Chapter 8 - Random Numbers and Simple Games.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 8.1 Drawing random numbers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import random"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.3858569577847155"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "random.random()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.8343129788220142"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "random.random()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 8.1.1 The Seed"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.24, 0.54, 0.37, 0.60, 0.63, 0.07, 0.01, 0.84, 0.26, 0.23, 1.00\n",
      "0.47, 0.84, 0.48, 0.64, 0.15, 0.63, 0.87, 0.52, 0.74, 0.67, 0.06\n",
      "0.76, 0.59, 0.30, 0.03, 0.87, 0.47, 0.72, 0.88, 0.71, 0.92, 0.39\n"
     ]
    }
   ],
   "source": [
    "random.seed(3)\n",
    "print(', '.join(['%.2f' % random.random() for i in range(11)]))\n",
    "print(', '.join(['%.2f' % random.random() for i in range(11)]))\n",
    "print(', '.join(['%.2f' % random.random() for i in range(11)]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.24, 0.54, 0.37, 0.60, 0.63, 0.07, 0.01, 0.84, 0.26, 0.23, 1.00\n",
      "0.47, 0.84, 0.48, 0.64, 0.15, 0.63, 0.87, 0.52, 0.74, 0.67, 0.06\n",
      "0.76, 0.59, 0.30, 0.03, 0.87, 0.47, 0.72, 0.88, 0.71, 0.92, 0.39\n"
     ]
    }
   ],
   "source": [
    "random.seed(3)\n",
    "print(', '.join(['%.2f' % random.random() for i in range(11)]))\n",
    "print(', '.join(['%.2f' % random.random() for i in range(11)]))\n",
    "print(', '.join(['%.2f' % random.random() for i in range(11)]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 8.1.2 Uniformly Distributed Random Numbers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt \n",
    "\n",
    "plt.figure()\n",
    "plt.plot([random.uniform(-1, 1) for i in range(300)], '+')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# normal distribution\n",
    "plt.figure()\n",
    "plt.plot([random.gauss(0, 1) for i in range(300)], '+')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 8.1.3 Visualizing the Distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/onnoeberhard/anaconda3/lib/python3.6/site-packages/matplotlib/axes/_axes.py:6462: UserWarning: The 'normed' kwarg is deprecated, and has been replaced by the 'density' kwarg.\n",
      "  warnings.warn(\"The 'normed' kwarg is deprecated, and has been \"\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "plt.subplot(211)\n",
    "plt.hist([random.random() for i in range(1000)], bins=50, normed=True)\n",
    "plt.subplot(212)\n",
    "plt.hist([random.random() for i in range(100000)], bins=50, normed=True)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 8.1.4 Vectorized Drawing of random numbers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.6261707558351127"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.random()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.41875195, 0.22002842, 0.37352794, 0.07059993, 0.62934343,\n",
       "       0.60844382, 0.16715554, 0.96157769, 0.46538005, 0.15534173,\n",
       "       0.16649141, 0.38051623, 0.76307046, 0.91083554, 0.65517142,\n",
       "       0.68555901, 0.28538579, 0.86987374, 0.08422141, 0.72905601])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = np.random.random(size=20)\n",
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 8.1.5 Computing the mean and standard deviation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.4800165752869951"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.2810610390159891"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.std(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.2810610390159891"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sqrt(np.var(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The Gaussian or normal distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.6854581253108192"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "random.gauss(0, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.04892370386886112"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.normal()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.15075558 -0.51104611 -0.96761321]]\n",
      "[[0.42596201 0.39819833 0.93113491]]\n",
      "0.23512074765480273\n",
      "4.548813503927325\n",
      "4.548813503927325\n",
      "[[0.71518937 0.60276338 0.54488318]]\n",
      "[[0.4236548  0.64589411 0.43758721]]\n",
      "[[0.891773   0.96366276 0.38344152]]\n",
      "[[0.79172504 0.52889492 0.56804456]]\n",
      "[[4.96862431 5.15828096 3.63683485]]\n",
      "[[ 1.41020463 -0.37447169  0.27519832]]\n",
      "3.882026172983832\n",
      "3.882026172983832\n",
      "[[6 6]\n",
      " [6 4]]\n",
      "[4 3 5]\n"
     ]
    }
   ],
   "source": [
    "# numpy random madness\n",
    "\n",
    "# uniformly distributed numbers\n",
    "print(np.random.uniform(-1, 1, (1, 3)))    # here, the interval can be specified [low, high)\n",
    "print(np.random.rand(1, 3))                # interval [0, 1), convenience function for uniform.\n",
    "print(np.random.rand())\n",
    "# uniform(a, b) is the same as (b - a) * rand() + a:\n",
    "np.random.seed(0)\n",
    "print(np.random.uniform(4, 5))\n",
    "np.random.seed(0)\n",
    "print((5 - 4) * np.random.rand() + 4)    # however, this is longer and more confusing in this case!\n",
    "\n",
    "# all identical, all like rand, but longer. (unuseful!)\n",
    "print(np.random.random([1, 3]))\n",
    "print(np.random.random_sample([1, 3]))\n",
    "print(np.random.ranf([1, 3]))\n",
    "print(np.random.sample([1, 3]))\n",
    "\n",
    "# normally distributed numbers\n",
    "print(np.random.normal(4, 2, (1, 3)))    # here, mu and sigma can be specified\n",
    "print(np.random.randn(1, 3))             # mu = 0, sigma = 1; convenience function for normal.\n",
    "# normal(mu, sigma) is the same as sigma * randn() + mu:\n",
    "np.random.seed(0)\n",
    "print(np.random.normal(3, 0.5))\n",
    "np.random.seed(0)\n",
    "print(0.5 * np.random.randn() + 3)    # however, this is longer and more confusing in this case!\n",
    "\n",
    "# integers\n",
    "print(np.random.randint(3, 9, size=(2, 2)))    # draw int from [low, high)\n",
    "\n",
    "# sample\n",
    "print(np.random.choice([1, 2, 3, 4, 5], 3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/onnoeberhard/anaconda3/lib/python3.6/site-packages/matplotlib/axes/_axes.py:6462: UserWarning: The 'normed' kwarg is deprecated, and has been replaced by the 'density' kwarg.\n",
      "  warnings.warn(\"The 'normed' kwarg is deprecated, and has been \"\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "plt.hist([np.random.randn() for i in range(1000)], bins=25, normed=True)\n",
    "from scipy.stats import norm\n",
    "x = np.linspace(-4, 4)\n",
    "plt.plot(x, norm.pdf(x))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 8.2 Drawing Integers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "random.randint(2, 6)    # upper limit is includet! (unlike in np.random.randint)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 8.2.8 Drawing a random element from a list"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.arange(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "random.choice(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.choice(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2, 9, 0, 5, 4, 6, 3, 8, 1, 7])"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "random.shuffle(a)    # changes the list (does not copy)\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([9, 2, 5, 6, 1, 7, 8, 0, 4, 3])"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.shuffle(a)\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.10231682, 0.29217737, 0.98299011, 0.13974578, 0.3305963 ,\n",
       "       0.05105306, 0.33126888, 0.32032629, 0.94680717, 0.84515409,\n",
       "       0.38276422, 0.02476906, 0.83103111, 0.66053618, 0.15236448,\n",
       "       0.99607127, 0.10023344, 0.86711454, 0.29426616, 0.43535347])"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.fromiter((np.random.rand() for i in range(20)), dtype=float)    # turn generator into 1d array."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 8.3 Computing Probabilities"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'b'"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "random.choice(\"abc\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 8.6 Random Walk in One Space Dimension"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from math import sqrt\n",
    "from scipy.stats import norm\n",
    "\n",
    "n_steps = 10000\n",
    "n_particles = 500\n",
    "x = np.sum(np.random.choice((-1, 1), (n_steps, n_particles)), axis=1)\n",
    "plt.figure()\n",
    "plt.hist(x, bins=25, density=True)\n",
    "plt.plot(x, np.zeros(x.shape), 'kx')\n",
    "x = np.linspace(-2 * sqrt(n_steps), 2 * sqrt(n_steps))\n",
    "plt.plot(x, norm.pdf(x, 0, sqrt(n_steps)))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "22.4291545984239 100.0\n"
     ]
    }
   ],
   "source": [
    "# sigma obviously is not sqrt(n_steps!)\n",
    "# -> find sigma using a monte carlo simulation!\n",
    "\n",
    "x = np.sum(np.random.choice((-1, 1), (n_steps, n_particles)), axis=1)    # random values\n",
    "xm = x.mean()\n",
    "variance = 1 / len(x) * np.sum((x - xm)**2)\n",
    "sigma = sqrt(variance)\n",
    "print(sigma, sqrt(n_steps))    # aha!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# again:\n",
    "plt.figure()\n",
    "plt.hist(x, bins=25, density=True)\n",
    "plt.plot(x, np.zeros(x.shape), 'kx')\n",
    "x_ = np.linspace(-sqrt(n_steps), sqrt(n_steps), 200)\n",
    "plt.plot(x_, norm.pdf(x_, xm, sigma))    # xm does really approach 0.\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": "3.6.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}