transient": {
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        "execution": {
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          "shell.execute_reply": "2022-05-10T00:54:27.387Z"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "from scipy.stats import binom"
      ],
      "outputs": [],
      "execution_count": 7,
      "metadata": {
        "collapsed": true,
        "jupyter": {
          "source_hidden": false,
          "outputs_hidden": false
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        "nteract": {
          "transient": {
            "deleting": false
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        },
        "execution": {
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        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "p = 0.5 # 50% Success\n",
        "k = 100 # Number of Trials\n",
        "\n",
        "# define the distribution\n",
        "dist = binom(k, p)\n",
        "for n in repeat_numbers:\n",
        "\tprint('P of {} success: {}'.format(n, round(dist.pmf(n)*100),6))"
      ],
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "P of 79.05999755859375 success: 0\n",
            "P of 82.76000213623047 success: 0\n",
            "P of 78.55000305175781 success: 0\n",
            "P of 106.1500015258789 success: 0\n"
          ]
        }
      ],
      "execution_count": 8,
      "metadata": {
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          "source_hidden": false,
          "outputs_hidden": false
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          "transient": {
            "deleting": false
          }
        },
        "execution": {
          "iopub.status.busy": "2022-05-10T00:54:27.199Z",
          "iopub.execute_input": "2022-05-10T00:54:27.204Z",
          "iopub.status.idle": "2022-05-10T00:54:27.216Z",
          "shell.execute_reply": "2022-05-10T00:54:27.394Z"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "p = 0.5 # 50% Success\n",
        "k = 100 # Number of Trials\n",
        "\n",
        "# define the distribution\n",
        "dist = binom(k, p)\n",
        "for n in repeat_numbers:\n",
        "\tprint('P of {} success: {}'.format(n, round(dist.cdf(n)*100),6))"
      ],
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "P of 79.05999755859375 success: 100\n",
            "P of 82.76000213623047 success: 100\n",
            "P of 78.55000305175781 success: 100\n",
            "P of 106.1500015258789 success: 100\n"
          ]
        }
      ],
      "execution_count": 9,
      "metadata": {
        "collapsed": true,
        "jupyter": {
          "source_hidden": false,
          "outputs_hidden": false
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          "transient": {
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          }
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        "execution": {
          "iopub.status.busy": "2022-05-10T00:54:27.225Z",
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          "shell.execute_reply": "2022-05-10T00:54:27.397Z"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "min_n = min(repeat_numbers)\n",
        "max_n = max(repeat_numbers)"
      ],
      "outputs": [],
      "execution_count": 10,
      "metadata": {
        "collapsed": true,
        "jupyter": {
          "source_hidden": false,
          "outputs_hidden": false
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          "transient": {
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        "execution": {
          "iopub.status.busy": "2022-05-10T00:54:27.248Z",
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          "shell.execute_reply": "2022-05-10T00:54:27.400Z"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "p = 0.5 # 50% Success\n",
        "k = 100 # Number of Trials\n",
        "\n",
        "# define the distribution\n",
        "dist = binom(k, p)\n",
        "for n in range(int(min_n), int(max_n), 2):\n",
        "\tprint('P of {} success: {}'.format(n, round(dist.pmf(n)*100),6))"
      ],
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "P of 78 success: 0\n",
            "P of 80 success: 0\n",
            "P of 82 success: 0\n",
            "P of 84 success: 0\n",
            "P of 86 success: 0\n",
            "P of 88 success: 0\n",
            "P of 90 success: 0\n",
            "P of 92 success: 0\n",
            "P of 94 success: 0\n",
            "P of 96 success: 0\n",
            "P of 98 success: 0\n",
            "P of 100 success: 0\n",
            "P of 102 success: 0\n",
            "P of 104 success: 0\n"
          ]
        }
      ],
      "execution_count": 11,
      "metadata": {
        "collapsed": true,
        "jupyter": {
          "source_hidden": false,
          "outputs_hidden": false
        },
        "nteract": {
          "transient": {
            "deleting": false
          }
        },
        "execution": {
          "iopub.status.busy": "2022-05-10T00:54:27.269Z",
          "iopub.execute_input": "2022-05-10T00:54:27.274Z",
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          "shell.execute_reply": "2022-05-10T00:54:27.403Z"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "probabilities = [0.2, 0.3, 0.3, 0.2]\n",
        "\n",
        "np.random.choice(repeat_numbers, 10, p=probabilities)"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 12,
          "data": {
            "text/plain": "array([ 78.55000305, 106.15000153, 106.15000153,  78.55000305,\n        79.05999756,  78.55000305,  82.76000214,  82.76000214,\n        82.76000214, 106.15000153])"
          },
          "metadata": {}
        }
      ],
      "execution_count": 12,
      "metadata": {
        "collapsed": true,
        "jupyter": {
          "source_hidden": false,
          "outputs_hidden": false
        },
        "nteract": {
          "transient": {
            "deleting": false
          }
        },
        "execution": {
          "iopub.status.busy": "2022-05-10T00:54:27.291Z",
          "iopub.execute_input": "2022-05-10T00:54:27.296Z",
          "iopub.status.idle": "2022-05-10T00:54:27.306Z",
          "shell.execute_reply": "2022-05-10T00:54:27.407Z"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "from scipy.stats import rv_discrete  \n",
        "\n",
        "distrib = rv_discrete(values=(repeat_numbers, probabilities))\n",
        "distrib.rvs(size=10)\n"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 13,
          "data": {
            "text/plain": "array([106,  78, 106,  78,  79,  82, 106, 106,  82,  82])"
          },
          "metadata": {}
        }
      ],
      "execution_count": 13,
      "metadata": {
        "collapsed": true,
        "jupyter": {
          "source_hidden": false,
          "outputs_hidden": false
        },
        "nteract": {
          "transient": {
            "deleting": false
          }
        },
        "execution": {
          "iopub.status.busy": "2022-05-10T00:54:27.313Z",
          "iopub.execute_input": "2022-05-10T00:54:27.321Z",
          "iopub.status.idle": "2022-05-10T00:54:27.334Z",
          "shell.execute_reply": "2022-05-10T00:54:27.410Z"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "ax=sns.distplot(repeat_numbers,\n",
        "                kde=True,\n",
        "                color='blue',\n",
        "                hist_kws={\"linewidth\":10,'alpha':0.50})\n",
        "ax.set(xlabel='Binomial',ylabel='Frequency')"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 14,
          "data": {
            "text/plain": "[Text(0.5, 0, 'Binomial'), Text(0, 0.5, 'Frequency')]"
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": "<Figure size 432x288 with 1 Axes>",
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "execution_count": 14,
      "metadata": {
        "collapsed": true,
        "jupyter": {
          "source_hidden": false,
          "outputs_hidden": false
        },
        "nteract": {
          "transient": {
            "deleting": false
          }
        },
        "execution": {
          "iopub.status.busy": "2022-05-10T00:54:27.341Z",
          "iopub.execute_input": "2022-05-10T00:54:27.346Z",
          "shell.execute_reply": "2022-05-10T00:54:27.413Z",
          "iopub.status.idle": "2022-05-10T00:54:27.419Z"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import scipy.stats as stats\n",
        "\n",
        "binomial=stats.binom(n=0.05,p=0.002)\n",
        "plt.scatter(repeat_numbers,binomial.pmf(repeat_numbers),label=\"PMF\",color=\"Orange\")\n",
        "plt.title(\"Binomial Distribution\")\n",
        "plt.ylabel(\"Density\")\n",
        "plt.legend()\n",
        "#plt.xticks(repeat_numbers)\n",
        "plt.show()"
      ],
      "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"
          }
        }
      ],
      "execution_count": 15,
      "metadata": {
        "collapsed": true,
        "jupyter": {
          "source_hidden": false,
          "outputs_hidden": false
        },
        "nteract": {
          "transient": {
            "deleting": false
          }
        },
        "execution": {
          "iopub.status.busy": "2022-05-10T00:54:27.426Z",
          "iopub.execute_input": "2022-05-10T00:54:27.430Z",
          "iopub.status.idle": "2022-05-10T00:54:27.515Z",
          "shell.execute_reply": "2022-05-10T00:54:27.471Z"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "plt.hist(repeat_numbers,bins=14,density=True,align='left',rwidth=0.75,label=\"Sample\",alpha=0.75)\n",
        "plt.title(\"Binomial Distribution\")\n",
        "plt.ylabel(\"Density\")\n",
        "plt.legend()\n",
        "#plt.xticks(repeat_numbers)\n",
        "plt.show()"
      ],
      "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"
          }
        }
      ],
      "execution_count": 16,
      "metadata": {
        "collapsed": true,
        "jupyter": {
          "source_hidden": false,
          "outputs_hidden": false
        },
        "nteract": {
          "transient": {
            "deleting": false
          }
        },
        "execution": {
          "iopub.status.busy": "2022-05-10T00:54:27.527Z",
          "iopub.execute_input": "2022-05-10T00:54:27.533Z",
          "iopub.status.idle": "2022-05-10T00:54:27.615Z",
          "shell.execute_reply": "2022-05-10T00:54:27.634Z"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "plt.hist(repeat_numbers,bins=14,density=True,align='left',rwidth=0.75,label=\"Sample\",alpha=0.75)\n",
        "plt.scatter(repeat_numbers,binomial.pmf(repeat_numbers),label=\"PMF\",color=\"Orange\")\n",
        "plt.title(\"Binomial Distribution\")\n",
        "plt.ylabel(\"Density\")\n",
        "plt.legend()\n",
        "#plt.xticks(repeat_numbers)\n",
        "plt.show()"
      ],
      "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"
          }
        }
      ],
      "execution_count": 17,
      "metadata": {
        "collapsed": true,
        "jupyter": {
          "source_hidden": false,
          "outputs_hidden": false
        },
        "nteract": {
          "transient": {
            "deleting": false
          }
        },
        "execution": {
          "iopub.status.busy": "2022-05-10T00:54:27.624Z",
          "iopub.execute_input": "2022-05-10T00:54:27.628Z",
          "iopub.status.idle": "2022-05-10T00:54:27.734Z",
          "shell.execute_reply": "2022-05-10T00:54:27.820Z"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Calculate Returns "
      ],
      "metadata": {
        "nteract": {
          "transient": {
            "deleting": false
          }
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# View Columns\n",
        "dataset['Returns'] = dataset['Adj Close'].pct_change()\n",
        "dataset = dataset.dropna()\n",
        "dataset.head()"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 18,
          "data": {
            "text/plain": "                 Open       High        Low      Close  Adj Close    Volume  \\\nDate                                                                          \n2021-01-05  92.099998  93.209999  91.410004  92.769997  92.769997  34208000   \n2021-01-06  91.620003  92.279999  89.459999  90.330002  90.330002  51911700   \n2021-01-07  91.330002  95.510002  91.199997  95.160004  95.160004  42897200   \n2021-01-08  95.980003  96.400002  93.269997  94.580002  94.580002  39816400   \n2021-01-11  94.029999  99.230003  93.760002  97.250000  97.250000  48600200   \n\n             Returns  \nDate                  \n2021-01-05  0.005092  \n2021-01-06 -0.026302  \n2021-01-07  0.053471  \n2021-01-08 -0.006095  \n2021-01-11  0.028230  ",
            "text/html": "<div>\n<style scoped>\n    .dataframe tbody tr th:only-of-type {\n        vertical-align: middle;\n    }\n\n    .dataframe tbody tr th {\n        vertical-align: top;\n    }\n\n    .dataframe thead th {\n        text-align: right;\n    }\n</style>\n<table border=\"1\" class=\"dataframe\">\n  <thead>\n    <tr style=\"text-align: right;\">\n      <th></th>\n      <th>Open</th>\n      <th>High</th>\n      <th>Low</th>\n      <th>Close</th>\n      <th>Adj Close</th>\n      <th>Volume</th>\n      <th>Returns</th>\n    </tr>\n    <tr>\n      <th>Date</th>\n      <th></th>\n      <th></th>\n      <th></th>\n      <th></th>\n      <th></th>\n      <th></th>\n      <th></th>\n    </tr>\n  </thead>\n  <tbody>\n    <tr>\n      <th>2021-01-05</th>\n      <td>92.099998</td>\n      <td>93.209999</td>\n      <td>91.410004</td>\n      <td>92.769997</td>\n      <td>92.769997</td>\n      <td>34208000</td>\n      <td>0.005092</td>\n    </tr>\n    <tr>\n      <th>2021-01-06</th>\n      <td>91.620003</td>\n      <td>92.279999</td>\n      <td>89.459999</td>\n      <td>90.330002</td>\n      <td>90.330002</td>\n      <td>51911700</td>\n      <td>-0.026302</td>\n    </tr>\n    <tr>\n      <th>2021-01-07</th>\n      <td>91.330002</td>\n      <td>95.510002</td>\n      <td>91.199997</td>\n      <td>95.160004</td>\n      <td>95.160004</td>\n      <td>42897200</td>\n      <td>0.053471</td>\n    </tr>\n    <tr>\n      <th>2021-01-08</th>\n      <td>95.980003</td>\n      <td>96.400002</td>\n      <td>93.269997</td>\n      <td>94.580002</td>\n      <td>94.580002</td>\n      <td>39816400</td>\n      <td>-0.006095</td>\n    </tr>\n    <tr>\n      <th>2021-01-11</th>\n      <td>94.029999</td>\n      <td>99.230003</td>\n      <td>93.760002</td>\n      <td>97.250000</td>\n      <td>97.250000</td>\n      <td>48600200</td>\n      <td>0.028230</td>\n    </tr>\n  </tbody>\n</table>\n</div>"
          },
          "metadata": {}
        }
      ],
      "execution_count": 18,
      "metadata": {
        "collapsed": true,
        "jupyter": {
          "source_hidden": false,
          "outputs_hidden": false
        },
        "nteract": {
          "transient": {
            "deleting": false
          }
        },
        "execution": {
          "iopub.status.busy": "2022-05-10T00:54:27.742Z",
          "iopub.execute_input": "2022-05-10T00:54:27.747Z",
          "iopub.status.idle": "2022-05-10T00:54:27.759Z",
          "shell.execute_reply": "2022-05-10T00:54:27.824Z"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "r = np.array(dataset['Returns']).squeeze()\n",
        "r"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 19,
          "data": {
            "text/plain": "array([ 0.00509202, -0.02630155,  0.05347063, -0.00609502,  0.02823005,\n       -0.01943444, -0.03754197, -0.01078664, -0.02841725,  0.01405734,\n       -0.00782557,  0.03132393,  0.013766  ,  0.01444117,  0.00616171,\n       -0.0619787 , -0.01485817, -0.02148077,  0.02358716,  0.01368922,\n       -0.01091606, -0.00056893,  0.00068312,  0.04061433, -0.0061222 ,\n        0.01583978,  0.00335685,  0.0119792 , -0.02463472, -0.01661925,\n       -0.01445411,  0.01060472, -0.04699709, -0.0073797 ,  0.02596182,\n       -0.05198993,  0.02535797,  0.02224585, -0.02616046, -0.03886838,\n       -0.03846155,  0.00990349, -0.05807435,  0.06179015, -0.01286135,\n        0.04785871, -0.00221593,  0.01789015,  0.0030303 , -0.00145018,\n       -0.0545806 ,  0.0120327 ,  0.01568436, -0.02391041, -0.0242408 ,\n       -0.00339961,  0.01561273, -0.00348798, -0.01477832,  0.03289474,\n        0.03299358,  0.00419292,  0.00012283,  0.00933196,  0.01399029,\n       -0.00707854, -0.05050749,  0.02048868, -0.02045142,  0.05677911,\n       -0.0103602 , -0.01265978, -0.02268529,  0.02951942, -0.03124621,\n        0.04679996,  0.03202032, -0.0023417 , -0.01396553, -0.00130913,\n       -0.02729115, -0.03761333,  0.00076381, -0.00992239,  0.00077088,\n        0.01181151, -0.03578226,  0.01105414, -0.02850452, -0.02076639,\n        0.02052264,  0.00080447, -0.00281312,  0.02404622,  0.02400622,\n       -0.01140148,  0.00349882,  0.00542353,  0.00616486,  0.00102121,\n        0.02116812,  0.00911583,  0.0143547 , -0.02061733,  0.01619336,\n       -0.00281936, -0.00565457, -0.0114971 ,  0.02000999, -0.00306523,\n        0.00295173, -0.01324343, -0.00447372,  0.05554858,  0.00106438,\n       -0.02433556,  0.01198699,  0.00287147,  0.02720113, -0.00557486,\n        0.01705208,  0.02802015,  0.04926278, -0.00660069,  0.01489658,\n       -0.00242868, -0.04160051, -0.00883591,  0.01292627, -0.00099014,\n       -0.00605655, -0.01340571, -0.02380688, -0.01196366,  0.00803356,\n        0.00612149,  0.02640343,  0.02013192,  0.01030591, -0.00358114,\n       -0.0086038 ,  0.0757992 ,  0.05126107,  0.03147164,  0.02297763,\n        0.03617786,  0.05517057, -0.05405404, -0.01993768, -0.02297701,\n       -0.01022493,  0.01126969, -0.0109584 ,  0.0380282 , -0.02777024,\n        0.00074427, -0.03830416,  0.00251348,  0.00916109,  0.03936928,\n       -0.01029691,  0.0060381 , -0.00951068,  0.03850103, -0.00071815,\n       -0.00538985, -0.00659324, -0.00718248,  0.00659342, -0.00700506,\n       -0.02730191, -0.00018835, -0.00894964, -0.00380222,  0.00887405,\n       -0.00122959,  0.00587124, -0.02202979, -0.02242967,  0.01250612,\n        0.01517212,  0.01695731, -0.00329721,  0.02230624, -0.0613906 ,\n       -0.01152481,  0.02541109, -0.00437322, -0.02059542,  0.0146502 ,\n        0.01797468,  0.02711306, -0.01305777, -0.00361695,  0.00343906,\n        0.03922318,  0.02592519,  0.00116086,  0.03844093, -0.00085887,\n        0.00051575,  0.02525992,  0.00410624,  0.02119847,  0.00465838,\n       -0.00528757, -0.00915927, -0.0076758 ,  0.04158696,  0.01916469,\n        0.02272194,  0.0533977 , -0.00843639,  0.10136429, -0.00825789,\n       -0.0607709 ,  0.0438979 ,  0.01287586, -0.00946645,  0.04068531,\n       -0.00728108,  0.02431616,  0.0025158 , -0.01859597, -0.01704698,\n        0.0525614 , -0.01894807,  0.04586271, -0.02186405, -0.05847064,\n        0.01052909, -0.04426598, -0.03437259,  0.04163677,  0.00269244,\n       -0.04916   ,  0.00325849, -0.03428365,  0.01345294,  0.08038343,\n       -0.05365188, -0.0064195 , -0.01415606,  0.06222383, -0.00256496,\n        0.0157075 ,  0.05624744, -0.00783886, -0.03192948, -0.02097667,\n       -0.00861178])"
          },
          "metadata": {}
        }
      ],
      "execution_count": 19,
      "metadata": {
        "collapsed": true,
        "jupyter": {
          "source_hidden": false,
          "outputs_hidden": false
        },
        "nteract": {
          "transient": {
            "deleting": false
          }
        },
        "execution": {
          "iopub.status.busy": "2022-05-10T00:54:27.768Z",
          "iopub.execute_input": "2022-05-10T00:54:27.773Z",
          "iopub.status.idle": "2022-05-10T00:54:27.784Z",
          "shell.execute_reply": "2022-05-10T00:54:27.827Z"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "\n",
        "repeat_r = [item for item, count in collections.Counter(r).items() if count > -2]\n",
        "repeat_r\n"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 21,
          "data": {
            "text/plain": "[0.0050920214059477775,\n -0.02630155115128041,\n 0.05347062695834204,\n -0.0060950169055702386,\n 0.028230049875814567,\n -0.01943444102466263,\n -0.03754196526993381,\n -0.010786640629078414,\n -0.028417246448264377,\n 0.014057339039104555,\n -0.007825567044426163,\n 0.03132392990757049,\n 0.013766001890469504,\n 0.01444117172830417,\n 0.006161710910212781,\n -0.06197870133381089,\n -0.014858169171954705,\n -0.021480773829153366,\n 0.023587158884369464,\n 0.013689218550202664,\n -0.010916061377903352,\n -0.0005689276607697691,\n 0.0006831191995666508,\n 0.04061433029410311,\n -0.006122199093914538,\n 0.015839783896212767,\n 0.0033568510353050307,\n 0.01197920286086629,\n -0.024634719433623387,\n -0.0166192505825693,\n -0.014454114036796195,\n 0.010604720756756114,\n -0.04699708638555955,\n -0.0073796985187249975,\n 0.02596182007431147,\n -0.051989925759517,\n 0.02535797153260111,\n 0.02224585499817411,\n -0.026160460148136888,\n -0.03886837929182929,\n 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0.03850102556220647,\n -0.0007181492814980484,\n -0.005389853357581242,\n -0.006593238338919893,\n -0.007182479596970381,\n 0.006593417956269665,\n -0.007005064191166865,\n -0.027301908522896823,\n -0.0001883455157886127,\n -0.008949642618753195,\n -0.0038022234609110805,\n 0.008874048455098649,\n -0.0012295931020981365,\n 0.00587123821534874,\n -0.022029786672880136,\n -0.02242967138299079,\n 0.012506121170859252,\n 0.015172121797548233,\n 0.01695731288595126,\n -0.003297206491662319,\n 0.022306243310759033,\n -0.06139059536079783,\n -0.011524805039729302,\n 0.02541109208302994,\n -0.004373222263981624,\n -0.020595418967338186,\n 0.014650202056545325,\n 0.017974677093980818,\n 0.027113060354517993,\n -0.013057768243283974,\n -0.003616954713957954,\n 0.0034390581706347234,\n 0.039223178890633426,\n 0.025925193355801124,\n 0.0011608615527489707,\n 0.03844093340182453,\n -0.0008588720592543897,\n 0.0005157530959285506,\n 0.02525992316198722,\n 0.004106241986514547,\n 0.021198472058058693,\n 0.004658382575849718,\n -0.005287574426627084,\n -0.009159266669678146,\n -0.0076758028810346435,\n 0.04158695716872107,\n 0.019164687631955557,\n 0.02272194302504582,\n 0.05339769620689383,\n -0.008436390269886407,\n 0.10136429291056093,\n -0.008257894665175458,\n -0.06077090493575521,\n 0.04389790236643787,\n 0.012875864348618205,\n -0.009466454136603186,\n 0.04068531115835228,\n -0.007281079911916399,\n 0.02431616243966417,\n 0.0025158004057526107,\n -0.018595967579615014,\n -0.017046984203271998,\n 0.05256139927331449,\n -0.018948069932440692,\n 0.04586271051925017,\n -0.021864050798921175,\n -0.058470637067229214,\n 0.010529086305433921,\n -0.0442659841595372,\n -0.0343725931328136,\n 0.04163676575991748,\n 0.0026924361285129983,\n -0.04916000495458872,\n 0.0032584860851119224,\n -0.03428365135600575,\n 0.01345293729972119,\n 0.08038343219662858,\n -0.053651881299328075,\n -0.0064194993765622765,\n -0.014156057700487779,\n 0.062223834744846274,\n -0.0025649574848353174,\n 0.01570749534430904,\n 0.05624744255531566,\n -0.007838861810590347,\n -0.03192947818825009,\n -0.020976667513691094,\n -0.008611781278416442]"
          },
          "metadata": {}
        }
      ],
      "execution_count": 21,
      "metadata": {
        "collapsed": true,
        "jupyter": {
          "source_hidden": false,
          "outputs_hidden": false
        },
        "nteract": {
          "transient": {
            "deleting": false
          }
        },
        "execution": {
          "iopub.status.busy": "2022-05-10T00:55:50.381Z",
          "iopub.execute_input": "2022-05-10T00:55:50.388Z",
          "iopub.status.idle": "2022-05-10T00:55:50.405Z",
          "shell.execute_reply": "2022-05-10T00:55:50.421Z"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def Average(lst):\n",
        "    return sum(lst) / len(lst)"
      ],
      "outputs": [],
      "execution_count": 26,
      "metadata": {
        "collapsed": true,
        "jupyter": {
          "source_hidden": false,
          "outputs_hidden": false
        },
        "nteract": {
          "transient": {
            "deleting": false
          }
        },
        "execution": {
          "iopub.status.busy": "2022-05-10T00:58:40.037Z",
          "iopub.execute_input": "2022-05-10T00:58:40.047Z",
          "iopub.status.idle": "2022-05-10T00:58:40.059Z",
          "shell.execute_reply": "2022-05-10T00:58:40.071Z"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "n = len(repeat_r)\n",
        "p = Average(repeat_r)\n",
        "k = np.arange(0,21)\n",
        "binomial = binom.pmf(k,n,p)\n",
        "\n",
        "plt.plot(k, binomial, 'o-')\n",
        "plt.title(\"Binomial: n=%i, p=%.2f\" % (n,p), fontsize=15)\n",
        "plt.xlabel(\"Number of successes\", fontsize=15)\n",
        "plt.ylabel(\"Probability of successes\", fontsize=15)\n",
        "plt.show()"
      ],
      "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"
          }
        }
      ],
      "execution_count": 27,
      "metadata": {
        "collapsed": true,
        "jupyter": {
          "source_hidden": false,
          "outputs_hidden": false
        },
        "nteract": {
          "transient": {
            "deleting": false
          }
        },
        "execution": {
          "iopub.status.busy": "2022-05-10T00:58:40.727Z",
          "iopub.execute_input": "2022-05-10T00:58:40.734Z",
          "iopub.status.idle": "2022-05-10T00:58:40.808Z",
          "shell.execute_reply": "2022-05-10T00:58:40.823Z"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "ax=sns.distplot(repeat_r,\n",
        "                kde=True,\n",
        "                color='blue',\n",
        "                hist_kws={\"linewidth\":10,'alpha':0.50})\n",
        "ax.set(xlabel='Binomial',ylabel='Frequency')"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 28,
          "data": {
            "text/plain": "[Text(0.5, 0, 'Binomial'), Text(0, 0.5, 'Frequency')]"
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": "<Figure size 432x288 with 1 Axes>",
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "execution_count": 28,
      "metadata": {
        "collapsed": true,
        "jupyter": {
          "source_hidden": false,
          "outputs_hidden": false
        },
        "nteract": {
          "transient": {
            "deleting": false
          }
        },
        "execution": {
          "iopub.status.busy": "2022-05-10T00:59:17.585Z",
          "iopub.execute_input": "2022-05-10T00:59:17.595Z",
          "shell.execute_reply": "2022-05-10T00:59:17.689Z",
          "iopub.status.idle": "2022-05-10T00:59:17.703Z"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "binomial=stats.binom(n=10,p=0.05)\n",
        "plt.scatter(repeat_r,binomial.pmf(repeat_r),label=\"PMF\",color=\"Orange\")\n",
        "plt.title(\"Binomial Distribution\")\n",
        "plt.ylabel(\"Density\")\n",
        "plt.legend()\n",
        "#plt.xticks(repeat_numbers)\n",
        "plt.show()"
      ],
      "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"
          }
        }
      ],
      "execution_count": 30,
      "metadata": {
        "collapsed": true,
        "jupyter": {
          "source_hidden": false,
          "outputs_hidden": false
        },
        "nteract": {
          "transient": {
            "deleting": false
          }
        },
        "execution": {
          "iopub.status.busy": "2022-05-10T00:59:57.995Z",
          "iopub.execute_input": "2022-05-10T00:59:58.004Z",
          "iopub.status.idle": "2022-05-10T00:59:58.085Z",
          "shell.execute_reply": "2022-05-10T00:59:58.100Z"
        }
      }
    }
  ],
  "metadata": {
    "kernel_info": {
      "name": "python3"
    },
    "language_info": {
      "name": "python",
      "version": "3.6.13",
      "mimetype": "text/x-python",
      "codemirror_mode": {
        "name": "ipython",
        "version": 3
      },
      "pygments_lexer": "ipython3",
      "nbconvert_exporter": "python",
      "file_extension": ".py"
    },
    "kernelspec": {
      "argv": [
        "C:/Users/Tin Hang/Anaconda3\\python.exe",
        "-m",
        "ipykernel_launcher",
        "-f",
        "{connection_file}"
      ],
      "display_name": "Python 3",
      "language": "python",
      "name": "python3"
    },
    "nteract": {
      "version": "0.28.0"
    }
  },
  "nbformat": 4,
  "nbformat_minor": 0
}