FDoS Exercise 9.3

ex9.3.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.5,  0.3,  0.2],\n",
       "       [ 0.5,  0.4,  0.1],\n",
       "       [ 0.2,  0.8,  0. ]])"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q = np.array([[0.5, 0.3, 0.2], # watching series\n",
    "              [0.5, 0.4, 0.1], # studying\n",
    "              [0.2, 0.8, 0]])  # cleaning\n",
    "Q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "p0 = np.matrix(np.array([1/3,1/3,1/3]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "<table><tr><td>1</td><td>[ 0.4  0.5  0.1]</td><td>[ 0.4  0.5  0.1]</td></tr><tr><td>2</td><td>[ 0.47  0.4   0.13]</td><td>[ 0.435  0.45   0.115]</td></tr><tr><td>3</td><td>[ 0.461  0.405  0.134]</td><td>[ 0.444  0.435  0.121]</td></tr><tr><td>4</td><td>[ 0.46   0.408  0.133]</td><td>[ 0.448  0.428  0.124]</td></tr><tr><td>5</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.45   0.424  0.126]</td></tr><tr><td>6</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.452  0.421  0.127]</td></tr><tr><td>7</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.453  0.419  0.128]</td></tr><tr><td>8</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.454  0.418  0.128]</td></tr><tr><td>9</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.455  0.416  0.129]</td></tr><tr><td>10</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.455  0.415  0.129]</td></tr><tr><td>11</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.456  0.415  0.13 ]</td></tr><tr><td>12</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.456  0.414  0.13 ]</td></tr><tr><td>13</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.456  0.414  0.13 ]</td></tr><tr><td>14</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.457  0.413  0.13 ]</td></tr><tr><td>15</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.457  0.413  0.13 ]</td></tr><tr><td>16</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.457  0.412  0.131]</td></tr><tr><td>17</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.457  0.412  0.131]</td></tr><tr><td>18</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.457  0.412  0.131]</td></tr><tr><td>19</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.458  0.412  0.131]</td></tr><tr><td>20</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.458  0.411  0.131]</td></tr><tr><td>21</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.458  0.411  0.131]</td></tr><tr><td>22</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.458  0.411  0.131]</td></tr><tr><td>23</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.458  0.411  0.131]</td></tr><tr><td>24</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.458  0.411  0.131]</td></tr><tr><td>25</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.458  0.41   0.131]</td></tr><tr><td>26</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.458  0.41   0.131]</td></tr><tr><td>27</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.458  0.41   0.131]</td></tr><tr><td>28</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.458  0.41   0.132]</td></tr><tr><td>29</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.458  0.41   0.132]</td></tr><tr><td>30</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.41   0.132]</td></tr><tr><td>31</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.41   0.132]</td></tr><tr><td>32</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.41   0.132]</td></tr><tr><td>33</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.41   0.132]</td></tr><tr><td>34</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.41   0.132]</td></tr><tr><td>35</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.409  0.132]</td></tr><tr><td>36</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.409  0.132]</td></tr><tr><td>37</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.409  0.132]</td></tr><tr><td>38</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.409  0.132]</td></tr><tr><td>39</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.409  0.132]</td></tr><tr><td>40</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.409  0.132]</td></tr><tr><td>41</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.409  0.132]</td></tr><tr><td>42</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.409  0.132]</td></tr><tr><td>43</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.409  0.132]</td></tr><tr><td>44</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.409  0.132]</td></tr><tr><td>45</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.409  0.132]</td></tr><tr><td>46</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.409  0.132]</td></tr><tr><td>47</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.409  0.132]</td></tr><tr><td>48</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.409  0.132]</td></tr><tr><td>49</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.409  0.132]</td></tr><tr><td>50</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.409  0.132]</td></tr><tr><td>51</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.409  0.132]</td></tr><tr><td>52</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.409  0.132]</td></tr><tr><td>53</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.409  0.132]</td></tr><tr><td>54</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.409  0.132]</td></tr><tr><td>55</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.409  0.132]</td></tr><tr><td>56</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.409  0.132]</td></tr><tr><td>57</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.409  0.132]</td></tr><tr><td>58</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.409  0.132]</td></tr><tr><td>59</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.409  0.132]</td></tr><tr><td>60</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.408  0.132]</td></tr><tr><td>61</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.408  0.132]</td></tr><tr><td>62</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.408  0.132]</td></tr><tr><td>63</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.408  0.132]</td></tr><tr><td>64</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.408  0.132]</td></tr><tr><td>65</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.408  0.132]</td></tr><tr><td>66</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.408  0.132]</td></tr><tr><td>67</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.408  0.132]</td></tr><tr><td>68</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.408  0.132]</td></tr><tr><td>69</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.408  0.132]</td></tr><tr><td>70</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.408  0.132]</td></tr><tr><td>71</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.408  0.132]</td></tr><tr><td>72</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.408  0.132]</td></tr><tr><td>73</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.459  0.408  0.132]</td></tr><tr><td>74</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.46   0.408  0.132]</td></tr><tr><td>75</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.46   0.408  0.132]</td></tr><tr><td>76</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.46   0.408  0.132]</td></tr><tr><td>77</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.46   0.408  0.132]</td></tr><tr><td>78</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.46   0.408  0.132]</td></tr><tr><td>79</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.46   0.408  0.132]</td></tr><tr><td>80</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.46   0.408  0.132]</td></tr><tr><td>81</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.46   0.408  0.132]</td></tr><tr><td>82</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.46   0.408  0.132]</td></tr><tr><td>83</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.46   0.408  0.132]</td></tr><tr><td>84</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.46   0.408  0.132]</td></tr><tr><td>85</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.46   0.408  0.132]</td></tr><tr><td>86</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.46   0.408  0.132]</td></tr><tr><td>87</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.46   0.408  0.132]</td></tr><tr><td>88</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.46   0.408  0.132]</td></tr><tr><td>89</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.46   0.408  0.132]</td></tr><tr><td>90</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.46   0.408  0.132]</td></tr><tr><td>91</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.46   0.408  0.132]</td></tr><tr><td>92</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.46   0.408  0.132]</td></tr><tr><td>93</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.46   0.408  0.132]</td></tr><tr><td>94</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.46   0.408  0.132]</td></tr><tr><td>95</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.46   0.408  0.132]</td></tr><tr><td>96</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.46   0.408  0.132]</td></tr><tr><td>97</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.46   0.408  0.132]</td></tr><tr><td>98</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.46   0.408  0.132]</td></tr><tr><td>99</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.46   0.408  0.132]</td></tr><tr><td>100</td><td>[ 0.46   0.407  0.133]</td><td>[ 0.46   0.408  0.132]</td></tr></table>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import tabletext\n",
    "from IPython.display import HTML, display\n",
    "\n",
    "results = []\n",
    "sum_t = 0\n",
    "for t in range(1,101):\n",
    "    p_t = p0*np.linalg.matrix_power(Q, t)\n",
    "    sum_t += p_t\n",
    "    a_t = sum_t/t\n",
    "    results.append([t, p_t.tolist()[0], a_t.tolist()[0]])\n",
    "\n",
    "display(HTML(\n",
    "    '<table><tr>{}</tr></table>'.format(\n",
    "        '</tr><tr>'.join(\n",
    "            '<td>{}</td>'.format('</td><td>'.join(str(np.around(_,decimals=3)) for _ in row)) for row in results)\n",
    "        )\n",
    " ))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Results\n",
    "Stationary distribution $\\pi = \\big(0.46, 0.408, 0.132\\big)$.\n",
    "Therefore spends the student approximately $40.8 \\%$ of his time studying."
   ]
  }
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