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6 |

"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"die_bins = np.arange(0.5, 6.6, 1)\n",
"die.hist(bins = die_bins)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Variables whose successive values are separated by the same fixed amount, such as the values on rolls of a die (successive values separated by 1), fall into a class of variables that are called *discrete*. The histogram above is called a *discrete* histogram. Its bins are specified by the array `die_bins` and ensure that each bar is centered over the corresponding integer value. \n",
"\n",
"It is important to remember that the die can't show 1.3 spots, or 5.2 spots – it always shows an integer number of spots. But our visualization spreads the probability of each value over the area of a bar. While this might seem a bit arbitrary at this stage of the course, it will become important later when we overlay smooth curves over discrete histograms.\n",
"\n",
"Before going further, let's make sure that the numbers on the axes make sense. The probability of each face is 1/6, which is 16.67% when rounded to two decimal places. The width of each bin is 1 unit. So the height of each bar is 16.67% per unit. This agrees with the horizontal and vertical scales of the graph."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Empirical Distributions\n",
"The distribution above consists of the theoretical probability of each face. It is called a *probability distribution* and is not based on observed data. It can be studied and understood without any dice being rolled.\n",
"\n",
"*Empirical distributions,* on the other hand, are distributions of observed data. They can be visualized by *empirical histograms*. \n",
"\n",
"Let us get some data by simulating rolls of a die. This can be done by sampling at random with replacement from the integers 1 through 6. We have used `np.random.choice` for such simulations before. But now we will introduce a Table method for doing this. This will make it easier for us to use our familiar Table methods for visualization.\n",
"\n",
"The Table method is called `sample`. It draws at random with replacement from the rows of a table. Its argument is the sample size, and it returns a table consisting of the rows that were selected. An optional argument `with_replacement=False` specifies that the sample should be drawn without replacement. But that does not apply to rolling a die.\n",
"\n",
"Here are the results of 10 rolls of a die."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
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"text/plain": [
"Face\n",
"2\n",
"4\n",
"5\n",
"5\n",
"1\n",
"6\n",
"1\n",
"4\n",
"6\n",
"5"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"die.sample(10)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can use the same method to simulate as many rolls as we like, and then draw empirical histograms of the results. Because we are going to do this repeatedly, we define a function `empirical_hist_die` that takes the sample size as its argument, rolls a die as many times as the argument, and then draws a histogram of the observed results."
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [],
"source": [
"def empirical_hist_die(n):\n",
" die.sample(n).hist(bins = die_bins)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Empirical Histograms\n",
"\n",
"Here is an empirical histogram of 10 rolls. It doesn't look very much like the probability histogram above. Run the cell a few times to see how it varies."
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
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0adPoAKmpqVqzZo3Wrl2rK6+80r0+IiJCknT8+HGP7U+cOFHn6AwA8MvU4JHY66+/rlWrVrmX58+fr8zMzDrblZWVad++fbr11lsbNfm0adOUnZ2tnJwcde3a1WOsc+fOioiI0KZNm3TttddKkqqqqrRjxw7NmjWrUfMAAJqnBkussrLS/dmUJJWXl8vpdHpsYxiGgoKC9Mc//lHTp083PfHUqVP1xhtvaMWKFQoNDXXPExwcrJCQEBmGocTERC1YsEAOh0PR0dGaP3++goODNWbMmMbsIwCgmWqwxCZNmqRJkyZJknr16qXnnnuu3gs7GmvZsmWS5L58vta0adOUmpoqSXr44Yd15swZpaSkqKysTLGxscrOzv5Jpy0BAM2P6Qs7PvvsM69OXFZWdsFtDMNQamqqu9QAAPihRt/s/O233+rIkSM6deqUXC5XnfFBgwZ5JRgAABdiusROnTqladOm6a233lJNTU2d8dpL30+ePOnVgAAA1Md0iT3yyCPKycnRpEmTNGjQIPeTNQAAsIrpEvvwww81efJkPfvss77MAwCAaaZvdg4MDFRUVJQvswAA0CimS2z48OH64IMPfJkFAIBGMV1iDz74oIqLizVlyhTl5eWpuLhYX3/9dZ1fAAD4i+nPxGJjY2UYhnbv3q2srKx6t+PqRACAv5gusccff7zBBwADAOBvpkuMp2YAAJqan/TNzjU1NTp58qTOnfP/l/IBAFCrUSW2a9cujRgxQu3bt1d0dLS2bdsmSSotLdWdd96pf/3rXz4JCQDA+ZgusY8++khDhw7VwYMHddddd3k8NzEsLEwVFRX6+9//7pOQAACcj+kSe+aZZxQVFaXc3Fw9+eSTdcZvuOEG/ec///FqOAAAGmK6xHbt2qXf//73atWq1XmvUuzQoYPHF2gCAOBrpkusRYsWatGi/s1LSkrUunVrr4QCAMAM0yXWu3dvrV+//rxj1dXVevPNN9WvXz+vBQMA4EJMl9ijjz6qf//733rggQe0Z88eSVJxcbE+/PBD3XHHHTp48KAee+wxnwUFAODHTN/sfPPNN2vp0qVKSUnR66+/LklKTEyUy+XSr371Ky1btkzXXXedz4ICAPBjpktMksaMGaOhQ4dq06ZN+vLLL+V0OtWlSxcNGTJEISEhvsoIAMB5NarEJCkoKEjx8fG+yAIAQKOY/kzsvffeU0pKSr3jKSkp9V74AQCAL5guscWLF+v06dP1jldVVWnRokVeCQUAgBmmS2zfvn3q3bt3vePXXHON9u/f75VQAACYYbrEzp07pzNnztQ7fubMGZ09e9YroQAAMMN0ifXo0UNr166V0+msM+Z0OrV27Vp1797dq+EAAGiI6RKbMmWKPv74YyUkJGj37t06e/aszp49q927d+vuu+/Wxx9/rMmTJ/syKwAAHkxfYj969GgdPHhQaWlp+uCDDyRJhmHI5XLJMAxNmzZN48aN81lQAAB+rFH3iU2dOlVjxozRu+++q8LCQrlcLnXp0kXDhg3TlVde6aOIAACcn6kSO3PmjO68806NGzdOv//97/Xggw/6OhcAABdk6jOx1q1b69NPP1VNTY2v8wAAYJrpCzt+85vfaPv27b7MAgBAo5gusTlz5mjXrl2aMWOGCgsLz3upPQAA/mT6wo7rrrtOLpdL6enpSk9PV4sWLXTRRRd5bGMYho4ePer1kAAAnI/pEhs5cqQMw/BlFgAAGsV0iWVkZPgyBwAAjWb6MzEAAJqaRpXYoUOH9NBDD6l3796KjIzU1q1bJUmlpaV67LHHtHv3bp+EBADgfEyfTvzf//6n2267TU6nU3379tWhQ4fc942FhYUpLy9PZ8+e1YsvvuizsAAA/JDpEps5c6batGmjDz/8UAEBAYqOjvYYj4uL09tvv+31gAAA1Mf06cTt27dr4sSJuvzyy897lWJkZKSOHTvm1XAAADSkUV+KGRwcXO/4qVOnFBAQ4JVQAACY0agvxdyyZct5x1wul95991317t3ba8EAALgQ0yWWmJiod955R3PnztXJkyclff+Nzl988YXuueceffLJJzzdHgDgV6ZLbPTo0Zo5c6bmzZunfv36udcNGDBAOTk5mj17tm655ZZGTb5t2zbddddduvrqqxUaGqqVK1d6jLtcLqWlpal79+5q166d4uPj9fnnnzdqDgBA89WoL8VMTk7WmDFjtHbtWhUUFMjpdKpLly6644471Llz50ZPXllZqR49eighIUFTpkypM75o0SL3sxodDofmzp2rkSNHKi8vT23atGn0fACA5uWCJXb27Fm99957Kiws1KWXXqpbb71VSUlJXpk8Li5OcXFxklTnNV0ulzIyMpScnKzhw4dL+v7RVw6HQ6tXr9aECRO8kgEAYF8NllhJSYmGDh2qgwcPyuVySZKCg4P1xhtvaNCgQT4NVlRUpJKSEg0ePNi9rnXr1ho4cKByc3MpMQBAw5+JzZ49W4WFhUpKStIbb7yhtLQ0XXzxxXr88cd9HqykpESSFB4e7rE+PDxcx48f9/n8AICmr8EjsY0bNyohIUGzZ892r7v88ss1ceJEffXVV+rQoYPPA/74xmqXy9XgV8Lk5+f7OpLX+TNzRWW1Kisr/TZfLX/Oea7mXPPfx3PfKfeTvX6br5Y/5/xVSIjKKyr8Nl8t//5/NZr9e7WisuJn/R3ncDgaHL/g6cT+/ft7rBswYIBcLpeOHDni0xKLiIiQJB0/flwdO3Z0rz9x4kSdo7MfutAONzX5+fl+zbz3i6IGb1r3hcrKSr/O2TKgZbPfx9NV3+n/XnnHb/NJ/t/H5HtG/iL2sbm/V0OCQ+RwNP7CP7MaPJ1YU1OjVq1aeayrXa6qqvJZKEnq3LmzIiIitGnTJve6qqoq7dixo06xAgB+mS54dWJhYaE+/vhj9/I333wj6fsjiJCQkDrbx8bGmp68oqJCBQUFkr6/cfrIkSP67LPP1LZtW0VGRioxMVELFiyQw+FQdHS05s+fr+DgYI0ZM8b0HACA5uuCJZaWlqa0tLQ66398cUftZ1W1T/Mw45NPPtGwYcPqzJWQkKCMjAw9/PDDOnPmjFJSUlRWVqbY2FhlZ2dzjxgAQNIFSiw9Pd2nk99www0qKyurd9wwDKWmpio1NdWnOQAA9tRgid19993+ygEAQKOZfnYiAABNDSUGALAtSgwAYFuUGADAtigxAIBtUWIAANuixAAAtkWJAQBsixIDANgWJQYAsC1KDABgW5QYAMC2KDEAgG1RYgAA26LEAAC2RYkBAGyLEgMA2BYlBgCwLUoMAGBblBgAwLYoMQCAbVFiAADbosQAALZFiQEAbIsSAwDYFiUGALAtSgwAYFuUGADAtigxAIBtUWIAANuixAAAtkWJAQBsixIDANgWJQYAsC1KDABgW5QYAMC2KDEAgG1RYgAA26LEAAC2RYkBAGyLEgMA2BYlBgCwLUoMAGBblBgAwLZsUWLLli1Tr169FBERoZtuuknbt2+3OhIAoAlo8iWWnZ2t6dOn67HHHtO///1v9evXT2PHjtXhw4etjgYAsFiTL7H09HTdfffd+uMf/6hu3bpp3rx5ioiI0CuvvGJ1NACAxYyysjKX1SHqU11drSuuuELLly/XiBEj3OunTp2qffv26b333rMwHQDAak36SKy0tFQ1NTUKDw/3WB8eHq7jx49blAoA0FQ06RKrZRiGx7LL5aqzDgDwy9OkSywsLEwBAQF1jrpOnDhR5+gMAPDL06RLLDAwUL1799amTZs81m/atEn9+/e3KBUAoKloaXWAC7n//vs1efJkxcbGqn///nrllVdUXFysCRMmWB0NAGCxJn0kJkmjRo1SWlqa5s2bpxtuuEE7d+5UVlaWOnXqZHW0n2zbtm266667dPXVVys0NFQrV660OpLXLVy4UDfffLMiIyMVFRWlcePGad++fVbH8qqXX35ZAwcOVGRkpCIjI3XLLbdow4YNVsfymQULFig0NFQpKSlWR/GatLQ0hYaGevzq2rWr1bG8rri4WFOmTFFUVJQiIiLUv39/bd261epYXtHkj8QkaeLEiZo4caLVMbymsrJSPXr0UEJCgqZMmWJ1HJ/YunWr7r33Xl177bVyuVz661//qhEjRig3N1dt27a1Op5XtG/fXk8//bSioqLkdDq1atUqjR8/Xps3b9avf/1rq+N5VV5enjIzMxUTE2N1FK9zOBzKyclxLwcEBFiYxvvKysp06623asCAAcrKylJYWJiKioqazXUFtiix5iYuLk5xcXGSpKSkJIvT+EZ2drbH8tKlS9WpUyft3LlTt99+u0WpvCs+Pt5jecaMGVq+fLny8vKaVYmVl5dr0qRJWrx4sebOnWt1HK9r2bKlIiIirI7hMy+88ILatWunpUuXutddeeWV1gXysiZ/OhHNQ0VFhZxOp0JDQ62O4hM1NTVas2aNKisr1a9fP6vjeFVycrKGDx+um266yeooPlFYWKirr75avXr10j333KPCwkKrI3nVunXrFBsbqwkTJig6Olq/+c1v9NJLL8nlarLPuWgUjsTgF9OnT1fPnj2b3V/we/fuVVxcnKqqqhQcHKwVK1Y0q1NumZmZKigo8PhXfHPSt29fLVmyRA6HQydOnNC8efMUFxennTt36tJLL7U6nlcUFhZq+fLlSkpKUnJysvbs2aNp06ZJku677z6L0/18lBh87s9//rN27typ9evXN7vPGxwOh7Zs2aLy8nKtXbtWiYmJysnJUY8ePayO9rPl5+dr1qxZev/99xUYGGh1HJ+45ZZbPJb79u2r3r176/XXX9cDDzxgUSrvcjqd6tOnj2bOnClJuuaaa1RQUKBly5ZRYsCFpKamKjs7W++++26zOg9fKzAwUFdddZUkqU+fPtq1a5eWLFmiF1980eJkP99HH32k0tJSXX/99e51NTU12r59u1555RUdPXpUF198sYUJvS8kJETdu3dXQUGB1VG8JiIiQt26dfNY17VrVx05csSiRN5FicFnpk2bpuzsbOXk5DTLy5bPx+l0qrq62uoYXhEfH68+ffp4rLv//vsVFRWlRx99tFkenVVVVSk/P1833HCD1VG8ZsCAATpw4IDHugMHDigyMtKiRN5FiVmgoqLC/S89p9OpI0eO6LPPPlPbtm2bzRtr6tSpeuONN7RixQqFhoaqpKREkhQcHKyQkBCL03nHU089pbi4OHXo0EEVFRVavXq1tm7dqqysLKujeUXtfVM/FBQUpLZt2zaL06WS9Je//EW33XabOnbs6P5M7PTp00pISLA6mtckJSUpLi5O8+fP16hRo/TZZ5/ppZde0owZM6yO5hVN+qtYmqstW7Zo2LBhddYnJCQoIyPDgkTeV99ViNOmTVNqaqqf0/hGYmKitmzZouPHj+uSSy5RTEyMHnroIQ0ZMsTqaD4THx+vHj16aN68eVZH8Yp77rlH27dvV2lpqS677DL17dtXTzzxhLp37251NK/asGGDZs2apQMHDqhjx46aNGmSJk+e3CwepE6JAQBsi/vEAAC2RYkBAGyLEgMA2BYlBgCwLUoMAGBblBgAwLYoMcACK1eurPNljLW/Nm/ebHU8wDZ4YgdgoczMTLVv395j3Y+fcwegfpQYYKGePXu6HyAMoPE4nQg0MRs3btTYsWPVrVs3XXHFFbr++uu1ePFi1dTU1Nk2MzNTN954o9q1a6fOnTtr6NChys3NdY+fPn1aM2fOVK9evRQeHq5evXpp/vz5cjqd/twlwGc4EgMsVFNTo3PnzrmXDcNQYWGhbrzxRt133326+OKLtXv3bs2ZM0elpaV66qmn3Nv+5S9/0Ysvvqg//OEPSk1NVYsWLZSXl6cjR46of//+OnfunEaPHq39+/crJSVFMTExysvL07x583Tq1Ck9++yzFuwx4F2UGGCh6667zmN5wIABWr9+vXvZ5XJp4MCBqq6u1uLFi/Xkk0+qRYsWKigo0JIlS5SUlKS//vWv7u1vvfVW93+vXr1aO3bs0Lp16zRo0CBJ0k033SRJmjNnjpKTkxUeHu7L3QN8jhIDLLRixQp16NDBvRwSEqLi4mI999xz+vDDD1VcXOxxpPb1118rIiJCmzdvltPp1J/+9Kd6X36FTXUAAAHtSURBVPuf//ynIiMj3UdltQYPHqzZs2crLy9PQ4cO9cl+Af5CiQEW6tGjh8eFHU6nU0OGDFFxcbGmT58uh8Oh1q1ba926dZo/f76qqqokSSdPnpSkOlc2/tDXX3+tw4cP67LLLjvveO1rAHZGiQFNyMGDB/XJJ59o6dKlGjdunHv9+++/77FdWFiYJOnYsWNyOBznfa1LL71UnTt31quvvnre8U6dOnknNGAhSgxoQk6fPi1Juuiii9zrvvvuO7355pse2/32t79VixYt9Oqrr9Z7gcaQIUO0du1aBQcHq2vXrr4LDViIEgOakG7duikyMlLPPPOMAgIC1LJlSy1ZsqTOdl26dFFSUpLS09NVUVGh22+/XQEBAfr444/VtWtXjRo1SnfeeadWrlyp4cOH6/7771fPnj1VXV2tgwcP6v3339fKlSsVFBRkwV4C3kOJAU1IYGCgVq5cqccff1xTpkxR27ZtNX78eEVGRuqhhx7y2Hb27Nm66qqrtGzZMq1atUpBQUGKiYnR4MGDJX1/NJedna3nn39emZmZKioqUlBQkLp06aK4uDgFBgZasYuAVxllZWUuq0MAAPBT8MQOAIBtUWIAANuixAAAtkWJAQBsixIDANgWJQYAsC1KDABgW5QYAMC2KDEAgG39P1xoFA3L54jDAAAAAElFTkSuQmCC\n",
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