L2.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lab 2, 5.02"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"https://imgs.xkcd.com/comics/exercise.png\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Python has a lot of life-saving (life simplifying?) features. \n",
    "\n",
    "This lab is about getting to know common tools to do stuff done quickly"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## List + tuple"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[], [], []]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lists = [[]] * 3\n",
    "lists"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[3], [3], [3]]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lists[0].append(3)\n",
    "lists"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[3], [5], [7]]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lists = [[], [], []]\n",
    "lists[0].append(3)\n",
    "lists[1].append(5)\n",
    "lists[2].append(7)\n",
    "lists"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "()"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tuples = tuple() \n",
    "tuples"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Slicing and indexing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'He'"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "word = \"Hello\"\n",
    "word[0:2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'o'"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "word[-1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Everything is a pointer\n",
    "\n",
    "<img src=\"img/python_adressing.gif\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Dict"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{'blue': 3, 'yellow': 2}\n",
      "{'blue': 3, 'yellow': 2}\n",
      "{'blue': 3, 'yellow': 2}\n",
      "{'blue': 3, 'yellow': 2}\n",
      "{'blue': 3, 'yellow': 2}\n",
      "{'blue': 3, 'yellow': 2}\n"
     ]
    }
   ],
   "source": [
    "print {'yellow': 2, 'blue': 3}\n",
    "print dict(yellow=2, blue=3)\n",
    "print dict({'yellow': 2, 'blue': 3})\n",
    "print dict(zip(('yellow', 'blue'), (2, 3)))\n",
    "print dict([['blue', 3], ['yellow', 2]])\n",
    "print dict((('blue', 3), ('yellow', 2)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[('blue', 3), ('yellow', 2)]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = {'yellow': 2, 'blue': 3}\n",
    "m.items()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Ex. 1\n",
    "a) Write function that takes a list and returns sum of elements in the list. Test it using following code:\n",
    "\n",
    "\n",
    "Hint: If you don't remember syntax for functions you can refer to lecture slides"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'f' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-29-e8227275c6c1>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0;32massert\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0;36m16\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m: name 'f' is not defined"
     ]
    }
   ],
   "source": [
    "assert f([1,2,3]) == 16"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Ex. 2\n",
    "\n",
    "* Use list and list indexing to simplify printing out tic-tac-toe board.\n",
    "* Add storing board as a list of lists (``` board = [['', '', ''], ['','',''], ['','','']] ```)\n",
    "* Wrap printing as a function ```print_board(board)```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "A = [['', 'O', ''], ['','',''], ['','','X']]\n",
    "x = 1\n",
    "y = 1\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "A[x][y] = 'X'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'board' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-30-2545a47c0cc7>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0;32mprint\u001b[0m \u001b[0mboard\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m''\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'O'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m''\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m''\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m''\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m''\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m''\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m''\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m'X'\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m: name 'board' is not defined"
     ]
    }
   ],
   "source": [
    "print board([['', 'O', ''], ['','',''], ['','','X']])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Numpy and matplotlib"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Section adapted from https://github.com/jrjohansson/scientific-python-lectures"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The numpy package (module) is used in almost all numerical computation using Python. \n",
    "It is a package that provide high-performance vector, matrix and higher-dimensional data structures for Python. It is implemented in C and Fortran so when calculations are vectorized (formulated with vectors and matrices), performance is very good.\n",
    "To use numpy you need to import the module, using for example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/kudkudak/anaconda/lib/python2.7/site-packages/matplotlib/font_manager.py:273: UserWarning: Matplotlib is building the font cache using fc-list. This may take a moment.\n",
      "  warnings.warn('Matplotlib is building the font cache using fc-list. This may take a moment.')\n"
     ]
    }
   ],
   "source": [
    "%matplotlib inline \n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are a number of ways to initialize new numpy arrays, for example from\n",
    "\n",
    "* a Python list or tuples\n",
    "* using functions that are dedicated to generating numpy arrays, such as `arange`, `linspace`, etc.\n",
    "* reading data from files"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### From lists"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 2])"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# a vector: the argument to the array function is a Python list\n",
    "v = np.array([1,2])\n",
    "\n",
    "v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 2],\n",
       "       [3, 4]])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# a matrix: the argument to the array function is a nested Python list\n",
    "M = np.array([[1, 2], [3, 4]])\n",
    "\n",
    "M"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Same interface as list"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1\n",
      "4\n"
     ]
    }
   ],
   "source": [
    "# v is a vector, and has only one dimension, taking one index\n",
    "print v[0]\n",
    "\n",
    "# M is a matrix, or a 2 dimensional array, taking two indices \n",
    "print M[1,1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So far the `numpy.ndarray` looks awefully much like a Python list (or nested list). Why not simply use Python lists for computations instead of creating a new array type? \n",
    "\n",
    "There are several reasons:\n",
    "\n",
    "* Python lists are very general. They can contain any kind of object. They are dynamically typed. They do not support mathematical functions such as matrix and dot multiplications, etc. Implementing such functions for Python lists would not be very efficient because of the dynamic typing.\n",
    "* Numpy arrays are **statically typed** and **homogeneous**. The type of the elements is determined when the array is created.\n",
    "* Numpy arrays are memory efficient.\n",
    "* Because of the static typing, fast implementation of mathematical functions such as multiplication and addition of `numpy` arrays can be implemented in a compiled language (C and Fortran is used).\n",
    "\n",
    "Using the `dtype` (data type) property of an `ndarray`, we can see what type the data of an array has:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "int64\n",
      "(2, 2)\n"
     ]
    }
   ],
   "source": [
    "# What is M?\n",
    "print M.dtype\n",
    "print M.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[2 4]\n",
      "[[ 7 10]\n",
      " [15 22]]\n"
     ]
    }
   ],
   "source": [
    "# What has happened here?\n",
    "print M[0] + v\n",
    "# Every operation like this is executed using fast C/Fortran code!\n",
    "print np.dot(M, M)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sum: 3\n",
      "Avg: 1.5\n"
     ]
    }
   ],
   "source": [
    "# There is all sort of things you can do with numpy array\n",
    "dir(v)\n",
    "\n",
    "# For instance:\n",
    "print \"Sum:\", v.sum()\n",
    "print \"Avg:\", v.mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Random module"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.577476027676\n"
     ]
    }
   ],
   "source": [
    "print np.random.uniform(0, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "8\n"
     ]
    }
   ],
   "source": [
    "print np.random.choice(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1, 1]\n"
     ]
    }
   ],
   "source": [
    "# Selecting random elemn\n",
    "L = [[1,1], [1,2]]\n",
    "print L[np.random.choice(len(L))]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Ex 3\n",
    "\n",
    "Solve a set of linear equations\n",
    "\n",
    "$3x + 2y = 4$\n",
    "\n",
    "$1x + 10y = 2$\n",
    "\n",
    "Using numpy\n",
    "\n",
    "Hint:\n",
    "* Initialize matrix A and b\n",
    "* Use np.linalg.inv\n",
    "* Use np.dot, \"*\" is not multiplying matrices correctly"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Ex 4\n",
    "\n",
    "Evaluate sinus for range [0, 1] (100 consecutive numbers)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Matplotlib"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Section adapted from https://github.com/jrjohansson/scientific-python-lectures/blob/master/Lecture-4-Matplotlib.ipynb"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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99cqOSCVq+XHQEbENcHNm7tj4ehawV2bOi4gtgN7M7LPM0OOgpQE0diycdBJcdhkcdFDZ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\nlAlAkmrKBCBJNfX/AXxH+abiQFdBAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10487c650>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(0, 5, 10)\n",
    "y = x ** 2\n",
    "\n",
    "plt.plot(x, y, 'r')\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y')\n",
    "plt.title('title')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "X = "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([ 22.,  27.,  23.,  16.,  31.,  19.,  20.,  19.,  12.,  11.]),\n",
       " array([ 0.0052952 ,  0.1024081 ,  0.19952101,  0.29663391,  0.39374682,\n",
       "         0.49085972,  0.58797263,  0.68508553,  0.78219844,  0.87931134,\n",
       "         0.97642425]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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hE+wycm5OK9S7nIh09D7fPcY+LRj1pKy3A1+aaUX9WLMfkvwicHFV/R0w8tStk0iX34kX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\ndUlqiKEuSQ35P6t2kfBDMtJlAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x106d95110>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.random.uniform(0, 1, 200)\n",
    "plt.hist(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Ex 5. (Coin simulation)\n",
    "\n",
    "A lot of data analysis can be conducted in bayesian framework. In such framework we postulate parametrized model and then find parameters that best explain data.\n",
    "\n",
    "Once we have those parameters we can do multiple things, one is assesing correctness of some hypothesis.\n",
    "\n",
    "In this exercise you will just postulate model and simulate its outcome for given variables.\n",
    "\n",
    "## Problem\n",
    "\n",
    "We have a coin from factory that produces biased coins. Repeat 10000 times:\n",
    "\n",
    "1. Assign $\\theta$ using beta distribution (https://en.wikipedia.org/wiki/Beta_distribution) with a=10, b=10, by ```theta=np.random.beta(a=10, b=10)```\n",
    "\n",
    "2. Run 1000 trials using ```np.random.binomial(n, theta)```\n",
    "\n",
    "\n",
    "Then:\n",
    "\n",
    "1. Plot histogram of results (result is a list of 100000 numbers)\n",
    "\n",
    "2. Plot histogram of results for a=10, b=30. What has changed?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([  25.,  148.,  298.,  279.,  165.,   52.,   29.,    2.,    1.,    1.]),\n",
       " array([ 0.02217915,  0.04369789,  0.06521663,  0.08673537,  0.10825411,\n",
       "         0.12977285,  0.15129159,  0.17281033,  0.19432907,  0.21584781,\n",
       "         0.23736655]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Hx16ZJGnFJjLtU1X/m+RdwAEW7yt83OCXpPVjIlf+kqT1bVKv81/2DV5J/ibJsSSHkly+\nkn03klX0xasHyo8n+UaSh5McnF6rJ2O5vkjy8iT/muRnSd67kn03mhH7orVx8ebufL+R5P4krxp2\n341mxL5Y2bioqrEuLP5C+U9gB3AOcAh4xWnbvBH4p279tcADw+67kZZR+qL7+r+AbWt9HlPsi5cA\nvw78BfDeley7kZZR+qLRcXElcH63fk3jebFkX6xmXEziyn+YN3jtBj4JUFUPAucnmRly341klL4A\nCJvn+UvL9kVV/aCqHgKeXum+G8wofQHtjYsHquqn3ZcPcOo9Qy2OizP1BaxwXExiAA3zBq8zbbPZ\n3hy2mr44MbBNAfck+VqSP5pYK6djlP/bFsfF82l5XLwD+OIq913vRukLWOG4WC9v8hr5Naub1FVV\n9XiSl7L4n3q0qu5f60ZpzTU5LpK8Dngb8Btr3Za1doa+WNG4mMSV/wngVwe+3t6Vnb7NRUtsM8y+\nG8kofUFVPd79+33gcyz+WbhRjfJ/2+K4OKMWx0V3Y/PvgGur6scr2XcDGaUvVj4uJnDT4mxO3bQ4\nl8WbFjtP2+a3OXWT80pO3cBZdt+NtIzYF+cBL+zWXwB8Fbh6rc9pkn0xsO0+4M9Ws+9GWEbsi+bG\nBYuBeAy4crX9uBGWEftixeNiUidxDYvv8D0G7OnK3gn88cA2H+lO9BvAa55v3428rLYvgEu6//yH\ngcMt9AUww+Kc50+AHwHfGRjQTY2LM/VFo+Pio8APga93533w+fbdyMtq+2I148I3eUlSgzbLy8Uk\nSStg+EtSgwx/SWqQ4S9JDTL8JalBhr8kNcjwl6QGGf6S1KD/A5ywpWYuABD/AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x107c6a810>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(np.random.beta(10, 100, 1000)) # Beta distribution for bias of the coin"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Ex. 6\n",
    "\n",
    "Solve Ex.3 using Matlab"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.11"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}