code/pmf_1.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sebastian Raschka \n",
      "\n",
      "CPython 3.6.1\n",
      "IPython 6.0.0\n",
      "\n",
      "numpy 1.13.0\n",
      "scipy 0.19.0\n",
      "matplotlib 2.0.2\n"
     ]
    }
   ],
   "source": [
    "%load_ext watermark\n",
    "%watermark -a 'Sebastian Raschka' -v -p numpy,scipy,matplotlib"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy import stats"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Probability Mass Functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Goal of probability and uncertainty computations is to estimate population parameters from samples"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Bernoulli Trial"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Bernoulli trial (or binomial trial): random experiment with 2 possible outcomes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "520"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rng = np.random.RandomState(123)\n",
    "\n",
    "coin_flips = rng.randint(0, 2, size=1000)\n",
    "heads = np.sum(coin_flips)\n",
    "heads"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "480"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tails = coin_flips.shape[0] - heads\n",
    "tails"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Heads probability: 0.00\n",
      "Heads probability: 40.00\n",
      "Heads probability: 47.00\n",
      "Heads probability: 53.70\n",
      "Heads probability: 49.53\n",
      "Heads probability: 49.80\n",
      "Heads probability: 50.03\n"
     ]
    }
   ],
   "source": [
    "rng = np.random.RandomState(123)\n",
    "\n",
    "for i in range(7):\n",
    "    num = 10**i\n",
    "    coin_flips = rng.randint(0, 2, size=num)\n",
    "    heads_proba = np.mean(coin_flips)\n",
    "    print('Heads probability: %.2f' % (heads_proba*100))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Do 100 coin flips 1000 times:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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gfnmdDFxa3h8DXmX7kfJUvy9L+gfb68d0dBERsd/UDG/+8FCyKLbRDEDYy0nAoO1t5RLW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\nRkREVEnCiIiIKkkYERFR5f8DoxS2UfdXtPQAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f78e710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n_experiments = 1000\n",
    "n_bernoulli_trials = 100\n",
    "\n",
    "rng = np.random.RandomState(123)\n",
    "outcomes = np.empty(n_experiments, dtype=np.float)\n",
    "\n",
    "for i in range(n_experiments):\n",
    "    num = 10**i\n",
    "    coin_flips = rng.randint(0, 2, size=n_bernoulli_trials)\n",
    "    head_counts = np.sum(coin_flips)\n",
    "    outcomes[i] = head_counts\n",
    "\n",
    "plt.hist(outcomes, normed=True)\n",
    "plt.xlabel('Number of heads in 100 coin flips')\n",
    "plt.ylabel('Probability of outcome')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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SOQDcvXk3RyPokFg6f3rWbmaN5VCwtnfjkjkOAbMmcShYw/Vs7fd9DcwKKtdzCpIulLRT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\nHApmZpZxKJiZWeb/A+XW0UJvI9WqAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x110148da0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.sort(outcomes)\n",
    "y = np.arange(1, x.shape[0] + 1) / x.shape[0]\n",
    "\n",
    "plt.plot(x, y, marker='o', linestyle='')\n",
    "plt.ylabel('ECDF')\n",
    "plt.xlabel('Number of heads in 100 coin flips')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- repeat with biased coin"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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JI+pPpXNmP//6JAf27XsnuTzJzUluSnLKkHFKkmY2WMJIsgg4i+4Mq/2A1yTZb9JihwHL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\nQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x110240e10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1103a2048>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = 0.7\n",
    "n_experiments = 1000\n",
    "n_bernoulli_trials = 100\n",
    "\n",
    "rng = np.random.RandomState(123)\n",
    "outcomes = np.empty(n_experiments, dtype=np.float)\n",
    "\n",
    "for i in range(n_experiments):\n",
    "    num = 10**i\n",
    "    coin_flips = rng.rand(n_bernoulli_trials)\n",
    "    head_counts = np.sum(coin_flips < p)\n",
    "    outcomes[i] = head_counts\n",
    "\n",
    "plt.hist(outcomes, normed=True)\n",
    "plt.xlabel('Number of heads in 100 coin flips')\n",
    "plt.ylabel('Probability of outcome')\n",
    "plt.show()\n",
    "\n",
    "\n",
    "x = np.sort(outcomes)\n",
    "y = np.arange(1, x.shape[0] + 1) / x.shape[0]\n",
    "\n",
    "plt.plot(x, y, marker='o', linestyle='')\n",
    "plt.ylabel('ECDF')\n",
    "plt.xlabel('Number of heads in 100 coin flips')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Binomial Distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Bernoulli trial (or binomial trial): random experiment with 2 possible outcomes\n",
    "- a binomial distribution describes a binomial variable $B(n, p)$ of \n",
    "*n* of Bernoulli trials (which are statistically independent); *p* is the probability of success (and *q* is the probability of failure, 1-*p*)\n",
    "- Probability of *k* successes:\n",
    "\n",
    "$$P(k) = \\binom{n}{k} p^k q^{n-k}$$\n",
    "\n",
    "where $\\binom{n}{k}$ (\"*n choose k*\") is the binomial coefficient\n",
    "\n",
    "$$\\binom{n}{k} = \\frac{n!}{k!(n-k)!}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Compute probability of 500 heads in 1000 bernoulli trials flipping a fair coin:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def factorial(n):\n",
    "    if n == 0:\n",
    "        return 1\n",
    "    else:\n",
    "        return n * factorial(n-1)\n",
    "\n",
    "def combin(n, k): # \"n choose k*\n",
    "    return factorial(n) / factorial(k) / factorial(n - k)    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.07958923738717877"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = 0.5 # probability of success\n",
    "n = 100 # n_trials\n",
    "k = 50 # k_successes \n",
    "\n",
    "proba = combin(n, k) * p**k * (1 - p)**(n - k)\n",
    "proba"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.079589237387178879"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rv = stats.binom(n, p)\n",
    "rv.pmf(50)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Remember, through simulation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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ERHRRzWW1C4GTgNvK6yRJ/9B2YBER0S01PYxDgD1sPwIgaTFwE/ChNgOLiIhuqblKCmB6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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1100b4ef0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n_repetitions = 5000\n",
    "\n",
    "rng = np.random.RandomState(123)\n",
    "outcomes = np.empty(n_repetitions, dtype=np.float)\n",
    "\n",
    "for i in range(n_repetitions):\n",
    "    num = 10**i\n",
    "    coin_flips = rng.randint(0, 2, size=n)\n",
    "    head_counts = np.sum(coin_flips)\n",
    "    outcomes[i] = head_counts\n",
    "\n",
    "plt.hist(outcomes, normed=True)\n",
    "plt.xlabel('Number of heads in 100 coin flips')\n",
    "plt.ylabel('Probability of outcome')\n",
    "plt.axvline(50, color='orange')\n",
    "plt.axhline(proba, color='orange')\n",
    "plt.ylim([0, 0.1])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Binomial Probability Mass Function (PMF)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- probabilities of discrete variables\n",
    "- previous coin flip (histogram above) is a discrete uniform PMF (if coin is unbiased)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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gSODGEsPv08xaDHBk6eXcCZwOvLyUv5qn5v25qrcdgLPUnPd5he2H+8SzExg+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\nYV0JAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x110356f60>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rng = np.random.RandomState(123)\n",
    "\n",
    "p = 0.5\n",
    "n = 100\n",
    "outcomes = rng.binomial(n, p, size=10000)\n",
    "\n",
    "plt.hist(outcomes, normed=True)\n",
    "plt.xlabel('Number of heads in 100 coin flips')\n",
    "plt.ylabel('Probability of outcome')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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6EitH61YrKR+4EXgz0AhskbTBzJLn5r4MWBa+VgM3hT8TXu8lG5dN9rV1s7g6nmlZh1Nb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\nuiXNAz4JrDKzswimRVhLbl7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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1102d7cf8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n = 100\n",
    "\n",
    "probas = []\n",
    "for k in range(n):\n",
    "    probas.append(combin(n, k) * p**k * (1 - p)**(n - k))\n",
    "    \n",
    "plt.plot(range(100), probas)\n",
    "plt.xlabel('Number of successes in 100 coin flips')\n",
    "plt.ylabel('Probability of outcome, PDF')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Binomial Cumulative Distribution Function (CDF)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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UjSfIMcuDkuholnQVcBXAuHHjDuocA3pXM2XEgK4MqyIJ5bLT655KyVE9BD0k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\nWZv/DzgcdmAlRW+IAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x110115710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n = 100\n",
    "\n",
    "probas = []\n",
    "for k in range(n):\n",
    "    probas.append(combin(n, k) * p**k * (1 - p)**(n - k))\n",
    "    \n",
    "\n",
    "plt.plot(range(100), np.cumsum(probas))\n",
    "plt.xlabel('Number of successes in 100 coin flips')\n",
    "plt.ylabel('Probability of outcome, CDF')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Rolling a die"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Another discrete uniform PMF would be a dice rolls with a fair die"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x11025d6d8>"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1100d2198>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n = 10000\n",
    "rng = np.random.RandomState(123)\n",
    "dice_rolls = rng.randint(1, 7, n)\n",
    "\n",
    "outcome_proba = np.bincount(dice_rolls)[1:] / n * 100 \n",
    "plt.bar(range(1, 7), outcome_proba)\n",
    "plt.ylabel('Probability of event')\n",
    "plt.xlabel('Die roll outcome')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- A random variable that counts the number of a specific outcome, say rolling a 3, follows a binomial distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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E4OASx14LdEiaKOlA4FSgs6FOJ3BafjpsBvBYROyQJOAq4P6I+FKTNqfn9dOB60vEYmZm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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x110552390>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = 0.1 # probability_rolling a 3\n",
    "n = 1000 # number of die rolls\n",
    "\n",
    "rng = np.random.RandomState(1)\n",
    "outcomes = rng.binomial(n, p, size=5000)\n",
    "\n",
    "plt.hist(outcomes, normed=True)\n",
    "plt.xlabel('Number of rolling a 3 in 1000 die rolls')\n",
    "plt.ylabel('Probability of outcome')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Poisson Distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- similar to binomial distribution but different ;)\n",
    "    - binomial measures the number of successes based on discrete events (*k* successes in *n* trials\n",
    "    - poisson is based on \"continuous events\", but note that it is still a discrete probability distribution\n",
    "    \n",
    "- in poisson, we have $np \\rightarrow \\lambda$, assuming that we have infinite events ($n \\rightarrow \\infty)$ so that the probability of a singular event is infinitely small ($p \\rightarrow 0$)\n",
    "- $\\mathcal{Bin}(n, p) \\approx \\mathcal{Poisson}(np), \\text{ for } n \\rightarrow  \\infty$\n",
    "- in that sense, the poisson distribution approximates the binomial distribution for a rare number of events that happen in a large number of trials\n",
    "\n",
    "$$P(k \\text{ events in an interval}) = e^{-\\lambda} \\frac{\\lambda^k}{k!}$$\n",
    "\n",
    "where $\\lambda$ is the average number of events per interval\n",
    "- poisson distributions are used to model rare events with large *n* and small *p* (binomial distributions are also appropriate but poisson is easier to compute)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.00289778712376148"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = 100\n",
    "k = 5\n",
    "p = 0.01\n",
    "\n",
    "binomial_proba = combin(n, k) * p**k * (1 - p)**(n - k)\n",
    "binomial_proba"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0030656620097620196"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lambda_ = n*p\n",
    "\n",
    "poisson_proba = np.e**(-lambda_)*lambda_**k/factorial(k)\n",
    "poisson_proba"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.00306566200976202"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rv = stats.poisson(mu=lambda_)\n",
    "rv.pmf(k=k)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x11085edd8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n = 10000\n",
    "p = 0.05\n",
    "\n",
    "rng = np.random.RandomState(123)\n",
    "binomial_samples = rng.binomial(n, p, 5000)\n",
    "plt.hist(binomial_samples, normed=True)\n",
    "plt.ylabel('Probability')\n",
    "plt.xlabel('k successes')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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iKcAZPPugrU2SzkrPf/85SQPAAMCcOXPGPUgzs8mizCuS/cDs3PqsVNZMm3p99wFfScNh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\n/H8DRDcso6smiQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x110a16518>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rng = np.random.RandomState(123)\n",
    "poisson_samples = rng.poisson(n*p, 5000)\n",
    "plt.hist(poisson_samples, normed=True)\n",
    "plt.ylabel('Probability')\n",
    "plt.xlabel('k successes')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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McoCLqFVczKwIeBr4qrsv3I/3kBBURqq58e8zycww7vjKEZqVTSTFJNIpPBK4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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x110a10e48>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = 0.1\n",
    "n = 100\n",
    "lambda_ = n*p\n",
    "\n",
    "probas = []\n",
    "for k in range(25):\n",
    "    probas.append(np.e**(-lambda_)*lambda_**k/factorial(k))\n",
    "    \n",
    "\n",
    "plt.plot(range(25), np.cumsum(probas))\n",
    "plt.xlabel('k successes')\n",
    "plt.ylabel('Probability of outcome, CDF')\n",
    "plt.show()"
   ]
  }
 ],
 "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": "3.6.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}