Introducing Random Variables.ipynb

{
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Random Variables\n",
    "\n",
    "Frequently, when an experiment is performed, we are interested mainly in some function of the outcome as opposed to the actual outcome itself.\n",
    "\n",
    "For instance,<br>\n",
    "1) In recent flipping a coin experiment, we may be interested in the total number of heads that occur and not care at all about the actual **Head(H)–Tail(T)** sequence that results. <br>\n",
    "\n",
    "2) In throwing dice, we are often interested in the sum of the two dice and are not really concerned about the separate values of each die. That is, we may be interested in knowingthat the sum is 7 and may not be concerned over whether the actual outcome was: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), or (6, 1). <br>\n",
    "\n",
    "Also, These quantities of interest, or, more formally, these real-valued functions defined on the sample space, are known as '**Random Variables**'."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Lets do an experiment with Python to demostrate \n",
    "> ### *Why we need Random Variables?*\n",
    "\n",
    "## & show its importance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "from itertools import product\n",
    "# from IPython.core.display import HTML\n",
    "# css = open('media/style-table.css').read() + open('media/style-notebook.css').read()\n",
    "# HTML('<style>{}</style>'.format(css))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "one_toss = np.array(['H', 'T'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[('H', 'H'), ('H', 'T'), ('T', 'H'), ('T', 'T')]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "two_tosses = list(product(one_toss, repeat=2))\n",
    "two_tosses"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[('H', 'H', 'H'),\n",
       " ('H', 'H', 'T'),\n",
       " ('H', 'T', 'H'),\n",
       " ('H', 'T', 'T'),\n",
       " ('T', 'H', 'H'),\n",
       " ('T', 'H', 'T'),\n",
       " ('T', 'T', 'H'),\n",
       " ('T', 'T', 'T')]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# For three tosses, just change the number of repetitions:\n",
    "three_tosses = list(product(one_toss, repeat=3))\n",
    "three_tosses"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As shown earlier in slide,<br>\n",
    "A *probability space* $(\\Omega, P)$ is an outcome space accompanied by the probabilities of all the outcomes.\n",
    "<br>If you assume all eight outcomes of three tosses are equally likely, the probabilities are all 1/8:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "three_toss_probs = (1/8)*np.ones(8)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
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       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Omega</th>\n",
       "      <th>P(omega)</th>\n",
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       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>(H, H, H)</td>\n",
       "      <td>0.125</td>\n",
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       "      <th>1</th>\n",
       "      <td>(H, H, T)</td>\n",
       "      <td>0.125</td>\n",
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       "      <th>2</th>\n",
       "      <td>(H, T, H)</td>\n",
       "      <td>0.125</td>\n",
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       "      <th>3</th>\n",
       "      <td>(H, T, T)</td>\n",
       "      <td>0.125</td>\n",
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       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>(T, H, H)</td>\n",
       "      <td>0.125</td>\n",
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       "      <th>5</th>\n",
       "      <td>(T, H, T)</td>\n",
       "      <td>0.125</td>\n",
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       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>(T, T, H)</td>\n",
       "      <td>0.125</td>\n",
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       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>(T, T, T)</td>\n",
       "      <td>0.125</td>\n",
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      "text/plain": [
       "       Omega  P(omega)\n",
       "0  (H, H, H)     0.125\n",
       "1  (H, H, T)     0.125\n",
       "2  (H, T, H)     0.125\n",
       "3  (H, T, T)     0.125\n",
       "4  (T, H, H)     0.125\n",
       "5  (T, H, T)     0.125\n",
       "6  (T, T, H)     0.125\n",
       "7  (T, T, T)     0.125"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "three_toss_space = pd.DataFrame({\n",
    "    'Omega':three_tosses,\n",
    "    'P(omega)':three_toss_probs\n",
    "})\n",
    "three_toss_space"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As you can see above, Product spaces(Probability spaces) get large very quickly. \n",
    "If we are tossing 10 times, the outcome space would consist of the $2^{10}$  sequences of 10 elements where each element is H or T. <br>\n",
    "The outcomes are a pain to list by hand, but computers are good at saving us that kind of pain.\n",
    "\n",
    "Lets take example of rolling die,<br>\n",
    "If we roll a die 5 times, there are almost 8,000 possible outcomes:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
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       "      <th></th>\n",
       "      <th>Omega</th>\n",
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       "      <th>0</th>\n",
       "      <td>(1, 1, 1, 1, 1)</td>\n",
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       "      <td>0.000129</td>\n",
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       "      <td>0.000129</td>\n",
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       "      <td>0.000129</td>\n",
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       "      <td>0.000129</td>\n",
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       "      <th>23</th>\n",
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       "      <td>0.000129</td>\n",
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       "      <td>0.000129</td>\n",
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       "      <td>0.000129</td>\n",
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       "      <td>0.000129</td>\n",
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       "      <th>27</th>\n",
       "      <td>(1, 1, 1, 5, 4)</td>\n",
       "      <td>0.000129</td>\n",
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       "      <td>(1, 1, 1, 5, 5)</td>\n",
       "      <td>0.000129</td>\n",
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       "      <td>(1, 1, 1, 5, 6)</td>\n",
       "      <td>0.000129</td>\n",
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       "      <td>0.000129</td>\n",
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       "      <th>7747</th>\n",
       "      <td>(6, 6, 6, 2, 2)</td>\n",
       "      <td>0.000129</td>\n",
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       "      <th>7748</th>\n",
       "      <td>(6, 6, 6, 2, 3)</td>\n",
       "      <td>0.000129</td>\n",
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       "      <th>7749</th>\n",
       "      <td>(6, 6, 6, 2, 4)</td>\n",
       "      <td>0.000129</td>\n",
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       "      <td>0.000129</td>\n",
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       "      <td>0.000129</td>\n",
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       "      <td>0.000129</td>\n",
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       "      <td>0.000129</td>\n",
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       "      <th>7755</th>\n",
       "      <td>(6, 6, 6, 3, 4)</td>\n",
       "      <td>0.000129</td>\n",
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       "      <th>7756</th>\n",
       "      <td>(6, 6, 6, 3, 5)</td>\n",
       "      <td>0.000129</td>\n",
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       "      <th>7757</th>\n",
       "      <td>(6, 6, 6, 3, 6)</td>\n",
       "      <td>0.000129</td>\n",
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       "      <th>7758</th>\n",
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       "      <td>0.000129</td>\n",
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       "      <th>7759</th>\n",
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       "      <td>0.000129</td>\n",
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       "      <td>0.000129</td>\n",
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       "      <td>0.000129</td>\n",
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       "      <td>0.000129</td>\n",
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       "      <td>0.000129</td>\n",
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       "      <td>0.000129</td>\n",
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       "      <th>7765</th>\n",
       "      <td>(6, 6, 6, 5, 2)</td>\n",
       "      <td>0.000129</td>\n",
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       "      <th>7766</th>\n",
       "      <td>(6, 6, 6, 5, 3)</td>\n",
       "      <td>0.000129</td>\n",
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       "      <th>7767</th>\n",
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       "      <td>0.000129</td>\n",
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       "      <td>0.000129</td>\n",
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       "      <td>0.000129</td>\n",
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       "      <td>0.000129</td>\n",
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       "      <td>0.000129</td>\n",
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       "      <td>0.000129</td>\n",
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       "      <td>0.000129</td>\n",
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       "      <td>0.000129</td>\n",
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       "    <tr>\n",
       "      <th>7775</th>\n",
       "      <td>(6, 6, 6, 6, 6)</td>\n",
       "      <td>0.000129</td>\n",
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      "text/plain": [
       "                Omega  P(omega)\n",
       "0     (1, 1, 1, 1, 1)  0.000129\n",
       "1     (1, 1, 1, 1, 2)  0.000129\n",
       "2     (1, 1, 1, 1, 3)  0.000129\n",
       "3     (1, 1, 1, 1, 4)  0.000129\n",
       "4     (1, 1, 1, 1, 5)  0.000129\n",
       "5     (1, 1, 1, 1, 6)  0.000129\n",
       "6     (1, 1, 1, 2, 1)  0.000129\n",
       "7     (1, 1, 1, 2, 2)  0.000129\n",
       "8     (1, 1, 1, 2, 3)  0.000129\n",
       "9     (1, 1, 1, 2, 4)  0.000129\n",
       "10    (1, 1, 1, 2, 5)  0.000129\n",
       "11    (1, 1, 1, 2, 6)  0.000129\n",
       "12    (1, 1, 1, 3, 1)  0.000129\n",
       "13    (1, 1, 1, 3, 2)  0.000129\n",
       "14    (1, 1, 1, 3, 3)  0.000129\n",
       "15    (1, 1, 1, 3, 4)  0.000129\n",
       "16    (1, 1, 1, 3, 5)  0.000129\n",
       "17    (1, 1, 1, 3, 6)  0.000129\n",
       "18    (1, 1, 1, 4, 1)  0.000129\n",
       "19    (1, 1, 1, 4, 2)  0.000129\n",
       "20    (1, 1, 1, 4, 3)  0.000129\n",
       "21    (1, 1, 1, 4, 4)  0.000129\n",
       "22    (1, 1, 1, 4, 5)  0.000129\n",
       "23    (1, 1, 1, 4, 6)  0.000129\n",
       "24    (1, 1, 1, 5, 1)  0.000129\n",
       "25    (1, 1, 1, 5, 2)  0.000129\n",
       "26    (1, 1, 1, 5, 3)  0.000129\n",
       "27    (1, 1, 1, 5, 4)  0.000129\n",
       "28    (1, 1, 1, 5, 5)  0.000129\n",
       "29    (1, 1, 1, 5, 6)  0.000129\n",
       "...               ...       ...\n",
       "7746  (6, 6, 6, 2, 1)  0.000129\n",
       "7747  (6, 6, 6, 2, 2)  0.000129\n",
       "7748  (6, 6, 6, 2, 3)  0.000129\n",
       "7749  (6, 6, 6, 2, 4)  0.000129\n",
       "7750  (6, 6, 6, 2, 5)  0.000129\n",
       "7751  (6, 6, 6, 2, 6)  0.000129\n",
       "7752  (6, 6, 6, 3, 1)  0.000129\n",
       "7753  (6, 6, 6, 3, 2)  0.000129\n",
       "7754  (6, 6, 6, 3, 3)  0.000129\n",
       "7755  (6, 6, 6, 3, 4)  0.000129\n",
       "7756  (6, 6, 6, 3, 5)  0.000129\n",
       "7757  (6, 6, 6, 3, 6)  0.000129\n",
       "7758  (6, 6, 6, 4, 1)  0.000129\n",
       "7759  (6, 6, 6, 4, 2)  0.000129\n",
       "7760  (6, 6, 6, 4, 3)  0.000129\n",
       "7761  (6, 6, 6, 4, 4)  0.000129\n",
       "7762  (6, 6, 6, 4, 5)  0.000129\n",
       "7763  (6, 6, 6, 4, 6)  0.000129\n",
       "7764  (6, 6, 6, 5, 1)  0.000129\n",
       "7765  (6, 6, 6, 5, 2)  0.000129\n",
       "7766  (6, 6, 6, 5, 3)  0.000129\n",
       "7767  (6, 6, 6, 5, 4)  0.000129\n",
       "7768  (6, 6, 6, 5, 5)  0.000129\n",
       "7769  (6, 6, 6, 5, 6)  0.000129\n",
       "7770  (6, 6, 6, 6, 1)  0.000129\n",
       "7771  (6, 6, 6, 6, 2)  0.000129\n",
       "7772  (6, 6, 6, 6, 3)  0.000129\n",
       "7773  (6, 6, 6, 6, 4)  0.000129\n",
       "7774  (6, 6, 6, 6, 5)  0.000129\n",
       "7775  (6, 6, 6, 6, 6)  0.000129\n",
       "\n",
       "[7776 rows x 2 columns]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "die = np.arange(1, 7, 1)\n",
    "five_rolls = list(product(die, repeat=5))\n",
    "# five_rolls = [list(i) for i in product(die, repeat=5)]\n",
    "five_roll_probs = (1/6**5)**np.ones(6**5)\n",
    "five_roll_space = pd.DataFrame({\n",
    "   'Omega':five_rolls,\n",
    "    'P(omega)':five_roll_probs\n",
    "})\n",
    "five_roll_space"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A Function on the Outcome Space\n",
    "\n",
    "Suppose you roll a die five times and add up the number of spots you see. If that seems artificial, be patient for a moment and you'll soon see why it's interesting.\n",
    "\n",
    "The sum of the rolls is a numerical function on the outcome space $\\Omega$  of five rolls. The sum is thus a random variable. Let's call it  $S$ . Then, formally,\n",
    "$S: \\Omega \\rightarrow \\{ 5, 6, \\ldots, 30 \\}$\n",
    " \n",
    "The range of  $S$  is the integers 5 through 30, because each die shows at least one and at most six spots. We can also use the equivalent notation\n",
    "\n",
    "$\\Omega \\stackrel{S}{\\rightarrow} \\{ 5, 6, \\ldots, 30 \\}$\n",
    " \n",
    "From a computational perspective, the elements of  $\\Omega$  are in the column omega of five_roll_space. Let's apply this function and create a larger table."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Omega</th>\n",
       "      <th>P(omega)</th>\n",
       "      <th>S(omega)</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>(1, 1, 1, 1, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>(1, 1, 1, 1, 2)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>6</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>(1, 1, 1, 1, 3)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>7</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>(1, 1, 1, 1, 4)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>(1, 1, 1, 1, 5)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>9</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>(1, 1, 1, 1, 6)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>(1, 1, 1, 2, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>6</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>(1, 1, 1, 2, 2)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>7</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>(1, 1, 1, 2, 3)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>(1, 1, 1, 2, 4)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>9</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>(1, 1, 1, 2, 5)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>(1, 1, 1, 2, 6)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>11</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>(1, 1, 1, 3, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>7</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>(1, 1, 1, 3, 2)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>(1, 1, 1, 3, 3)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>9</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>(1, 1, 1, 3, 4)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>(1, 1, 1, 3, 5)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>11</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>(1, 1, 1, 3, 6)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>12</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>(1, 1, 1, 4, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>(1, 1, 1, 4, 2)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>9</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>(1, 1, 1, 4, 3)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>(1, 1, 1, 4, 4)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>11</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>(1, 1, 1, 4, 5)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>12</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>(1, 1, 1, 4, 6)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>13</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>(1, 1, 1, 5, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>9</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>(1, 1, 1, 5, 2)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>26</th>\n",
       "      <td>(1, 1, 1, 5, 3)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>11</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>27</th>\n",
       "      <td>(1, 1, 1, 5, 4)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>12</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28</th>\n",
       "      <td>(1, 1, 1, 5, 5)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>13</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>29</th>\n",
       "      <td>(1, 1, 1, 5, 6)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>14</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7746</th>\n",
       "      <td>(6, 6, 6, 2, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>21</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7747</th>\n",
       "      <td>(6, 6, 6, 2, 2)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>22</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7748</th>\n",
       "      <td>(6, 6, 6, 2, 3)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>23</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7749</th>\n",
       "      <td>(6, 6, 6, 2, 4)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>24</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7750</th>\n",
       "      <td>(6, 6, 6, 2, 5)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>25</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7751</th>\n",
       "      <td>(6, 6, 6, 2, 6)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>26</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7752</th>\n",
       "      <td>(6, 6, 6, 3, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>22</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7753</th>\n",
       "      <td>(6, 6, 6, 3, 2)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>23</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7754</th>\n",
       "      <td>(6, 6, 6, 3, 3)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>24</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7755</th>\n",
       "      <td>(6, 6, 6, 3, 4)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>25</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7756</th>\n",
       "      <td>(6, 6, 6, 3, 5)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>26</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7757</th>\n",
       "      <td>(6, 6, 6, 3, 6)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>27</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7758</th>\n",
       "      <td>(6, 6, 6, 4, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>23</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7759</th>\n",
       "      <td>(6, 6, 6, 4, 2)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>24</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7760</th>\n",
       "      <td>(6, 6, 6, 4, 3)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>25</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7761</th>\n",
       "      <td>(6, 6, 6, 4, 4)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>26</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7762</th>\n",
       "      <td>(6, 6, 6, 4, 5)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>27</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7763</th>\n",
       "      <td>(6, 6, 6, 4, 6)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>28</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7764</th>\n",
       "      <td>(6, 6, 6, 5, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>24</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7765</th>\n",
       "      <td>(6, 6, 6, 5, 2)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>25</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7766</th>\n",
       "      <td>(6, 6, 6, 5, 3)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>26</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7767</th>\n",
       "      <td>(6, 6, 6, 5, 4)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>27</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7768</th>\n",
       "      <td>(6, 6, 6, 5, 5)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>28</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7769</th>\n",
       "      <td>(6, 6, 6, 5, 6)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>29</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7770</th>\n",
       "      <td>(6, 6, 6, 6, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>25</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7771</th>\n",
       "      <td>(6, 6, 6, 6, 2)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>26</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7772</th>\n",
       "      <td>(6, 6, 6, 6, 3)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>27</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7773</th>\n",
       "      <td>(6, 6, 6, 6, 4)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>28</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7774</th>\n",
       "      <td>(6, 6, 6, 6, 5)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>29</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7775</th>\n",
       "      <td>(6, 6, 6, 6, 6)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>30</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>7776 rows × 3 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "                Omega  P(omega)  S(omega)\n",
       "0     (1, 1, 1, 1, 1)  0.000129         5\n",
       "1     (1, 1, 1, 1, 2)  0.000129         6\n",
       "2     (1, 1, 1, 1, 3)  0.000129         7\n",
       "3     (1, 1, 1, 1, 4)  0.000129         8\n",
       "4     (1, 1, 1, 1, 5)  0.000129         9\n",
       "5     (1, 1, 1, 1, 6)  0.000129        10\n",
       "6     (1, 1, 1, 2, 1)  0.000129         6\n",
       "7     (1, 1, 1, 2, 2)  0.000129         7\n",
       "8     (1, 1, 1, 2, 3)  0.000129         8\n",
       "9     (1, 1, 1, 2, 4)  0.000129         9\n",
       "10    (1, 1, 1, 2, 5)  0.000129        10\n",
       "11    (1, 1, 1, 2, 6)  0.000129        11\n",
       "12    (1, 1, 1, 3, 1)  0.000129         7\n",
       "13    (1, 1, 1, 3, 2)  0.000129         8\n",
       "14    (1, 1, 1, 3, 3)  0.000129         9\n",
       "15    (1, 1, 1, 3, 4)  0.000129        10\n",
       "16    (1, 1, 1, 3, 5)  0.000129        11\n",
       "17    (1, 1, 1, 3, 6)  0.000129        12\n",
       "18    (1, 1, 1, 4, 1)  0.000129         8\n",
       "19    (1, 1, 1, 4, 2)  0.000129         9\n",
       "20    (1, 1, 1, 4, 3)  0.000129        10\n",
       "21    (1, 1, 1, 4, 4)  0.000129        11\n",
       "22    (1, 1, 1, 4, 5)  0.000129        12\n",
       "23    (1, 1, 1, 4, 6)  0.000129        13\n",
       "24    (1, 1, 1, 5, 1)  0.000129         9\n",
       "25    (1, 1, 1, 5, 2)  0.000129        10\n",
       "26    (1, 1, 1, 5, 3)  0.000129        11\n",
       "27    (1, 1, 1, 5, 4)  0.000129        12\n",
       "28    (1, 1, 1, 5, 5)  0.000129        13\n",
       "29    (1, 1, 1, 5, 6)  0.000129        14\n",
       "...               ...       ...       ...\n",
       "7746  (6, 6, 6, 2, 1)  0.000129        21\n",
       "7747  (6, 6, 6, 2, 2)  0.000129        22\n",
       "7748  (6, 6, 6, 2, 3)  0.000129        23\n",
       "7749  (6, 6, 6, 2, 4)  0.000129        24\n",
       "7750  (6, 6, 6, 2, 5)  0.000129        25\n",
       "7751  (6, 6, 6, 2, 6)  0.000129        26\n",
       "7752  (6, 6, 6, 3, 1)  0.000129        22\n",
       "7753  (6, 6, 6, 3, 2)  0.000129        23\n",
       "7754  (6, 6, 6, 3, 3)  0.000129        24\n",
       "7755  (6, 6, 6, 3, 4)  0.000129        25\n",
       "7756  (6, 6, 6, 3, 5)  0.000129        26\n",
       "7757  (6, 6, 6, 3, 6)  0.000129        27\n",
       "7758  (6, 6, 6, 4, 1)  0.000129        23\n",
       "7759  (6, 6, 6, 4, 2)  0.000129        24\n",
       "7760  (6, 6, 6, 4, 3)  0.000129        25\n",
       "7761  (6, 6, 6, 4, 4)  0.000129        26\n",
       "7762  (6, 6, 6, 4, 5)  0.000129        27\n",
       "7763  (6, 6, 6, 4, 6)  0.000129        28\n",
       "7764  (6, 6, 6, 5, 1)  0.000129        24\n",
       "7765  (6, 6, 6, 5, 2)  0.000129        25\n",
       "7766  (6, 6, 6, 5, 3)  0.000129        26\n",
       "7767  (6, 6, 6, 5, 4)  0.000129        27\n",
       "7768  (6, 6, 6, 5, 5)  0.000129        28\n",
       "7769  (6, 6, 6, 5, 6)  0.000129        29\n",
       "7770  (6, 6, 6, 6, 1)  0.000129        25\n",
       "7771  (6, 6, 6, 6, 2)  0.000129        26\n",
       "7772  (6, 6, 6, 6, 3)  0.000129        27\n",
       "7773  (6, 6, 6, 6, 4)  0.000129        28\n",
       "7774  (6, 6, 6, 6, 5)  0.000129        29\n",
       "7775  (6, 6, 6, 6, 6)  0.000129        30\n",
       "\n",
       "[7776 rows x 3 columns]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "five_rolls_sum = pd.DataFrame({\n",
    "    'Omega':five_rolls,\n",
    "    'S(omega)':five_roll_space['Omega'].map(lambda val: sum(val)),\n",
    "    'P(omega)':five_roll_probs\n",
    "})\n",
    "five_rolls_sum"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Functions of Random Variables,\n",
    "A random variable is a numerical function on  $\\Omega$ . Therefore by composition, a numerical function of a random variable is also a random variable.\n",
    "\n",
    "For example,  $S^2$  is a random variable, calculated as follows:\n",
    "\n",
    "$S^2(\\omega) = \\big{(} S(\\omega)\\big{)}^2$\n",
    " \n",
    "Thus for example  $S^2(\\text{[6 6 6 6 6]}) = 30^2 = 900$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Events Determined by $S$ ###\n",
    "From the table `five_rolls_sum` it is hard to tell how many rows show a sum of 6, or 10, or any other value. To better understand the properties of $S$, we have to organize the information in `five_rolls_sum`.\n",
    "\n",
    "For any subset $A$ of the range of $S$, define the event $\\{S \\in A\\}$ as\n",
    "\n",
    "$$\n",
    "\\{S \\in A \\} = \\{\\omega: S(\\omega) \\in A \\}\n",
    "$$\n",
    "\n",
    "That is, $\\{ S \\in A\\}$ is the collection of all $\\omega$ for which $S(\\omega)$ is in $A$.\n",
    "\n",
    "If that definition looks unfriendly, try it out in a special case. Take $A = \\{5, 30\\}$. Then $\\{S \\in A\\}$ if and only if either all the rolls show 1 spot or all the rolls show 6 spots. So \n",
    "$$\n",
    "\\{S \\in A\\} = \\{\\text{[1 1 1 1 1], [6 6 6 6 6]}\\}\n",
    "$$\n",
    "\n",
    "It is natural to ask about the chance the sum is a particular value, say 10. That's not easy to read off the table, but we can access the corresponding rows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Omega</th>\n",
       "      <th>P(omega)</th>\n",
       "      <th>S(omega)</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>(1, 1, 1, 1, 6)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>(1, 1, 1, 2, 5)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>(1, 1, 1, 3, 4)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>(1, 1, 1, 4, 3)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>(1, 1, 1, 5, 2)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>30</th>\n",
       "      <td>(1, 1, 1, 6, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>40</th>\n",
       "      <td>(1, 1, 2, 1, 5)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>45</th>\n",
       "      <td>(1, 1, 2, 2, 4)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50</th>\n",
       "      <td>(1, 1, 2, 3, 3)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>55</th>\n",
       "      <td>(1, 1, 2, 4, 2)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>60</th>\n",
       "      <td>(1, 1, 2, 5, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75</th>\n",
       "      <td>(1, 1, 3, 1, 4)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>80</th>\n",
       "      <td>(1, 1, 3, 2, 3)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>85</th>\n",
       "      <td>(1, 1, 3, 3, 2)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>90</th>\n",
       "      <td>(1, 1, 3, 4, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>110</th>\n",
       "      <td>(1, 1, 4, 1, 3)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>115</th>\n",
       "      <td>(1, 1, 4, 2, 2)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>120</th>\n",
       "      <td>(1, 1, 4, 3, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>145</th>\n",
       "      <td>(1, 1, 5, 1, 2)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>150</th>\n",
       "      <td>(1, 1, 5, 2, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>180</th>\n",
       "      <td>(1, 1, 6, 1, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>220</th>\n",
       "      <td>(1, 2, 1, 1, 5)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>225</th>\n",
       "      <td>(1, 2, 1, 2, 4)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>230</th>\n",
       "      <td>(1, 2, 1, 3, 3)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>235</th>\n",
       "      <td>(1, 2, 1, 4, 2)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>240</th>\n",
       "      <td>(1, 2, 1, 5, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>255</th>\n",
       "      <td>(1, 2, 2, 1, 4)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>260</th>\n",
       "      <td>(1, 2, 2, 2, 3)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>265</th>\n",
       "      <td>(1, 2, 2, 3, 2)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>270</th>\n",
       "      <td>(1, 2, 2, 4, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2635</th>\n",
       "      <td>(3, 1, 2, 2, 2)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2640</th>\n",
       "      <td>(3, 1, 2, 3, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2665</th>\n",
       "      <td>(3, 1, 3, 1, 2)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2670</th>\n",
       "      <td>(3, 1, 3, 2, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2700</th>\n",
       "      <td>(3, 1, 4, 1, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2810</th>\n",
       "      <td>(3, 2, 1, 1, 3)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2815</th>\n",
       "      <td>(3, 2, 1, 2, 2)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2820</th>\n",
       "      <td>(3, 2, 1, 3, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2845</th>\n",
       "      <td>(3, 2, 2, 1, 2)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2850</th>\n",
       "      <td>(3, 2, 2, 2, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2880</th>\n",
       "      <td>(3, 2, 3, 1, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3025</th>\n",
       "      <td>(3, 3, 1, 1, 2)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3030</th>\n",
       "      <td>(3, 3, 1, 2, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3060</th>\n",
       "      <td>(3, 3, 2, 1, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3240</th>\n",
       "      <td>(3, 4, 1, 1, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3890</th>\n",
       "      <td>(4, 1, 1, 1, 3)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3895</th>\n",
       "      <td>(4, 1, 1, 2, 2)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3900</th>\n",
       "      <td>(4, 1, 1, 3, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3925</th>\n",
       "      <td>(4, 1, 2, 1, 2)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3930</th>\n",
       "      <td>(4, 1, 2, 2, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3960</th>\n",
       "      <td>(4, 1, 3, 1, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4105</th>\n",
       "      <td>(4, 2, 1, 1, 2)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4110</th>\n",
       "      <td>(4, 2, 1, 2, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4140</th>\n",
       "      <td>(4, 2, 2, 1, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4320</th>\n",
       "      <td>(4, 3, 1, 1, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5185</th>\n",
       "      <td>(5, 1, 1, 1, 2)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5190</th>\n",
       "      <td>(5, 1, 1, 2, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5220</th>\n",
       "      <td>(5, 1, 2, 1, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5400</th>\n",
       "      <td>(5, 2, 1, 1, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6480</th>\n",
       "      <td>(6, 1, 1, 1, 1)</td>\n",
       "      <td>0.000129</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>126 rows × 3 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "                Omega  P(omega)  S(omega)\n",
       "5     (1, 1, 1, 1, 6)  0.000129        10\n",
       "10    (1, 1, 1, 2, 5)  0.000129        10\n",
       "15    (1, 1, 1, 3, 4)  0.000129        10\n",
       "20    (1, 1, 1, 4, 3)  0.000129        10\n",
       "25    (1, 1, 1, 5, 2)  0.000129        10\n",
       "30    (1, 1, 1, 6, 1)  0.000129        10\n",
       "40    (1, 1, 2, 1, 5)  0.000129        10\n",
       "45    (1, 1, 2, 2, 4)  0.000129        10\n",
       "50    (1, 1, 2, 3, 3)  0.000129        10\n",
       "55    (1, 1, 2, 4, 2)  0.000129        10\n",
       "60    (1, 1, 2, 5, 1)  0.000129        10\n",
       "75    (1, 1, 3, 1, 4)  0.000129        10\n",
       "80    (1, 1, 3, 2, 3)  0.000129        10\n",
       "85    (1, 1, 3, 3, 2)  0.000129        10\n",
       "90    (1, 1, 3, 4, 1)  0.000129        10\n",
       "110   (1, 1, 4, 1, 3)  0.000129        10\n",
       "115   (1, 1, 4, 2, 2)  0.000129        10\n",
       "120   (1, 1, 4, 3, 1)  0.000129        10\n",
       "145   (1, 1, 5, 1, 2)  0.000129        10\n",
       "150   (1, 1, 5, 2, 1)  0.000129        10\n",
       "180   (1, 1, 6, 1, 1)  0.000129        10\n",
       "220   (1, 2, 1, 1, 5)  0.000129        10\n",
       "225   (1, 2, 1, 2, 4)  0.000129        10\n",
       "230   (1, 2, 1, 3, 3)  0.000129        10\n",
       "235   (1, 2, 1, 4, 2)  0.000129        10\n",
       "240   (1, 2, 1, 5, 1)  0.000129        10\n",
       "255   (1, 2, 2, 1, 4)  0.000129        10\n",
       "260   (1, 2, 2, 2, 3)  0.000129        10\n",
       "265   (1, 2, 2, 3, 2)  0.000129        10\n",
       "270   (1, 2, 2, 4, 1)  0.000129        10\n",
       "...               ...       ...       ...\n",
       "2635  (3, 1, 2, 2, 2)  0.000129        10\n",
       "2640  (3, 1, 2, 3, 1)  0.000129        10\n",
       "2665  (3, 1, 3, 1, 2)  0.000129        10\n",
       "2670  (3, 1, 3, 2, 1)  0.000129        10\n",
       "2700  (3, 1, 4, 1, 1)  0.000129        10\n",
       "2810  (3, 2, 1, 1, 3)  0.000129        10\n",
       "2815  (3, 2, 1, 2, 2)  0.000129        10\n",
       "2820  (3, 2, 1, 3, 1)  0.000129        10\n",
       "2845  (3, 2, 2, 1, 2)  0.000129        10\n",
       "2850  (3, 2, 2, 2, 1)  0.000129        10\n",
       "2880  (3, 2, 3, 1, 1)  0.000129        10\n",
       "3025  (3, 3, 1, 1, 2)  0.000129        10\n",
       "3030  (3, 3, 1, 2, 1)  0.000129        10\n",
       "3060  (3, 3, 2, 1, 1)  0.000129        10\n",
       "3240  (3, 4, 1, 1, 1)  0.000129        10\n",
       "3890  (4, 1, 1, 1, 3)  0.000129        10\n",
       "3895  (4, 1, 1, 2, 2)  0.000129        10\n",
       "3900  (4, 1, 1, 3, 1)  0.000129        10\n",
       "3925  (4, 1, 2, 1, 2)  0.000129        10\n",
       "3930  (4, 1, 2, 2, 1)  0.000129        10\n",
       "3960  (4, 1, 3, 1, 1)  0.000129        10\n",
       "4105  (4, 2, 1, 1, 2)  0.000129        10\n",
       "4110  (4, 2, 1, 2, 1)  0.000129        10\n",
       "4140  (4, 2, 2, 1, 1)  0.000129        10\n",
       "4320  (4, 3, 1, 1, 1)  0.000129        10\n",
       "5185  (5, 1, 1, 1, 2)  0.000129        10\n",
       "5190  (5, 1, 1, 2, 1)  0.000129        10\n",
       "5220  (5, 1, 2, 1, 1)  0.000129        10\n",
       "5400  (5, 2, 1, 1, 1)  0.000129        10\n",
       "6480  (6, 1, 1, 1, 1)  0.000129        10\n",
       "\n",
       "[126 rows x 3 columns]"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "five_rolls_sum[five_rolls_sum['S(omega)']==10]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are 126 values of $\\omega$ for which $S(\\omega) = 10$. Since all the $\\omega$ are equally likely, the chance that $S$ has the value 10 is 126/7776. \n",
    "\n",
    "We are informal with notation and write $\\{ S = 10 \\}$ instead of $\\{ S \\in \\{10\\} \\}$:\n",
    "$$\n",
    "P(S = 10) = \\frac{126}{7776} = 1.62\\%\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### This is how Random Variables help us quantify the results of experiments for the purpose of analysis. \n",
    "i.e., Random variables provide *numerical* summaries of the experiment in question. - Stats110 harvard (also below paragraph)\n",
    "\n",
    ">This definition is abstract but fundamental; one of the most important skills to\n",
    "develop when studying probability and statistics is the ability to go back and forth\n",
    "between abstract ideas and concrete examples. Relatedly, it is important to work\n",
    "on recognizing the essential pattern or structure of a problem and how it connectsto problems you have studied previously. We will often discuss stories that involve\n",
    "tossing coins or drawing balls from urns because they are simple, convenient sce-\n",
    "narios to work with, but many other problems are isomorphic: they have the same\n",
    "essential structure, but in a different guise.\n",
    "\n",
    "we can use mathematical opeartion on these variables since they are real valued function nowto problems you have studied previously. We will often discuss stories that involve\n",
    "tossing coins or drawing balls from urns because they are simple, convenient sce-\n",
    "narios to work with, but many other problems are isomorphic: they have the same\n",
    "essential structure, but in a di↵erent guise."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# Looking at Distributions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The table below shows all the possible values of  $S$  along with all their probabilities. It is called a **\"Probability Distribution Table\"** for  $S$ ."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>S(omega)</th>\n",
       "      <th>P(omega)</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>5</td>\n",
       "      <td>0.000129</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>6</td>\n",
       "      <td>0.000643</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>7</td>\n",
       "      <td>0.001929</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>8</td>\n",
       "      <td>0.004501</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>9</td>\n",
       "      <td>0.009002</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>10</td>\n",
       "      <td>0.016204</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>11</td>\n",
       "      <td>0.026363</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>12</td>\n",
       "      <td>0.039223</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>13</td>\n",
       "      <td>0.054012</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>14</td>\n",
       "      <td>0.069444</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>15</td>\n",
       "      <td>0.083719</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>16</td>\n",
       "      <td>0.094522</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>17</td>\n",
       "      <td>0.100309</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>18</td>\n",
       "      <td>0.100309</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>19</td>\n",
       "      <td>0.094522</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>20</td>\n",
       "      <td>0.083719</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>21</td>\n",
       "      <td>0.069444</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>22</td>\n",
       "      <td>0.054012</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>23</td>\n",
       "      <td>0.039223</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>24</td>\n",
       "      <td>0.026363</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>25</td>\n",
       "      <td>0.016204</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>26</td>\n",
       "      <td>0.009002</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>27</td>\n",
       "      <td>0.004501</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>28</td>\n",
       "      <td>0.001929</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>29</td>\n",
       "      <td>0.000643</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>30</td>\n",
       "      <td>0.000129</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    S(omega)  P(omega)\n",
       "0          5  0.000129\n",
       "1          6  0.000643\n",
       "2          7  0.001929\n",
       "3          8  0.004501\n",
       "4          9  0.009002\n",
       "5         10  0.016204\n",
       "6         11  0.026363\n",
       "7         12  0.039223\n",
       "8         13  0.054012\n",
       "9         14  0.069444\n",
       "10        15  0.083719\n",
       "11        16  0.094522\n",
       "12        17  0.100309\n",
       "13        18  0.100309\n",
       "14        19  0.094522\n",
       "15        20  0.083719\n",
       "16        21  0.069444\n",
       "17        22  0.054012\n",
       "18        23  0.039223\n",
       "19        24  0.026363\n",
       "20        25  0.016204\n",
       "21        26  0.009002\n",
       "22        27  0.004501\n",
       "23        28  0.001929\n",
       "24        29  0.000643\n",
       "25        30  0.000129"
      ]
     },
     "execution_count": 119,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dist_S = five_rolls_sum.drop('Omega', axis=1).groupby('S(omega)', as_index=False).sum()\n",
    "dist_S"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The contents of the table – all the possible values of the random variable, along with all their probabilities – are called the probability distribution of  $S$ , or just distribution of  $S$  for short. The distribution shows how the total probability of 100% is distributed over all the possible values of  $S$ .\n",
    "\n",
    "Let's check this, to make sure that all the  $\\omega$ 's in the outcome space have been accounted for in the column of probabilities.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.99999999999999911"
      ]
     },
     "execution_count": 123,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dist_S.ix[:,1].sum()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That's 1 in a computing environment, and it is true in general for the distribution of any random variable.\n",
    "\n",
    "**Probabilities in a distribution are non-negative and sum to 1.**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Visualising Distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 126,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0      5\n",
       " 1      6\n",
       " 2      7\n",
       " 3      8\n",
       " 4      9\n",
       " 5     10\n",
       " 6     11\n",
       " 7     12\n",
       " 8     13\n",
       " 9     14\n",
       " 10    15\n",
       " 11    16\n",
       " 12    17\n",
       " 13    18\n",
       " 14    19\n",
       " 15    20\n",
       " 16    21\n",
       " 17    22\n",
       " 18    23\n",
       " 19    24\n",
       " 20    25\n",
       " 21    26\n",
       " 22    27\n",
       " 23    28\n",
       " 24    29\n",
       " 25    30\n",
       " Name: S(omega), dtype: int64, 0     0.000129\n",
       " 1     0.000643\n",
       " 2     0.001929\n",
       " 3     0.004501\n",
       " 4     0.009002\n",
       " 5     0.016204\n",
       " 6     0.026363\n",
       " 7     0.039223\n",
       " 8     0.054012\n",
       " 9     0.069444\n",
       " 10    0.083719\n",
       " 11    0.094522\n",
       " 12    0.100309\n",
       " 13    0.100309\n",
       " 14    0.094522\n",
       " 15    0.083719\n",
       " 16    0.069444\n",
       " 17    0.054012\n",
       " 18    0.039223\n",
       " 19    0.026363\n",
       " 20    0.016204\n",
       " 21    0.009002\n",
       " 22    0.004501\n",
       " 23    0.001929\n",
       " 24    0.000643\n",
       " 25    0.000129\n",
       " Name: P(omega), dtype: float64)"
      ]
     },
     "execution_count": 126,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dist_S.ix[:,0], dist_S.ix[:,1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "metadata": {},
   "outputs": [],
   "source": [
    "s = dist_S.ix[:,0]\n",
    "p_s = dist_S.ix[:,1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 136,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>S(omega)</th>\n",
       "      <th>P(omega)</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>5</td>\n",
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       "      <th>2</th>\n",
       "      <td>7</td>\n",
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       "      <th>3</th>\n",
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       "      <td>0.004501</td>\n",
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       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>9</td>\n",
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       "      <th>6</th>\n",
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       "      <th>7</th>\n",
       "      <td>12</td>\n",
       "      <td>0.039223</td>\n",
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       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>13</td>\n",
       "      <td>0.054012</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>14</td>\n",
       "      <td>0.069444</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>15</td>\n",
       "      <td>0.083719</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>16</td>\n",
       "      <td>0.094522</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>17</td>\n",
       "      <td>0.100309</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>18</td>\n",
       "      <td>0.100309</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>19</td>\n",
       "      <td>0.094522</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>20</td>\n",
       "      <td>0.083719</td>\n",
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       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>21</td>\n",
       "      <td>0.069444</td>\n",
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       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>22</td>\n",
       "      <td>0.054012</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>23</td>\n",
       "      <td>0.039223</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>24</td>\n",
       "      <td>0.026363</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>25</td>\n",
       "      <td>0.016204</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>26</td>\n",
       "      <td>0.009002</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>27</td>\n",
       "      <td>0.004501</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>28</td>\n",
       "      <td>0.001929</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>29</td>\n",
       "      <td>0.000643</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>30</td>\n",
       "      <td>0.000129</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    S(omega)  P(omega)\n",
       "0          5  0.000129\n",
       "1          6  0.000643\n",
       "2          7  0.001929\n",
       "3          8  0.004501\n",
       "4          9  0.009002\n",
       "5         10  0.016204\n",
       "6         11  0.026363\n",
       "7         12  0.039223\n",
       "8         13  0.054012\n",
       "9         14  0.069444\n",
       "10        15  0.083719\n",
       "11        16  0.094522\n",
       "12        17  0.100309\n",
       "13        18  0.100309\n",
       "14        19  0.094522\n",
       "15        20  0.083719\n",
       "16        21  0.069444\n",
       "17        22  0.054012\n",
       "18        23  0.039223\n",
       "19        24  0.026363\n",
       "20        25  0.016204\n",
       "21        26  0.009002\n",
       "22        27  0.004501\n",
       "23        28  0.001929\n",
       "24        29  0.000643\n",
       "25        30  0.000129"
      ]
     },
     "execution_count": 136,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dist_S = pd.concat([s,p_s],axis=1)\n",
    "dist_S"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 149,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x7fa37c678dd8>"
      ]
     },
     "execution_count": 149,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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XZ57fxS0xe79yr03w67Or+EjfzCwhDn0zs4Q49M3MEuLQNzNLiEPfzCwhDn0zs4Q49M3M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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa37d292ef0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "dist_S.plot(x=\"S(omega)\",y=\"P(omega)\", kind=\"bar\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 166,
   "metadata": {},
   "outputs": [],
   "source": [
    "from prob140 import Plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 165,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Collecting sympy\n",
      "  Downloading sympy-1.1.1.tar.gz (4.6MB)\n",
      "\u001b[K    100% |████████████████████████████████| 4.6MB 85kB/s eta 0:00:011\n",
      "\u001b[?25hCollecting mpmath>=0.19 (from sympy)\n",
      "  Downloading mpmath-0.19.tar.gz (498kB)\n",
      "\u001b[K    100% |████████████████████████████████| 501kB 98kB/s eta 0:00:01\n",
      "\u001b[?25hBuilding wheels for collected packages: sympy, mpmath\n",
      "  Running setup.py bdist_wheel for sympy ... \u001b[?25ldone\n",
      "\u001b[?25h  Stored in directory: /home/nipun/.cache/pip/wheels/86/b6/ef/72760cc52418901e59f20a85454243f2c9503a8f89b5501f5f\n",
      "  Running setup.py bdist_wheel for mpmath ... \u001b[?25ldone\n",
      "\u001b[?25h  Stored in directory: /home/nipun/.cache/pip/wheels/02/2b/99/cd867d5da48d951118a8020e86c0c12a65022702426582d4b8\n",
      "Successfully built sympy mpmath\n",
      "Installing collected packages: mpmath, sympy\n",
      "Successfully installed mpmath-0.19 sympy-1.1.1\n"
     ]
    }
   ],
   "source": [
    "!pip install sympy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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