Engine_change_Optimal_policy.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "import random as rnd\n",
    "import scipy.stats as stats\n",
    "import scipy.optimize as opt\n",
    "import json as json\n",
    "import matplotlib as mpl\n",
    "from math import exp\n",
    "from matplotlib import pyplot as plt\n",
    "from IPython.core.pylabtools import figsize\n",
    "from IPython.display import display\n",
    "from IPython.core.display import HTML"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = open('C:\\\\Users\\\\jsruj\\\\Desktop\\\\637_Project\\\\a530875.asc', 'r') # 'r' = read\n",
    "data = np.genfromtxt(f, delimiter='\\n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [],
   "source": [
    "data1 = np.ndarray.tolist(data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5297.0"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data1[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "data1[-1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "nan"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data1.pop()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4736"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(data1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "128"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "first_bus = data1[0]\n",
    "second_bus = first_bus + 1\n",
    "second_bus_index = data1.index(second_bus)\n",
    "second_bus_index"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "128"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nb_data_points = second_bus_index - 1\n",
    "nb_buses = len(data1)/(nb_data_points+1)\n",
    "nb_buses"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [],
   "source": [
    "tab_data = [data1[i:i+nb_data_points+1] for i in range(0, len(data1), (nb_data_points+1))]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {},
   "outputs": [],
   "source": [
    "df_data = pd.DataFrame.from_records(tab_data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {},
   "outputs": [
    {
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       "      <th>bus_no.</th>\n",
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       "      <td>8.0</td>\n",
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       "      <td>...</td>\n",
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       "      <td>9.0</td>\n",
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       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>328889.0</td>\n",
       "      <td>331458.0</td>\n",
       "      <td>333600.0</td>\n",
       "      <td>335522.0</td>\n",
       "      <td>337600.0</td>\n",
       "      <td>339990.0</td>\n",
       "      <td>341930.0</td>\n",
       "      <td>343613.0</td>\n",
       "      <td>345402.0</td>\n",
       "      <td>346998.0</td>\n",
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       "      <td>3</td>\n",
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       "      <td>8.0</td>\n",
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       "      <td>9.0</td>\n",
       "      <td>80.0</td>\n",
       "      <td>215000.0</td>\n",
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       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
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       "      <td>359570.0</td>\n",
       "      <td>361394.0</td>\n",
       "      <td>363184.0</td>\n",
       "      <td>365448.0</td>\n",
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       "    <tr>\n",
       "      <td>4</td>\n",
       "      <td>5301.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>12.0</td>\n",
       "      <td>78.0</td>\n",
       "      <td>168600.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>335230.0</td>\n",
       "      <td>337605.0</td>\n",
       "      <td>339845.0</td>\n",
       "      <td>341834.0</td>\n",
       "      <td>343345.0</td>\n",
       "      <td>345491.0</td>\n",
       "      <td>347198.0</td>\n",
       "      <td>349063.0</td>\n",
       "      <td>351406.0</td>\n",
       "      <td>353104.0</td>\n",
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       "    <tr>\n",
       "      <td>5</td>\n",
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       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>2.0</td>\n",
       "      <td>78.0</td>\n",
       "      <td>128800.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>337645.0</td>\n",
       "      <td>339073.0</td>\n",
       "      <td>341056.0</td>\n",
       "      <td>342949.0</td>\n",
       "      <td>343973.0</td>\n",
       "      <td>345274.0</td>\n",
       "      <td>347195.0</td>\n",
       "      <td>348916.0</td>\n",
       "      <td>351212.0</td>\n",
       "      <td>352845.0</td>\n",
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       "    <tr>\n",
       "      <td>6</td>\n",
       "      <td>5303.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>12.0</td>\n",
       "      <td>81.0</td>\n",
       "      <td>285900.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>367522.0</td>\n",
       "      <td>368572.0</td>\n",
       "      <td>369151.0</td>\n",
       "      <td>371787.0</td>\n",
       "      <td>373797.0</td>\n",
       "      <td>375822.0</td>\n",
       "      <td>378410.0</td>\n",
       "      <td>380239.0</td>\n",
       "      <td>382154.0</td>\n",
       "      <td>384561.0</td>\n",
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       "    <tr>\n",
       "      <td>7</td>\n",
       "      <td>5304.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>2.0</td>\n",
       "      <td>82.0</td>\n",
       "      <td>290000.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>369178.0</td>\n",
       "      <td>371016.0</td>\n",
       "      <td>372620.0</td>\n",
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       "      <td>384550.0</td>\n",
       "      <td>386287.0</td>\n",
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       "    <tr>\n",
       "      <td>8</td>\n",
       "      <td>5305.0</td>\n",
       "      <td>7.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>6.0</td>\n",
       "      <td>83.0</td>\n",
       "      <td>387300.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>416642.0</td>\n",
       "      <td>419439.0</td>\n",
       "      <td>421473.0</td>\n",
       "      <td>423175.0</td>\n",
       "      <td>424388.0</td>\n",
       "      <td>426109.0</td>\n",
       "      <td>427598.0</td>\n",
       "      <td>428711.0</td>\n",
       "      <td>430723.0</td>\n",
       "      <td>432552.0</td>\n",
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       "    <tr>\n",
       "      <td>9</td>\n",
       "      <td>5306.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>7.0</td>\n",
       "      <td>79.0</td>\n",
       "      <td>235400.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>426247.0</td>\n",
       "      <td>429293.0</td>\n",
       "      <td>430517.0</td>\n",
       "      <td>432861.0</td>\n",
       "      <td>434710.0</td>\n",
       "      <td>436028.0</td>\n",
       "      <td>437787.0</td>\n",
       "      <td>439685.0</td>\n",
       "      <td>442313.0</td>\n",
       "      <td>444040.0</td>\n",
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       "    <tr>\n",
       "      <td>10</td>\n",
       "      <td>5307.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>81.0</td>\n",
       "      <td>251200.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>350894.0</td>\n",
       "      <td>353262.0</td>\n",
       "      <td>354826.0</td>\n",
       "      <td>357266.0</td>\n",
       "      <td>359558.0</td>\n",
       "      <td>360307.0</td>\n",
       "      <td>362902.0</td>\n",
       "      <td>365088.0</td>\n",
       "      <td>367823.0</td>\n",
       "      <td>370234.0</td>\n",
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       "    <tr>\n",
       "      <td>11</td>\n",
       "      <td>5308.0</td>\n",
       "      <td>7.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>80.0</td>\n",
       "      <td>226100.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>363240.0</td>\n",
       "      <td>365372.0</td>\n",
       "      <td>366706.0</td>\n",
       "      <td>369025.0</td>\n",
       "      <td>371273.0</td>\n",
       "      <td>373386.0</td>\n",
       "      <td>375667.0</td>\n",
       "      <td>377932.0</td>\n",
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       "      <td>382891.0</td>\n",
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       "      <td>12</td>\n",
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       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>82.0</td>\n",
       "      <td>300100.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>350668.0</td>\n",
       "      <td>352504.0</td>\n",
       "      <td>354781.0</td>\n",
       "      <td>357748.0</td>\n",
       "      <td>359338.0</td>\n",
       "      <td>361303.0</td>\n",
       "      <td>363343.0</td>\n",
       "      <td>365255.0</td>\n",
       "      <td>367222.0</td>\n",
       "      <td>369580.0</td>\n",
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       "    <tr>\n",
       "      <td>13</td>\n",
       "      <td>5310.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>12.0</td>\n",
       "      <td>80.0</td>\n",
       "      <td>246600.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>366360.0</td>\n",
       "      <td>368973.0</td>\n",
       "      <td>371187.0</td>\n",
       "      <td>373685.0</td>\n",
       "      <td>375935.0</td>\n",
       "      <td>378180.0</td>\n",
       "      <td>379761.0</td>\n",
       "      <td>382239.0</td>\n",
       "      <td>384341.0</td>\n",
       "      <td>386242.0</td>\n",
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       "    <tr>\n",
       "      <td>14</td>\n",
       "      <td>5311.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>7.0</td>\n",
       "      <td>81.0</td>\n",
       "      <td>272500.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>366107.0</td>\n",
       "      <td>369250.0</td>\n",
       "      <td>371344.0</td>\n",
       "      <td>373105.0</td>\n",
       "      <td>375003.0</td>\n",
       "      <td>376612.0</td>\n",
       "      <td>379399.0</td>\n",
       "      <td>381676.0</td>\n",
       "      <td>384072.0</td>\n",
       "      <td>385580.0</td>\n",
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       "    <tr>\n",
       "      <td>15</td>\n",
       "      <td>5312.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>84.0</td>\n",
       "      <td>328600.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>330508.0</td>\n",
       "      <td>333075.0</td>\n",
       "      <td>335252.0</td>\n",
       "      <td>337405.0</td>\n",
       "      <td>339215.0</td>\n",
       "      <td>341542.0</td>\n",
       "      <td>344913.0</td>\n",
       "      <td>347319.0</td>\n",
       "      <td>349501.0</td>\n",
       "      <td>351131.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>16</td>\n",
       "      <td>5313.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>6.0</td>\n",
       "      <td>81.0</td>\n",
       "      <td>260000.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>349893.0</td>\n",
       "      <td>352450.0</td>\n",
       "      <td>354673.0</td>\n",
       "      <td>356855.0</td>\n",
       "      <td>358615.0</td>\n",
       "      <td>361319.0</td>\n",
       "      <td>364393.0</td>\n",
       "      <td>365726.0</td>\n",
       "      <td>365811.0</td>\n",
       "      <td>368314.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>17</td>\n",
       "      <td>5314.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>85.0</td>\n",
       "      <td>358600.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>348876.0</td>\n",
       "      <td>350763.0</td>\n",
       "      <td>352623.0</td>\n",
       "      <td>353283.0</td>\n",
       "      <td>358615.0</td>\n",
       "      <td>361319.0</td>\n",
       "      <td>364393.0</td>\n",
       "      <td>365726.0</td>\n",
       "      <td>365811.0</td>\n",
       "      <td>368314.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>18</td>\n",
       "      <td>5315.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>12.0</td>\n",
       "      <td>80.0</td>\n",
       "      <td>274700.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>384880.0</td>\n",
       "      <td>387624.0</td>\n",
       "      <td>389602.0</td>\n",
       "      <td>391957.0</td>\n",
       "      <td>393740.0</td>\n",
       "      <td>395830.0</td>\n",
       "      <td>398028.0</td>\n",
       "      <td>399429.0</td>\n",
       "      <td>402019.0</td>\n",
       "      <td>404079.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>19</td>\n",
       "      <td>5316.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>11.0</td>\n",
       "      <td>77.0</td>\n",
       "      <td>121300.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>82.0</td>\n",
       "      <td>293400.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>346220.0</td>\n",
       "      <td>348264.0</td>\n",
       "      <td>350601.0</td>\n",
       "      <td>353186.0</td>\n",
       "      <td>354175.0</td>\n",
       "      <td>355608.0</td>\n",
       "      <td>357366.0</td>\n",
       "      <td>358528.0</td>\n",
       "      <td>360283.0</td>\n",
       "      <td>362564.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>20</td>\n",
       "      <td>5317.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>12.0</td>\n",
       "      <td>81.0</td>\n",
       "      <td>292400.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>365689.0</td>\n",
       "      <td>367594.0</td>\n",
       "      <td>369038.0</td>\n",
       "      <td>371526.0</td>\n",
       "      <td>373270.0</td>\n",
       "      <td>375105.0</td>\n",
       "      <td>377611.0</td>\n",
       "      <td>379184.0</td>\n",
       "      <td>382103.0</td>\n",
       "      <td>384186.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>21</td>\n",
       "      <td>5318.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>7.0</td>\n",
       "      <td>84.0</td>\n",
       "      <td>363200.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>365027.0</td>\n",
       "      <td>366708.0</td>\n",
       "      <td>369279.0</td>\n",
       "      <td>371891.0</td>\n",
       "      <td>375593.0</td>\n",
       "      <td>376582.0</td>\n",
       "      <td>379712.0</td>\n",
       "      <td>382469.0</td>\n",
       "      <td>384827.0</td>\n",
       "      <td>387153.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>22</td>\n",
       "      <td>5319.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>81.0</td>\n",
       "      <td>211500.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>291616.0</td>\n",
       "      <td>293143.0</td>\n",
       "      <td>295111.0</td>\n",
       "      <td>297142.0</td>\n",
       "      <td>298303.0</td>\n",
       "      <td>300220.0</td>\n",
       "      <td>302776.0</td>\n",
       "      <td>304776.0</td>\n",
       "      <td>307116.0</td>\n",
       "      <td>308589.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>23</td>\n",
       "      <td>5320.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>2.0</td>\n",
       "      <td>85.0</td>\n",
       "      <td>324900.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>314962.0</td>\n",
       "      <td>317056.0</td>\n",
       "      <td>318832.0</td>\n",
       "      <td>321599.0</td>\n",
       "      <td>323722.0</td>\n",
       "      <td>324934.0</td>\n",
       "      <td>326111.0</td>\n",
       "      <td>327650.0</td>\n",
       "      <td>329760.0</td>\n",
       "      <td>332586.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>24</td>\n",
       "      <td>5321.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>10.0</td>\n",
       "      <td>82.0</td>\n",
       "      <td>263100.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>301366.0</td>\n",
       "      <td>304411.0</td>\n",
       "      <td>306214.0</td>\n",
       "      <td>307997.0</td>\n",
       "      <td>309524.0</td>\n",
       "      <td>310713.0</td>\n",
       "      <td>313222.0</td>\n",
       "      <td>315081.0</td>\n",
       "      <td>316663.0</td>\n",
       "      <td>318717.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>25</td>\n",
       "      <td>5322.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>83.0</td>\n",
       "      <td>299200.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>333026.0</td>\n",
       "      <td>334695.0</td>\n",
       "      <td>336601.0</td>\n",
       "      <td>339838.0</td>\n",
       "      <td>341710.0</td>\n",
       "      <td>342888.0</td>\n",
       "      <td>343794.0</td>\n",
       "      <td>345259.0</td>\n",
       "      <td>347141.0</td>\n",
       "      <td>349474.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>26</td>\n",
       "      <td>5323.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>82.0</td>\n",
       "      <td>267600.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>317047.0</td>\n",
       "      <td>319364.0</td>\n",
       "      <td>321296.0</td>\n",
       "      <td>322940.0</td>\n",
       "      <td>325216.0</td>\n",
       "      <td>327046.0</td>\n",
       "      <td>329137.0</td>\n",
       "      <td>331335.0</td>\n",
       "      <td>333785.0</td>\n",
       "      <td>335973.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>27</td>\n",
       "      <td>5324.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>335555.0</td>\n",
       "      <td>337773.0</td>\n",
       "      <td>339291.0</td>\n",
       "      <td>341130.0</td>\n",
       "      <td>342565.0</td>\n",
       "      <td>343587.0</td>\n",
       "      <td>345185.0</td>\n",
       "      <td>346164.0</td>\n",
       "      <td>347344.0</td>\n",
       "      <td>348475.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>28</td>\n",
       "      <td>5325.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>82.0</td>\n",
       "      <td>270600.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>347470.0</td>\n",
       "      <td>350295.0</td>\n",
       "      <td>352217.0</td>\n",
       "      <td>354990.0</td>\n",
       "      <td>357386.0</td>\n",
       "      <td>359552.0</td>\n",
       "      <td>362018.0</td>\n",
       "      <td>363826.0</td>\n",
       "      <td>365974.0</td>\n",
       "      <td>368525.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>29</td>\n",
       "      <td>5326.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>82.0</td>\n",
       "      <td>264100.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>330737.0</td>\n",
       "      <td>333305.0</td>\n",
       "      <td>335566.0</td>\n",
       "      <td>338261.0</td>\n",
       "      <td>340950.0</td>\n",
       "      <td>343610.0</td>\n",
       "      <td>345513.0</td>\n",
       "      <td>347321.0</td>\n",
       "      <td>349443.0</td>\n",
       "      <td>351357.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>30</td>\n",
       "      <td>5327.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>331401.0</td>\n",
       "      <td>333903.0</td>\n",
       "      <td>336111.0</td>\n",
       "      <td>339009.0</td>\n",
       "      <td>341059.0</td>\n",
       "      <td>343533.0</td>\n",
       "      <td>346155.0</td>\n",
       "      <td>348400.0</td>\n",
       "      <td>350399.0</td>\n",
       "      <td>352450.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>31</td>\n",
       "      <td>5328.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>2.0</td>\n",
       "      <td>83.0</td>\n",
       "      <td>281500.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>322800.0</td>\n",
       "      <td>325469.0</td>\n",
       "      <td>327562.0</td>\n",
       "      <td>330175.0</td>\n",
       "      <td>331676.0</td>\n",
       "      <td>333988.0</td>\n",
       "      <td>336422.0</td>\n",
       "      <td>338398.0</td>\n",
       "      <td>340622.0</td>\n",
       "      <td>343149.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>32</td>\n",
       "      <td>5329.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>82.0</td>\n",
       "      <td>257400.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>321382.0</td>\n",
       "      <td>323325.0</td>\n",
       "      <td>325169.0</td>\n",
       "      <td>327095.0</td>\n",
       "      <td>329342.0</td>\n",
       "      <td>331505.0</td>\n",
       "      <td>333778.0</td>\n",
       "      <td>335685.0</td>\n",
       "      <td>337583.0</td>\n",
       "      <td>339677.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>33</td>\n",
       "      <td>5330.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>292266.0</td>\n",
       "      <td>294345.0</td>\n",
       "      <td>296505.0</td>\n",
       "      <td>298285.0</td>\n",
       "      <td>300443.0</td>\n",
       "      <td>302478.0</td>\n",
       "      <td>305009.0</td>\n",
       "      <td>306988.0</td>\n",
       "      <td>308699.0</td>\n",
       "      <td>310910.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>34</td>\n",
       "      <td>5331.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>2.0</td>\n",
       "      <td>84.0</td>\n",
       "      <td>333800.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>346208.0</td>\n",
       "      <td>349408.0</td>\n",
       "      <td>352181.0</td>\n",
       "      <td>355371.0</td>\n",
       "      <td>358279.0</td>\n",
       "      <td>361072.0</td>\n",
       "      <td>364553.0</td>\n",
       "      <td>366981.0</td>\n",
       "      <td>368981.0</td>\n",
       "      <td>370926.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>35</td>\n",
       "      <td>5332.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>6.0</td>\n",
       "      <td>79.0</td>\n",
       "      <td>177500.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>336782.0</td>\n",
       "      <td>339635.0</td>\n",
       "      <td>341621.0</td>\n",
       "      <td>344331.0</td>\n",
       "      <td>346348.0</td>\n",
       "      <td>348836.0</td>\n",
       "      <td>351104.0</td>\n",
       "      <td>352990.0</td>\n",
       "      <td>355428.0</td>\n",
       "      <td>357659.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>36</td>\n",
       "      <td>5333.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>75.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>...</td>\n",
       "      <td>334496.0</td>\n",
       "      <td>336212.0</td>\n",
       "      <td>337879.0</td>\n",
       "      <td>339786.0</td>\n",
       "      <td>341331.0</td>\n",
       "      <td>342004.0</td>\n",
       "      <td>343654.0</td>\n",
       "      <td>345631.0</td>\n",
       "      <td>347549.0</td>\n",
       "      <td>347549.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>37 rows × 128 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "    bus_no.  month_purchased  year_purchased  month of first replace  \\\n",
       "0    5297.0              8.0            75.0                     4.0   \n",
       "1    5298.0              8.0            75.0                     0.0   \n",
       "2    5299.0              8.0            75.0                     9.0   \n",
       "3    5300.0              8.0            75.0                     9.0   \n",
       "4    5301.0              8.0            75.0                    12.0   \n",
       "5    5302.0              8.0            75.0                     2.0   \n",
       "6    5303.0              8.0            75.0                    12.0   \n",
       "7    5304.0              8.0            75.0                     2.0   \n",
       "8    5305.0              7.0            75.0                     6.0   \n",
       "9    5306.0              8.0            75.0                     7.0   \n",
       "10   5307.0              8.0            75.0                     4.0   \n",
       "11   5308.0              7.0            75.0                     1.0   \n",
       "12   5309.0              8.0            75.0                     9.0   \n",
       "13   5310.0              8.0            75.0                    12.0   \n",
       "14   5311.0              8.0            75.0                     7.0   \n",
       "15   5312.0              8.0            75.0                     8.0   \n",
       "16   5313.0              8.0            75.0                     6.0   \n",
       "17   5314.0              8.0            75.0                     3.0   \n",
       "18   5315.0              8.0            75.0                    12.0   \n",
       "19   5316.0              8.0            75.0                    11.0   \n",
       "20   5317.0              8.0            75.0                    12.0   \n",
       "21   5318.0              8.0            75.0                     7.0   \n",
       "22   5319.0              8.0            75.0                     4.0   \n",
       "23   5320.0              8.0            75.0                     2.0   \n",
       "24   5321.0              8.0            75.0                    10.0   \n",
       "25   5322.0              8.0            75.0                     4.0   \n",
       "26   5323.0              8.0            75.0                     8.0   \n",
       "27   5324.0              8.0            75.0                     0.0   \n",
       "28   5325.0              8.0            75.0                     1.0   \n",
       "29   5326.0              8.0            75.0                     3.0   \n",
       "30   5327.0              9.0            75.0                     0.0   \n",
       "31   5328.0              8.0            75.0                     2.0   \n",
       "32   5329.0              8.0            75.0                     1.0   \n",
       "33   5330.0              8.0            75.0                     0.0   \n",
       "34   5331.0              8.0            75.0                     2.0   \n",
       "35   5332.0              8.0            75.0                     6.0   \n",
       "36   5333.0              8.0            75.0                     0.0   \n",
       "\n",
       "    year of first replace  odomoter at replace  month of second replace  \\\n",
       "0                    79.0             153400.0                      0.0   \n",
       "1                     0.0                  0.0                      0.0   \n",
       "2                    79.0             209100.0                      0.0   \n",
       "3                    80.0             215000.0                      0.0   \n",
       "4                    78.0             168600.0                      0.0   \n",
       "5                    78.0             128800.0                      0.0   \n",
       "6                    81.0             285900.0                      0.0   \n",
       "7                    82.0             290000.0                      0.0   \n",
       "8                    83.0             387300.0                      0.0   \n",
       "9                    79.0             235400.0                      0.0   \n",
       "10                   81.0             251200.0                      0.0   \n",
       "11                   80.0             226100.0                      0.0   \n",
       "12                   82.0             300100.0                      0.0   \n",
       "13                   80.0             246600.0                      0.0   \n",
       "14                   81.0             272500.0                      0.0   \n",
       "15                   84.0             328600.0                      0.0   \n",
       "16                   81.0             260000.0                      0.0   \n",
       "17                   85.0             358600.0                      0.0   \n",
       "18                   80.0             274700.0                      0.0   \n",
       "19                   77.0             121300.0                      5.0   \n",
       "20                   81.0             292400.0                      0.0   \n",
       "21                   84.0             363200.0                      0.0   \n",
       "22                   81.0             211500.0                      0.0   \n",
       "23                   85.0             324900.0                      0.0   \n",
       "24                   82.0             263100.0                      0.0   \n",
       "25                   83.0             299200.0                      0.0   \n",
       "26                   82.0             267600.0                      0.0   \n",
       "27                    0.0                  0.0                      0.0   \n",
       "28                   82.0             270600.0                      0.0   \n",
       "29                   82.0             264100.0                      0.0   \n",
       "30                    0.0                  0.0                      0.0   \n",
       "31                   83.0             281500.0                      0.0   \n",
       "32                   82.0             257400.0                      0.0   \n",
       "33                    0.0                  0.0                      0.0   \n",
       "34                   84.0             333800.0                      0.0   \n",
       "35                   79.0             177500.0                      0.0   \n",
       "36                    0.0                  0.0                      0.0   \n",
       "\n",
       "    year of second replace  odomoter at replace  month odomoter data began  \\\n",
       "0                      0.0                  0.0                        9.0   \n",
       "1                      0.0                  0.0                        9.0   \n",
       "2                      0.0                  0.0                        9.0   \n",
       "3                      0.0                  0.0                        9.0   \n",
       "4                      0.0                  0.0                        9.0   \n",
       "5                      0.0                  0.0                        9.0   \n",
       "6                      0.0                  0.0                        9.0   \n",
       "7                      0.0                  0.0                        9.0   \n",
       "8                      0.0                  0.0                        9.0   \n",
       "9                      0.0                  0.0                        9.0   \n",
       "10                     0.0                  0.0                        9.0   \n",
       "11                     0.0                  0.0                        9.0   \n",
       "12                     0.0                  0.0                        9.0   \n",
       "13                     0.0                  0.0                        9.0   \n",
       "14                     0.0                  0.0                        9.0   \n",
       "15                     0.0                  0.0                        9.0   \n",
       "16                     0.0                  0.0                        9.0   \n",
       "17                     0.0                  0.0                        9.0   \n",
       "18                     0.0                  0.0                        9.0   \n",
       "19                    82.0             293400.0                        9.0   \n",
       "20                     0.0                  0.0                        9.0   \n",
       "21                     0.0                  0.0                        9.0   \n",
       "22                     0.0                  0.0                        9.0   \n",
       "23                     0.0                  0.0                        9.0   \n",
       "24                     0.0                  0.0                        9.0   \n",
       "25                     0.0                  0.0                        9.0   \n",
       "26                     0.0                  0.0                        9.0   \n",
       "27                     0.0                  0.0                        9.0   \n",
       "28                     0.0                  0.0                        9.0   \n",
       "29                     0.0                  0.0                        9.0   \n",
       "30                     0.0                  0.0                        9.0   \n",
       "31                     0.0                  0.0                        9.0   \n",
       "32                     0.0                  0.0                        9.0   \n",
       "33                     0.0                  0.0                        9.0   \n",
       "34                     0.0                  0.0                        9.0   \n",
       "35                     0.0                  0.0                        9.0   \n",
       "36                     0.0                  0.0                        9.0   \n",
       "\n",
       "    ...       118       119       120       121       122       123       124  \\\n",
       "0   ...  295115.0  296436.0  297957.0  299865.0  301796.0  303258.0  305493.0   \n",
       "1   ...  308882.0  310369.0  312719.0  315040.0  316984.0  319330.0  320645.0   \n",
       "2   ...  328889.0  331458.0  333600.0  335522.0  337600.0  339990.0  341930.0   \n",
       "3   ...  345888.0  348561.0  350557.0  352565.0  354530.0  357341.0  359570.0   \n",
       "4   ...  335230.0  337605.0  339845.0  341834.0  343345.0  345491.0  347198.0   \n",
       "5   ...  337645.0  339073.0  341056.0  342949.0  343973.0  345274.0  347195.0   \n",
       "6   ...  367522.0  368572.0  369151.0  371787.0  373797.0  375822.0  378410.0   \n",
       "7   ...  369178.0  371016.0  372620.0  374811.0  376954.0  379104.0  380907.0   \n",
       "8   ...  416642.0  419439.0  421473.0  423175.0  424388.0  426109.0  427598.0   \n",
       "9   ...  426247.0  429293.0  430517.0  432861.0  434710.0  436028.0  437787.0   \n",
       "10  ...  350894.0  353262.0  354826.0  357266.0  359558.0  360307.0  362902.0   \n",
       "11  ...  363240.0  365372.0  366706.0  369025.0  371273.0  373386.0  375667.0   \n",
       "12  ...  350668.0  352504.0  354781.0  357748.0  359338.0  361303.0  363343.0   \n",
       "13  ...  366360.0  368973.0  371187.0  373685.0  375935.0  378180.0  379761.0   \n",
       "14  ...  366107.0  369250.0  371344.0  373105.0  375003.0  376612.0  379399.0   \n",
       "15  ...  330508.0  333075.0  335252.0  337405.0  339215.0  341542.0  344913.0   \n",
       "16  ...  349893.0  352450.0  354673.0  356855.0  358615.0  361319.0  364393.0   \n",
       "17  ...  348876.0  350763.0  352623.0  353283.0  358615.0  361319.0  364393.0   \n",
       "18  ...  384880.0  387624.0  389602.0  391957.0  393740.0  395830.0  398028.0   \n",
       "19  ...  346220.0  348264.0  350601.0  353186.0  354175.0  355608.0  357366.0   \n",
       "20  ...  365689.0  367594.0  369038.0  371526.0  373270.0  375105.0  377611.0   \n",
       "21  ...  365027.0  366708.0  369279.0  371891.0  375593.0  376582.0  379712.0   \n",
       "22  ...  291616.0  293143.0  295111.0  297142.0  298303.0  300220.0  302776.0   \n",
       "23  ...  314962.0  317056.0  318832.0  321599.0  323722.0  324934.0  326111.0   \n",
       "24  ...  301366.0  304411.0  306214.0  307997.0  309524.0  310713.0  313222.0   \n",
       "25  ...  333026.0  334695.0  336601.0  339838.0  341710.0  342888.0  343794.0   \n",
       "26  ...  317047.0  319364.0  321296.0  322940.0  325216.0  327046.0  329137.0   \n",
       "27  ...  335555.0  337773.0  339291.0  341130.0  342565.0  343587.0  345185.0   \n",
       "28  ...  347470.0  350295.0  352217.0  354990.0  357386.0  359552.0  362018.0   \n",
       "29  ...  330737.0  333305.0  335566.0  338261.0  340950.0  343610.0  345513.0   \n",
       "30  ...  331401.0  333903.0  336111.0  339009.0  341059.0  343533.0  346155.0   \n",
       "31  ...  322800.0  325469.0  327562.0  330175.0  331676.0  333988.0  336422.0   \n",
       "32  ...  321382.0  323325.0  325169.0  327095.0  329342.0  331505.0  333778.0   \n",
       "33  ...  292266.0  294345.0  296505.0  298285.0  300443.0  302478.0  305009.0   \n",
       "34  ...  346208.0  349408.0  352181.0  355371.0  358279.0  361072.0  364553.0   \n",
       "35  ...  336782.0  339635.0  341621.0  344331.0  346348.0  348836.0  351104.0   \n",
       "36  ...  334496.0  336212.0  337879.0  339786.0  341331.0  342004.0  343654.0   \n",
       "\n",
       "         125       126       127  \n",
       "0   307867.0  309853.0  311850.0  \n",
       "1   322814.0  324812.0  326724.0  \n",
       "2   343613.0  345402.0  346998.0  \n",
       "3   361394.0  363184.0  365448.0  \n",
       "4   349063.0  351406.0  353104.0  \n",
       "5   348916.0  351212.0  352845.0  \n",
       "6   380239.0  382154.0  384561.0  \n",
       "7   382664.0  384550.0  386287.0  \n",
       "8   428711.0  430723.0  432552.0  \n",
       "9   439685.0  442313.0  444040.0  \n",
       "10  365088.0  367823.0  370234.0  \n",
       "11  377932.0  380045.0  382891.0  \n",
       "12  365255.0  367222.0  369580.0  \n",
       "13  382239.0  384341.0  386242.0  \n",
       "14  381676.0  384072.0  385580.0  \n",
       "15  347319.0  349501.0  351131.0  \n",
       "16  365726.0  365811.0  368314.0  \n",
       "17  365726.0  365811.0  368314.0  \n",
       "18  399429.0  402019.0  404079.0  \n",
       "19  358528.0  360283.0  362564.0  \n",
       "20  379184.0  382103.0  384186.0  \n",
       "21  382469.0  384827.0  387153.0  \n",
       "22  304776.0  307116.0  308589.0  \n",
       "23  327650.0  329760.0  332586.0  \n",
       "24  315081.0  316663.0  318717.0  \n",
       "25  345259.0  347141.0  349474.0  \n",
       "26  331335.0  333785.0  335973.0  \n",
       "27  346164.0  347344.0  348475.0  \n",
       "28  363826.0  365974.0  368525.0  \n",
       "29  347321.0  349443.0  351357.0  \n",
       "30  348400.0  350399.0  352450.0  \n",
       "31  338398.0  340622.0  343149.0  \n",
       "32  335685.0  337583.0  339677.0  \n",
       "33  306988.0  308699.0  310910.0  \n",
       "34  366981.0  368981.0  370926.0  \n",
       "35  352990.0  355428.0  357659.0  \n",
       "36  345631.0  347549.0  347549.0  \n",
       "\n",
       "[37 rows x 128 columns]"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_data.rename(columns={0:'bus_no.',1:'month_purchased',2:'year_purchased',3:'month of first replace',\n",
    "                        4:'year of first replace',5:'odomoter at replace',6:'month of second replace',\n",
    "                        7:'year of second replace',8:'odomoter at replace',9:'month odomoter data began',\n",
    "                        10:'year odomoter data began',11:'odomoter reading'})\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {},
   "outputs": [],
   "source": [
    "def g(theta1):\n",
    "    u_n\n",
    "    alpha\n",
    "    N\n",
    "    G = (1/N) * (np.sum((J(x1)-J(x2))/alpha)*u_n)\n",
    "    return G"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def J(theta1):\n",
    "    np.sum(np.log(prob_dist))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def prob_dist(theta1):\n",
    "    a = np.exp(Q(s,a))\n",
    "    b = np.sum(np.exp(Q(s,a)))\n",
    "    c = a/b\n",
    "    return c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Q(s,a):\n",
    "    "
   ]
  }
 ],
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