Statistical Mechanics 2 Probability Distributions.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Probability Distributions\n",
    "\n",
    "Three most common probaiblity distributions are:\n",
    "* Uniform\n",
    "* Exponential\n",
    "* Gaussian\n",
    "\n",
    "For uniform distribution:\n",
    "\n",
    "$$\n",
    "\\rho_{uniform}(x) = \\dfrac{1}{b-a} \\quad \\mbox{ for } a \\leq x < b\n",
    "$$\n",
    "\n",
    "For exponential distribution:\n",
    "\n",
    "$$\n",
    "\\rho_{exponential}(t) = \\dfrac{ e^{-\\frac{t}{\\tau}} }{ \\tau } \\mbox{ for } t \\geq 0\n",
    "$$\n",
    "\n",
    "For Gaussian distribution:\n",
    "\n",
    "$$\n",
    "\\rho_{Gaussian}(v) = \\dfrac{1}{\\sqrt{2 \\pi \\sigma}} e^{-\\frac{v^2}{2\\sigma^2}}\n",
    "$$\n",
    "\n",
    "where $\\rho$ denotes the probability density function (PDF)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Part A: Likelihoods\n",
    "\n",
    "What is the probability that a random number uniform on $[0,1)$ will happen to lie between x = 0.7 and x = 0.75?\n",
    "\n",
    "**Answer:**\n",
    "\n",
    "$$\n",
    "P(0.7 \\leq 0.75) = \\dfrac{ \\int_{0.7}^{0.75} \\rho(x) dx }{ \\int_{0}^{1} \\rho(x) dx } = \\dfrac{ 0.75-0.7 }{ 1 - 0 } = 0.05\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Percentage = 493 / 10000 = 0.0493\n"
     ]
    }
   ],
   "source": [
    "n = 10000\n",
    "sim = np.random.rand(n,)\n",
    "inside = len(sim[(sim>0.7)&(sim<0.75)])\n",
    "print(\"Percentage = %d / %d = %0.4f\"%(inside, n, inside/n))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What is probability that waiting time for radioactive decay of nucleus is more than twice the exponential decay time $\\tau$?\n",
    "\n",
    "**Answer:**\n",
    "\n",
    "$$\n",
    "P(t > 2 \\tau) = \n",
    "\\dfrac{  \\int_{2 \\tau}^{\\infty} \\exp(-t/\\tau) / \\tau dt  }{  \\int_{0}^{\\infty} \\exp(-t/\\tau) / \\tau dt  }\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Denominator evaluates to:\n",
    "\n",
    "$$\n",
    "\\int_{0}^{\\infty} \\exp(-t/\\tau) / \\tau dt = 0 + e^0 = 1\n",
    "$$\n",
    "\n",
    "Numerator evaluates to:\n",
    "\n",
    "$$\n",
    "\\int_{2 \\tau}^{\\infty} \\exp(-t / \\tau) / \\tau dt \n",
    "= \\left[ -e^{-t/\\tau} \\right]_{2\\tau}^{\\infty} = 0 + e^{-2}\n",
    "$$\n",
    "\n",
    "for an overall probability of:\n",
    "\n",
    "$$\n",
    "P(t > 2 \\tau) = \\dfrac{1}{e^{2}} \\sim 0.135\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.135335283237\n"
     ]
    }
   ],
   "source": [
    "print(1.0/np.exp(2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Percentage = 1332 / 10000 = 0.1332\n"
     ]
    }
   ],
   "source": [
    "tau = 1\n",
    "n = 10000\n",
    "sim = np.random.exponential(scale=tau, size=(n,))\n",
    "inside = len(sim[sim>2*tau])\n",
    "print(\"Percentage = %d / %d = %0.4f\"%(inside, n, inside/n))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What is probability that your score on an exam with a Gaussian distribution is greater than $2 \\sigma$ above the mean?\n",
    "\n",
    "Note:\n",
    "\n",
    "$$\n",
    "\\int_{2}^{\\infty} \\frac{1}{\\sqrt{2 \\pi}} \\exp(-v^2/2) dv = \\dfrac{1 - \\mbox{erf}(\\sqrt{2}) }{ 2 } \\sim 0.023\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Answer:**\n",
    "\n",
    "The probability of a score greater than $2 \\sigma$ above the mean is given by:\n",
    "\n",
    "$$\n",
    "P(x > 2 \\sigma) = \\dfrac{  \\int_{2 \\sigma}^{\\infty} \\rho_{Gaussian}(x) dx  }{  \\int_{-\\infty}^{\\infty} \\rho_{Gaussian}(x) dx  }\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The expression in the denominator is equal to 1.\n",
    "\n",
    "The expression in the numerator is equal to:\n",
    "\n",
    "$$\n",
    "\\dfrac{1}{ \\sqrt{2 \\pi \\sigma^2}} \\int_{2 \\sigma}^{\\infty} \\exp \\left( -\\dfrac{p^2}{2\\sigma^2} \\right) dp = \n",
    "\\dfrac{1}{ \\sqrt{2 \\pi \\sigma^2}} \\left[ \\int_{-\\infty}^{\\infty} \\exp \\left( -\\dfrac{p^2}{2\\sigma^2} \\right) dp \n",
    "- \\int_{-\\infty}^{2 \\sigma} \\exp \\left( -\\dfrac{p^2}{2\\sigma^2} \\right) dp \\right] \\\\\n",
    "= \\dfrac{1}{ \\sqrt{2 \\pi \\sigma^2}} \\left[ \\int_{0}^{\\infty} \\exp \\left( -\\dfrac{p^2}{2\\sigma^2} \\right) dp \n",
    "- \\int_{0}^{2\\sigma} \\exp \\left( -\\dfrac{p^2}{2\\sigma^2} \\right) dp \\right]\n",
    "$$\n",
    "\n",
    "Now the first term becomes:\n",
    "\n",
    "$$\n",
    "\\dfrac{1}{ \\sqrt{2 \\pi \\sigma^2}} \\int_{0}^{\\infty} \\exp \\left( -\\dfrac{p^2}{2\\sigma^2} \\right) dp = \\frac{1}{2}\n",
    "$$\n",
    "\n",
    "and the second integral can be evaluated:\n",
    "\n",
    "$$\n",
    "\\int_{-\\infty}^{2 \\sigma} \\dfrac{1}{2 \\pi \\sigma^2} \\exp \\left( -\\dfrac{p^2}{2\\sigma^2} \\right) dp = \\dfrac{1}{\\sqrt{\\pi}} \\int_{0}^{\\sqrt{2}} \\exp( -t^2 ) dt \\\\\n",
    "= \\dfrac{\\mbox{erf}(\\sqrt{2})}{2}\n",
    "$$\n",
    "\n",
    "Together, these make the overall integral evaluate to:\n",
    "\n",
    "$$\n",
    "\\dfrac{1}{ \\sqrt{2 \\pi \\sigma^2}} \\int_{2 \\sigma}^{\\infty} \\exp \\left( -\\dfrac{p^2}{2\\sigma^2} \\right) dp = \\dfrac{1 - \\mbox{erf}(\\sqrt{2})}{2}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0227501319482\n"
     ]
    }
   ],
   "source": [
    "from scipy.special import erf\n",
    "print((1-erf(np.sqrt(2)))/2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Percentage = 238 / 10000 = 0.0238\n"
     ]
    }
   ],
   "source": [
    "n = 10000\n",
    "mu = 0\n",
    "sigma = 3\n",
    "sim = sigma*np.random.randn(n,) + mu\n",
    "inside = len(sim[sim>2*sigma])\n",
    "print(\"Percentage = %d / %d = %0.4f\"%(inside, n, inside/n))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Part B: Normalization, Mean, and Standard Deviation\n",
    "\n",
    "Show that these probability distributions are normalized: $\\int \\rho(x) dx = 1$.\n",
    "\n",
    "What is the mean $x_0$ of each distribution?\n",
    "\n",
    "What is the standard deviation $\\sigma$, where $\\sigma^2 = \\int \\left(x-x_0\\right)^2 \\rho(x) dx$?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Uniform Distribution\n",
    "\n",
    "Mean of uniform distribution is $\\dfrac{b+a}{2}$, and the variance is $V = \\dfrac{(b-a)^2}{12}$, making the standard deviation $\\sigma = \\dfrac{b - a}{2 \\sqrt{3}}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exponential Distribution\n",
    "\n",
    "The mean of the exponential distribution is $\\tau$, and the standard deviation $\\sigma = \\tau$ too.\n",
    "\n",
    "The variance is $\\tau^2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Gaussian Distribution\n",
    "\n",
    "Mean of Gaussian distribution is $\\mu$, standard deviation is $\\sigma$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sum of Uniform Random Variables"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [],
   "source": [
    "import seaborn as sns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "x1 = np.random.rand(10000,)\n",
    "x2 = np.random.rand(10000,)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {},
   "outputs": [
    {
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AAQAA4CgUYAAAADgKBRgAAACOQgEGAACAo1CAAQAA4CimF8IAAACwQ7QLNgCTwRFgAAAA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\n19NsagAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x115a38c88>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = x1 + x2\n",
    "\n",
    "fig = plt.figure(figsize=(12,4))\n",
    "ax1, ax2, ax3 = [fig.add_subplot(1,3,i+1) for i in range(3)]\n",
    "\n",
    "ax1.hist(x1, normed=True, bins=25)\n",
    "ax2.hist(x2, normed=True, bins=25)\n",
    "ax3.hist(x,  normed=True, bins=25)\n",
    "\n",
    "ax1.set_xlabel(r'$x_1$')\n",
    "ax2.set_xlabel(r'$x_2$')\n",
    "ax3.set_xlabel(r'$x_1 + x_2$')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 135,
   "metadata": {},
   "outputs": [],
   "source": [
    "def draw_uniform_sums(nx):\n",
    "\n",
    "    tot = 10000\n",
    "\n",
    "    xs = []\n",
    "    xsum = np.zeros(tot,)\n",
    "    for x in range(nx):\n",
    "        x = np.random.rand(tot,)\n",
    "        xsum += x\n",
    "        xs.append(x)\n",
    "\n",
    "    fig = plt.figure(figsize=(14,2))\n",
    "    axes = [fig.add_subplot(1,nx,i+1) for i in range(nx)]\n",
    "\n",
    "    for i, (ax, x) in enumerate(zip(axes,xs)):\n",
    "        ax.hist(x, normed=True, bins=25)\n",
    "        ax.set_xlabel(r'$x_%d$'%(i+1))\n",
    "\n",
    "    fig2 = plt.figure(figsize=(4,4))\n",
    "    ax2 = fig2.add_subplot(1,1,1)\n",
    "    ax2.hist(xsum, normed=True, bins=25)\n",
    "    ax2.set_xlabel(r'$\\Sigma x_i$')\n",
    "    \n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 136,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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AnJKenq7Vq1f32DZq1KjYx3//+9+/9vM2btwY11wA3GM71FAcAJxAlwAAgHhJdzsAAAAA\nAPQHQw0AAAAAozHUAAAAADAaQw0AAAAAozHUAAAAADAaQw0AAAAAozHUAAAAADAaQw0AAAAAozHU\nAAAAADAaQw0AAAAAozHUAAAAADAaQw0AAAAAozHUAAAAADAaQw0AAAAAozHUAAAAADCa1+4K0WhU\nlZWVOnbsmDIzM1VVVaURI0bE9jc2Nuqtt96S1+vVo48+qp/+9Kf67LPPtHTpUl2+fFlDhw7VmjVr\nlJWVFdc7AiC50SUAnEKfALia7ZmanTt3KhKJqKGhQUuWLFFNTU1s35kzZ1RfX6+33npLr7/+ul5+\n+WVFIhHV1dWppKREW7du1ejRo9XQ0BDXOwEg+dElAJxCnwC4mu1Q09LSookTJ0qSCgsL1dbWFtt3\n6NAhjRs3TpmZmQoEAgoGgzp69GiPzykuLta+ffviFB+AKegSAE6hTwBczfblZ6FQSH6/P3bZ4/Go\nq6tLXq9XoVBIgUAgts/n8ykUCvXY7vP51N7ebhskLy9gex1Jaqqd0afrAbi2vj7fnJRsXSLRJ4AT\n6BO6BHBCf7vE9kyN3+9XOByOXY5Go/J6vV+7LxwOKxAI9NgeDoc1aNCgfoUEYD66BIBT6BMAV7Md\naoqKitTc3CxJam1tVX5+fmxfQUGBWlpa1NnZqfb2dp04cUL5+fkqKirSnj17JEnNzc0aP358nOID\nMAVdAsAp9AmAq6VZlmX1doXudxg5fvy4LMtSdXW1mpubFQwGNXnyZDU2NqqhoUGWZWnx4sWaNm2a\nzp49q+XLlyscDisnJ0e1tbXKzs5O1H0CkIToEgBOoU8AXM12qAEAAACAZMY/3wQAAABgNIYaAAAA\nAEZjqAEAAABgtKQdaqLRqJ577jnNnj1bc+fO1cmTJ3vsb2xsVGlpqcrLy7V79+6kzLhlyxaVlZWp\nrKxMr776qisZJfuc3ddZuHCh3nzzTRcSfpmht5x79uxReXm5ysrKVFlZKbf+HMwu5xtvvKHS0lLN\nmjVLf/rTn1zJ2O3gwYOaO3fuV7bv2rVLs2bN0uzZs9XY2OhCssQxoUsk+sRpJvSJSV0i0ScSfZLI\njN3XoUucyXlD9ImVpP74xz9ay5cvtyzLsj766CPrV7/6VWzf6dOnrZKSEquzs9O6ePFi7ONkyviP\nf/zDmjlzptXV1WVFo1Fr9uzZ1pEjRxKe0S5nt9raWqusrMzaunVrouPF9Jazvb3duvfee61z585Z\nlmVZGzdujH2cTDn/+9//WnfddZfV2dlpXbhwwfrJT37iSkbL+uIxKikpscrKynpsj0Qi1pQpU6wL\nFy5YnZ2dVmlpqXXmzBmXUsafCV1il5M+uX4m9IkpXWJZ9Ek3+iQxGbvRJc7kvFH6JGnP1LS0tGji\nxImSpMLCQrW1tcX2HTp0SOPGjVNmZqYCgYCCwaCOHj2aVBlvvvlmbdq0SR6PR2lpaerq6tKAAQMS\nntEupyT94Q9/UFpaWuw6bukt50cffaT8/Hy98MILeuCBB3TTTTcpNzc36XJmZWXplltu0aVLl3Tp\n0iWlpaW5klGSgsGg1q1b95XtJ06cUDAY1ODBg5WZmanx48dr//79LiRMDBO6xC4nfXL9TOgTU7pE\nok+60SeJySjRJU7mvFH6xOtkSCeFQiH5/f7YZY/Ho66uLnm9XoVCIQUCgdg+n8+nUCiUVBkzMjKU\nm5sry7L04osvavTo0Ro5cmTCM9rlPH78uH7/+99r7dq1Wr9+vSv5uvWW8/z58/rb3/6mbdu2KTs7\nWw8++KAKCwtdeUx7yylJw4cP17333qsrV65o8eLFCc/Xbdq0afr000+/sj1Znj+JYkKXSPSJ00zo\nE1O6RKJPutEniclIlzibU7ox+iRphxq/369wOBy7HI1GY9+Yq/eFw+EeD0IyZJSkzs5OrVy5Uj6f\nT88//3zC83XrLee2bdt06tQpzZs3T//617+UkZGhW2+9VcXFxUmVc8iQIfr+97+vvLw8SdKECRN0\n5MgRV4qjt5zNzc06ffq0/vznP0uSFixYoKKiIhUUFCQ857Uky/MnUUzokq/LQp/EL2ey9InpXSIl\n13MoEegT59Alict5o/RJ0r78rKioSM3NzZKk1tZW5efnx/YVFBSopaVFnZ2dam9v14kTJ3rsT4aM\nlmXpscce0/e+9z2tXr1aHo8n4fm69ZbzmWee0dtvv636+nrNnDlT8+fPd6U07HKOGTNGx48f12ef\nfaauri4dPHhQ3/nOd5Iu5+DBgzVw4EBlZmZqwIABCgQCunjxois5r2XUqFE6efKkLly4oEgkogMH\nDmjcuHFux4obE7rELid94mzOZOkT07tEok/ok/hkpEuczXmj9EnSnqmZOnWq9u7dq4qKClmWperq\nam3evFnBYFCTJ0/W3Llz9cADD8iyLD311FOuvB60t4zRaFQffPCBIpGI3n//fUnS008/7UrZ2z2W\nycIu55IlS7Rw4UJJ0j333OPaLwu7nPv27VN5ebnS09NVVFSkH//4x67kvFpTU5M6Ojo0e/ZsrVix\nQgsWLJBlWZo1a5aGDRvmdry4MaFL7HLSJ9fPhD4xtUsk+oQ+iW9GusT5nDdCn6RZlkvvPQcAAAAA\nDkjal58BAAAAQF8w1AAAAAAwGkMNAAAAAKMx1AAAAAAwGkMNAAAAAKMx1AAAAAAwGkMNAAAAAKMx\n1KBfnn32Wa1bt06S9Mknn2jatGk6fPiwSktLU/q/SgNw3tf1yf79+1VeXq6Kigq98cYbLicEYIKv\n65JDhw6poqJCFRUVeuWVV1xOiHjgn2+iX06dOqXS0lK99tprWrp0qVavXq0f/OAHCofDevLJJ7Vl\nyxa3IwIwxNf1yW233aacnBxlZmZq7ty52rhxo7KystyOCiCJXevYJCMjQ5I0b948rV+/Xn6/3+Wk\ncJLX7QAw27Bhw/Tzn/9cDz74oNauXasJEyZIkoYMGeJyMgCmuVafdPN4PEpP5wUGAHrXW5dcuXJF\nQ4cO1cCBA11MiHjgtwP65dy5c2publZ2drZuueUWt+MAMFhvfbJ3714Fg0ENGDDApXQATHGtLmlq\natL06dM1aNAgeb2s66cahhp8YxcvXtQjjzyiJ554Qo8//rj+7//+z+1IAAzVW5/85z//0W9/+1st\nX77cxYQATNBbl9x3333asWOHTp8+rWPHjrmYEvHAUINv5NKlS1q8eLHmzJmju+++W2VlZfr444/1\n17/+1e1oAAzTW59EIhGtWLFClZWV8vl8bkcFkMTsukSS0tPT5fP5OOubgnijAMTF/PnzdeTIEd1x\nxx1auXKl8vPz3Y4EwEDvvvuuXnnlFY0cOVKS9NJLL2nYsGEupwJgmh07dmjr1q2KRqOaMGGCnnrq\nKbcjwWEMNQAAAACMxsvPAAAAABiNoQYAAACA0RhqAAAAABiNoQYAAACA0RhqAAAAABiNoQYAAACA\n0RhqAAAAABiNoQYAAACA0f4/WgonwRXbPJsAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x117446160>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1177cf978>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "draw_uniform_sums(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 137,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1159283c8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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9/PDD8R6nT3V1dWpvb9fSpUuVlpampKQkJScPvb8nO3fuND/2eDyqqKiQ0+mM40S927t3\nr1pbW1VRUaH29nYFAoGYzzr0foIj0JYtW3Tx4kVt2rTJfOS4q6sr3mPd0NSpU/Xrr79q3rx5Wrhw\nodasWTPkH4wb6mbNmqWOjg7NmTNHr732mrxeb8zv0nKIMQBWBAAIAQARAgAiBABECACIEAAQBxTh\nBoqKitTV1aWsrCzzuvfee0933XVXHKdCNHEcAa7z/fffq6qqSjt37hzyL9hBZHDXANeZMmWKlixZ\noqVLl/Y4T0VZWZn5as5Tp06puLhYx44dkyR99NFHCgaDcZkXg8eKAL3yer06c+aM+Z+/vb1dzz77\nrLZu3aqVK1eqsrJSDzzwQJynRCSwIsANXb58WYcOHVJRUZF53a233qqZM2dq3rx5KisrMyMQDoe1\nfPnyeI2KCCAEuKHKykrdf//9mj59unnduXPn1NDQoDFjxuj22283rz99+rTuvPPOeIyJCOFZA1yn\nrq5OR48e1b59+8zrLl68qMWLF2v58uU6d+6c1q9fry1btkiSTpw4wTMKCY4Q4DobN25UV1eX3G63\nJKmrq0t//PGHKisrNXXqVIVCIW3fvl2HDx9Wfn6+Tpw4ofz8/DhPjcHgwUIM2qpVq1ReXt7jrNlI\nLIQAAA8WAiAEAEQIAIgQABAhACBCAECEAIAIAQARAgCS/h+xkokCyAbogQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x113465278>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "draw_uniform_sums(6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 138,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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pQVFREebMmYNly5bJt9XV1WHz5s3YunUr3nnnHaxYsQItLS1Ys2YNxowZgy1b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\npGtsaoiIiIiISNf+P38xTkxAuFvyAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f1bf8d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x115a0e4e0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "draw_uniform_sums(8)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}