numpy-random.ipynb

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    "***\n",
    "\n",
    "# The numpy.random package\n",
    "\n",
    "***\n"
   ]
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   "source": [
    "![img](images/jupyter_python_numpy.width-808.png)"
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   "source": [
    "## What is Numpy?\n",
    "[Numpy Documentation](https://numpy.org/doc/stable/docs/index.html)"
   ]
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   "source": [
    "NumPy, which is short for Numerical Python,  is an open source project created in 2005 by Travis Oliphant. In Python, Numpy is the fundamental library in scientific computing [1]. It forms the foundations that some of the most important data analytics (pandas) and machine learning (scikit-learn) deploy.\n",
    "At the core of Numpy are multidimensional array object (ndarry) [2]. It also provides various derived objects (e.g. masked arrays and matrices), and an assortment of routines for fast operations on arrays, including mathematical, logical, shape manipulation, sorting, selecting, I/O, discrete Fourier transforms, basic linear algebra, basic) statistical operations, random simulation and much more [1].\n",
    "\n",
    "#### Arrays\n",
    "\n",
    "- An array class is called as ndarray\n",
    "- The data in an ndarray must be homogeneous, i.e. all elemets contained in it must be the same type.\n",
    "- They have a dtype attribute which is an object that describes the data type of the array.\n",
    "- Size = the total number of elements an array contains. \n",
    "- A tuple of integers giving the size of the array along each dimension is known as shape of the array.\n",
    "- Elements in Numpy arrays are accessed by using square brackets and can be initialized by using nested Python Lists\n",
    "- Arrays have a ndim attribute for the number of axes or dimensions of the array [3].\n"
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   "source": [
    "##### Why Use NumPy?\n"
   ]
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    "NumPy arrays are stored in one continuous place in memory unlike Python lists, this makes accessing and manipulate much more efficent. This is incredibly important when working with large amounts of data. Array's also provide many more methods and functions which make working with them \"easier\" [4].\n",
    "\n",
    "The advantages of using Numpy:\n",
    "- Speed: NumPy uses algorithms written in C that complete in nanoseconds rather than seconds.\n",
    "- Fewer loops: NumPy helps you to reduce loops and keep from getting tangled up in iteration indices.\n",
    "- Clearer code: Without loops, your code will look more like the equations you’re trying to calculate.\n",
    "- Better quality: There are thousands of contributors working to keep NumPy fast, friendly, and bug free. [4]"
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   "source": [
    "#### Getting Started"
   ]
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   "source": [
    "# Importing the libary\n",
    "import numpy as np\n",
    "\n",
    "# Plotting\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Plotting\n",
    "import seaborn as sns"
   ]
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    "# Standard plot size.\n",
    "plt.rcParams['figure.figsize'] = (15, 10)\n",
    "\n",
    "# Standard colour scheme.\n",
    "plt.style.use('ggplot')"
   ]
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    "<br>"
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   "source": [
    "### What is numpy.random?\n",
    "[Documentation for numpy.random](https://numpy.org/doc/stable/reference/random/index.html#:~:text=Random%20sampling%20(%20numpy.-,random%20),Objects%20that%20generate%20random%20numbers)"
   ]
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   "source": [
    "Randomness has many uses in science, art, statistics, cryptography, gaming, gambling, and other fields. For example, random assignment in randomized controlled trials helps scientists to test hypotheses, and random numbers or pseudorandom numbers help video games such as video poker. Many elements of statistical practice depend on randomness via random numbers. Where those random numbers fail to be actually random, any subsequent statistical analysis may suffer from systematic bias. Elements of statistical practice that depend on randomness include: choosing a representative sample of the population being examined [5].\n",
    "\n",
    "Random number does NOT mean a different number every time. Random means something that can not be predicted logically. numpy.random is a module in the numpy library used for arrays [6].\n",
    "Numpy.random is used to create arrays of random numbers . The reason we use this “fake data'' is that it enables people to test before working with real data. It can efficiently generate arrays of sample values from many kinds of probability distributions [7].\n",
    "\n",
    "Python has a random module, which allows for the generation of single random numbers with a few basic distributions. numpy.random has much more functionability; allowing the genration of whole arrays of random numbers, ways to work with these arrays and allows for a wider selection of probability distributions.\n",
    "\n",
    "\n",
    "\n",
    "numpy.random can be broken into:\n",
    "\n",
    "- Simple random data\n",
    "- Permutations\n",
    "- Distribution\n"
   ]
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    "<br>"
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   "source": [
    "### Switch to PCG-64"
   ]
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   "source": [
    "\n",
    "Numpy has recently updated the method it uses to generate random numbers. Previously, Numpy employed the Mersenne Twister pseudo-random number generator (MT19937) refered to as legacy random state - This method can still be used. It has updated to the Permutation Congruential generator (PCG-64) method. PCG-64 is a 128-bit implementation of O’Neill’s permutation congruential generator [33]. It was developed in 2014 [34].\n",
    "\n",
    "\n",
    "The Mersenne Twister was developed in 1997 by Makoto Matsumoto and Takuji Nishimura[36]. It is a which has has a period of 2^19937 before it begins repeating a squence of numbers. It works in 624 32bit vaules, where the 32bit value produced cannot be used to predict the next 32bit value. After 624 values it would repeat itself leaving the door open to hackers who where able to gather the 624 sequential values[37]. This compromises the PRNG's predictability.\n",
    "\n",
    "On the otherhand, The Permutation Congruential generator \"...has a period of 2128 and supports advancing an arbitrary number of steps as well as 2127 streams\" and works in 128bits[37]. Making the PCG-64 PRNG faster, more space efficent and better statistic properties than its MT19937 counterpart. \n",
    "\"The main difference between the two is that Generator relies on an additional BitGenerator to manage state and generate the random bits, which are then transformed into random values from useful distributions\" [38]. \n",
    "\n",
    "See the below image for a full overview.\n",
    "\n",
    "![img](images/RNG.PNG)"
   ]
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   "source": [
    "  <br>\n",
    "  \n",
    "  </br>"
   ]
  },
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   "source": [
    "# Setting the PRNG as PCG-64\n",
    "rng = np.random.default_rng()"
   ]
  },
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   "source": [
    "  <br>\n",
    "  \n",
    "  </br>"
   ]
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   "source": [
    "As a result of the change over there as some changes to Simple Random Data as outlined below."
   ]
  },
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   "source": [
    "  <br>\n",
    "\n",
    "  </br>"
   ]
  },
  {
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   "id": "e23bd61e-62d2-43d7-865c-9ca21493ebb1",
   "metadata": {},
   "source": [
    "![image.png](images/generator_changes.PNG)"
   ]
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    "<br>"
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   "source": [
    " ***\n",
    " ## Simple Random Data\n",
    " ***\n"
   ]
  },
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   "source": [
    "#### rng.intergers()\n",
    "\n",
    "An array of specified shape filled with random integers from low (inclusive) to high (exclusive)."
   ]
  },
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   "id": "98add9f4-a7c4-4e70-8bd7-94e30f82f5d4",
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   "source": [
    "<br>"
   ]
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   "cell_type": "code",
   "execution_count": 4,
   "id": "95435480-5143-4613-a874-cf24ccdc2640",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# A Random intergr between 0 and 4\n",
    "rng.integers(5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ff9e49ed-a19a-4545-9229-42d78648edec",
   "metadata": {},
   "source": [
    "<br>"
   ]
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   "cell_type": "code",
   "execution_count": 5,
   "id": "7b972735-0118-4f86-86ef-673f8f864921",
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# A random intergr between low 0 and 10 (exculsive).\n",
    "rng.integers(0,10)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1e0fffcd-cdd8-443c-9f0c-55c08e8f24b3",
   "metadata": {},
   "source": [
    "<br>"
   ]
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   "cell_type": "code",
   "execution_count": 6,
   "id": "139b9f6b-6e11-4936-8177-fd40e7bf9c25",
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 6,
     "metadata": {},
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    }
   ],
   "source": [
    "# A random intergr between low -10 and 10 (exculsive)\n",
    "np.random.randint(-10, 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "490da556-67d2-4242-ba0d-090b026cb175",
   "metadata": {},
   "source": [
    "<br>"
   ]
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   "execution_count": 7,
   "id": "93d55528-a3b5-44b6-b66b-d7ae64cde79e",
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([5, 8, 9, 8, 2], dtype=int64)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# A 1D array of 5 random intergrs between 0 (low) and 10 (exculsive).\n",
    "rng.integers(0, 10, 5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fc0323dc-4d66-4802-b029-2ef0629bd5d9",
   "metadata": {},
   "source": [
    "<br>"
   ]
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   "source": [
    "<br>\n",
    "  The above is a 1D array. numpy.random also allows us to make more dimensional arrays. See examples below"
   ]
  },
  {
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   "id": "c69f2560-c126-4c7c-b2c6-26871c2ccb97",
   "metadata": {},
   "source": [
    "<br>"
   ]
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  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "7842c0ab-1ed2-4fcf-bbd7-b4109ec3c163",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0, 1],\n",
       "       [0, 0]], dtype=int64)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 2D array of random intergers \n",
    "rng.integers(low=2, size=(2,2))"
   ]
  },
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   "cell_type": "markdown",
   "id": "333b7fbb-4ebd-4913-8896-a2e2266224ad",
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   "source": [
    "<br>"
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   "execution_count": 9,
   "id": "94438c01-8214-43e2-89c2-6897cfda6715",
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    {
     "data": {
      "text/plain": [
       "array([[[0, 2],\n",
       "        [1, 4]],\n",
       "\n",
       "       [[0, 1],\n",
       "        [3, 2]]], dtype=int64)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 3D array of random numbers\n",
    "rng.integers(0, 5, (2, 2, 2))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b12d8e02-366e-4b75-b055-93461846332a",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "3e8022c6-8ba4-4182-ab23-bc66513834ea",
   "metadata": {},
   "outputs": [],
   "source": [
    "#  Array of 5 intergers \n",
    "x = rng.integers(0, 10, 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "4ae1d159-8513-432a-b0b5-b9e6d9c808cc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 8, 6, 8, 5], dtype=int64)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "91072f37-3099-4e37-9af3-47f0d987349d",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "3d56be4c-c99f-4b70-9c3a-963672e51392",
   "metadata": {},
   "outputs": [],
   "source": [
    "# We can manipulte the arrays. \n",
    "y = x * 10 # Multiplying each element in the array by 10."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "f88a6294-532e-445f-bf33-c3e9803bcfbd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0, 80, 60, 80, 50], dtype=int64)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y"
   ]
  },
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   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "### rng.random()\n",
    "An array of a given shape of random (float) values between 0.0 and 1.0, including 0.0 and not 1 from a uniform distribution. The first value passed is the number of \"rows\" in the array. The second is the amount of values each row contains. Any, other arguments passed indicate the number of ndarrays. Deminsions should be posistive"
   ]
  },
  {
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   "id": "448b8511-94d9-4a6e-9f49-4892c4ea407e",
   "metadata": {},
   "source": [
    "As random.randn is not supported in PCG64 to access negative values, we must use the following formula. \n",
    "[Negative rng.random values Documentation](https://numpy.org/doc/stable/reference/random/generated/numpy.random.Generator.random.html#numpy.random.Generator.random)"
   ]
  },
  {
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   "id": "3270e5ea-d1d0-429e-8d47-e436e82bae0a",
   "metadata": {},
   "source": [
    "                   (b - a) * random() + a\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "8132f71a-30bc-4635-a3be-447751ba21c5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-1.71350413, -2.59248963])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Generating an array of negative values between -3 and 0.\n",
    "3 * rng.random(2) - 3"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0f7039b8-f520-4214-98ed-0ad1abaeed33",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "6fbee888-9043-47e2-b0f7-794c6a990a39",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.12088121620392078"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# If no argument is given a single random Python float is returned.\n",
    "rng.random()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3184e260-7894-4cca-a9a6-03101274987b",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "cda93bde-1ba4-4c1a-9ac7-e1b7479ecdba",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Generating 4 random values. 1D array\n",
    "x = rng.random(4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "5d43e55a-89b1-4c62-988c-e4b19d1df360",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.9860467 , 0.34604513, 0.50692006, 0.88812468])"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1434a559-2b2e-4f01-b1b8-b0cfbc0d1add",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "16b1f206-ba1d-4df7-9664-ce18f7580a0f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.49080214, 0.98261033, 0.14432308, 0.39969727],\n",
       "       [0.57354146, 0.24411714, 0.4447777 , 0.81725202]])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# A 2D array containing 4 vaules each\n",
    "rng.random((2,4)) "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "abd27096-3c22-4c78-bd44-86e597cb33c2",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e966c6aa-39a1-4752-b4e8-5161fc38e435",
   "metadata": {},
   "source": [
    "  <br>\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "f9136473-85b2-4482-b74f-8f403458a0c4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[[0.31284527, 0.74833465, 0.25329876],\n",
       "        [0.52002461, 0.52768928, 0.48025393]],\n",
       "\n",
       "       [[0.62980782, 0.10345195, 0.47677529],\n",
       "        [0.42588932, 0.64789233, 0.7308958 ]]])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# A 3D array \n",
    "rng.random((2, 2, 3))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9605fafb-229d-4c90-840d-d9c7dc3079a1",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0f94917b-0bc1-4a7a-aa83-25f0185d7d1b",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "24f6769b-2615-4f71-8434-a6211f068cec",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([10049.,  9963., 10122.,  9922., 10062.,  9981., 10027.,  9958.,\n",
       "         9897., 10019.]),\n",
       " array([3.79546664e-06, 1.00002854e-01, 2.00001913e-01, 3.00000972e-01,\n",
       "        4.00000030e-01, 4.99999089e-01, 5.99998148e-01, 6.99997207e-01,\n",
       "        7.99996265e-01, 8.99995324e-01, 9.99994383e-01]),\n",
       " <BarContainer object of 10 artists>)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x720 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plotting a histogram to show the uniform distribution\n",
    "hist_plot = rng.random(100000)\n",
    "plt.hist(hist_plot)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "13646993-11c1-46bc-834d-c6c4d612dfbf",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d05f3741-74e5-4e96-ab4b-5bb7b883d251",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1044e112-2aa3-49c8-8abd-b5e96888cb2e",
   "metadata": {},
   "source": [
    "The histogram shows a uniform (rectangule) distribution. There spread of elements in each bin are similar in number.  If a smaller number was chosen the uniform shape would be less evident. See the histogram below"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "73daaf6b-6cd0-43ab-970b-6e43258768e5",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "8fa27e68-5117-4fce-924a-9491a20691e3",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Plotting a histogram \n",
    "h = rng.random(100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "65e02602-bb12-454c-9bdb-5e9127a0384e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([14.,  7.,  6., 10., 13., 12.,  8., 11.,  6., 13.]),\n",
       " array([0.0018198 , 0.097714  , 0.19360821, 0.28950242, 0.38539663,\n",
       "        0.48129083, 0.57718504, 0.67307925, 0.76897345, 0.86486766,\n",
       "        0.96076187]),\n",
       " <BarContainer object of 10 artists>)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 1080x720 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(h)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aef70c01-7846-491c-938f-21784338710f",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0bdac8b2-6ac4-4da8-b982-ffeed1dfbe4e",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3c8511b4-231a-4f40-a7e2-602fc2c34570",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ecff5677-e6d6-474d-a2a1-0db95709e9c2",
   "metadata": {
    "tags": []
   },
   "source": [
    "#### rng.choice\n",
    "\n",
    "The rng.choice method is used to get a random value from an array. It is used with one dimensional arrays and returns a new numpy array with randomly chosen values [9].\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b6f87383-fd70-428b-9471-8c2577c4848e",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "987922f6-264f-472f-8ec1-820d65d753b2",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "# Retuning a random value from an array\n",
    "matrix = ['red pill', 'blue pill']"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d6b44ddc-69d4-4b43-8c46-a06b7f116030",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "94783833-9110-4ade-a609-7bea8a4c0d49",
   "metadata": {},
   "source": [
    "![img](images/matrix.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1491bc80-9afb-4d35-b73b-2352df0fa1c8",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "739714a4-48eb-4717-8a78-e33757538077",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "blue pill\n"
     ]
    }
   ],
   "source": [
    "print(rng.choice(matrix))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6acd8c09-a8ad-4949-bcd4-e426ca6f1771",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "8c35a879-ca58-4158-b0b8-f81976425e5f",
   "metadata": {},
   "outputs": [],
   "source": [
    "# An array of 100 random intergers between 0 and 4\n",
    "x = rng.choice(5, 100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "4658ab21-045d-42ab-95ae-b59318f68de4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2, 0, 4, 3, 0, 3, 1, 1, 2, 3, 1, 1, 0, 3, 2, 3, 3, 4, 1, 4, 4, 1,\n",
       "       1, 3, 2, 3, 0, 3, 4, 1, 3, 2, 3, 4, 0, 1, 0, 2, 0, 4, 2, 2, 1, 0,\n",
       "       3, 4, 1, 1, 4, 2, 0, 3, 0, 2, 1, 3, 4, 3, 3, 3, 0, 1, 4, 2, 4, 2,\n",
       "       1, 2, 2, 0, 0, 3, 2, 1, 3, 1, 1, 0, 1, 3, 1, 0, 2, 3, 1, 0, 2, 2,\n",
       "       3, 1, 0, 3, 4, 1, 0, 2, 2, 0, 4, 0], dtype=int64)"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b06c01ee-ac3c-4cc5-a8ba-74b94376d946",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2917d9d7-af17-44b4-bea8-8d39257daf89",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "4b971982-d0e9-4eb6-a4b7-379d80218695",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Choice')"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 1080x720 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(x)\n",
    "plt.title('Choice')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "49307a7d-1ddd-418f-8295-9da242d95ebc",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1ae66870-cbdb-42b5-91ee-1f43e51359ed",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "310485ce-ddf2-4c58-b663-13bba648030c",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Probability of each element of in the array can also be set\n",
    "prob = rng.choice(5, 100, p =[0.2, 0.1, 0.3, 0.4, 0]) # a and p must be the same size."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "f0c2d448-0497-47c4-8e50-a095a40a2c91",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2, 1, 3, 3, 3, 3, 3, 3, 2, 3, 1, 3, 3, 3, 2, 2, 0, 0, 2, 3, 0, 0,\n",
       "       0, 2, 2, 3, 2, 0, 3, 0, 0, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 1, 2, 2,\n",
       "       0, 2, 3, 0, 3, 2, 1, 0, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 1,\n",
       "       3, 2, 3, 2, 0, 3, 1, 2, 1, 0, 0, 3, 2, 0, 2, 1, 2, 2, 2, 3, 0, 2,\n",
       "       0, 3, 2, 2, 3, 0, 0, 0, 2, 2, 3, 3], dtype=int64)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prob"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "960a1dfa-3f6c-447a-a992-90b7f44b73bd",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a97796bc-58d9-4f48-bf77-68f7c709bbac",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2de53015-8936-4223-bb76-aa4a89e1eec2",
   "metadata": {
    "tags": []
   },
   "source": [
    "#### rng.bytes\n",
    "\n",
    "This function outputs a string equalling the parameter(lenght) passed to it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "0937bf78-6a16-43e0-b054-cb47475a225a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "b'^\\xa9\\x81'"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Generates a string of 3 random bytes\n",
    "rng.bytes(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2b84c447-91f7-4c11-80a6-4a306531cc1a",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "354d1502-229c-4a4a-8853-18c62930eec4",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f444d69a-8c6f-4cd8-8616-a920eb49f9c0",
   "metadata": {},
   "source": [
    "\n",
    "## Permutations\n",
    "\n",
    "\n",
    "***"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "46b82fce-8d3c-4964-9cb6-81cdc63ac90e",
   "metadata": {},
   "source": [
    "A permutation is the arrangement of its values into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements [11]. Randomly reordering an arragement of elements is refered to permutating [7]. A permutation refers to an arrangement of elements e.g. [3, 2, 1] is a permutation of [1, 2, 3] and vice-versa. \n",
    "\n",
    "![img](images/permutations.png)\n",
    "\n",
    "Within numpy.random there are two methods for permutation:\n",
    "\n",
    "- Shuffle\n",
    "- Permutation\n",
    "- Permuted\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3583bc23-571f-4c9c-ad03-0ab708bfff98",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a94caf19-8ad3-496f-ad3b-c2b99287f8d6",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a21d55e8-e169-413b-9ab0-6fbd9146d0d7",
   "metadata": {},
   "source": [
    "#### rng.shuffle\n",
    "\n",
    "This method changes(shuffles) the values in an array, in place."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "55a6fb02-cda2-4c4a-a29e-4df817c2eca2",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "afb10782-4167-486d-a97a-967cdece7b6d",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Creating a 1D array \n",
    "single = rng.integers(0, 10, 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "842aebf8-3928-4355-b3ed-61c1909bfe1f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([5, 7, 8, 6, 4], dtype=int64)"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "single"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b9468d42-72d4-4183-a152-400db602aee1",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f055a484-727c-467b-af86-0838e4d4e79d",
   "metadata": {},
   "source": [
    "<br> For multi-demensional arrays, it shuffles the arrays in the matrix not their values."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "24347eb1-643f-4a46-be66-cb8b7ff1854a",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "a9ab67a8-a05f-40e7-80db-939535df5c6c",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Creating a 3D array\n",
    "multi = rng.integers(5, size=(3,5))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "85ea9550-9d4f-46bc-8444-8788c3b826ce",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2, 3, 4, 0, 0],\n",
       "       [0, 4, 4, 1, 2],\n",
       "       [3, 4, 0, 1, 4]], dtype=int64)"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "multi"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8b5167eb-5be0-4848-a91e-2ae464125a09",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c4b6c5eb-d2cf-452e-ab66-e95e5a678b02",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "aee7e3f2-91c0-4d78-9df0-d1be5b9e687a",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Randomly shuffling the 3D array\n",
    "rng.shuffle(multi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "527d9a3e-3e6d-48c1-a0cd-b9bc17b79491",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[3, 4, 0, 1, 4],\n",
       "       [2, 3, 4, 0, 0],\n",
       "       [0, 4, 4, 1, 2]], dtype=int64)"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# New arrangement of the 3D array. *Note only order of matrixes changed\n",
    "multi"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0f21317e-33af-4557-97f8-dfa114b7d8b3",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b4ffdd80-de43-45dd-8e3d-3fdddeb3c209",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "#### rng.permutation\n",
    "\n",
    "This method differs from random.shuffle in that it does not overwrite the orginal array, it returns a copy. Additionally, random.permutation does not shuffle in place i.e. it shuffles up to but not inculding the number passed to it. "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "19116b3a-7d15-4c8d-bb8e-391448347597",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "a79168b7-3fb8-4da3-b460-d0ac3fddfbd9",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Creating a randomly permuted array of elements up to 9\n",
    "orignal = rng.permutation(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "9daef638-22da-41c5-bc60-e2bacad9be80",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([5, 9, 3, 6, 7, 1, 2, 8, 0, 4])"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "orignal"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bec3323f-e1c9-4fb6-986e-0d4bde27012b",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0927e47e-cc48-4fc7-8c6f-01720227501a",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "id": "d2376c47-903a-4385-bd89-108a0181e106",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Permutating the order of the orignal order\n",
    "new = rng.permutation(orignal)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "id": "787d41bb-2de6-4ff6-978a-7d0a336c279a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2, 4, 7, 3, 5, 1, 8, 0, 6, 9])"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# A copy of the orginal array in a different order.\n",
    "new"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d82f935d-21bd-4032-af57-84ad5876f8a8",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1f8aaed2-19e9-427c-b8d5-db974e8593bd",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "id": "9af0cab9-d23c-42a3-a4ac-274049996f51",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([5, 9, 3, 6, 7, 1, 2, 8, 0, 4])"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Showing that the order of the orignal order has not changed.\n",
    "orignal"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d52cb2aa-972b-4e34-8bdd-5f14991f62c4",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5d39c3bd-cf23-49e6-9c4d-7d3364c8df21",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "35ca508a-6c66-4ba6-a565-1c4fabaf4b48",
   "metadata": {
    "tags": []
   },
   "source": [
    "#### rng.permuted\n",
    "\n",
    "Unlike shuffle, each slice along the given axis is shuffled independently of the others [39]. If no axis is specified, it is automatically shuffled on the x-axis.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "id": "a4178a53-0882-4df8-a2ff-0eb4502dc115",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Creating a 3d array\n",
    "x =  rng.integers(5, size=(3,2))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "10499f8a-cc10-483e-bf7d-7caa9904d217",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "id": "49599295-8797-4351-8ab4-91c3b4629b7a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[3, 0],\n",
       "       [3, 2],\n",
       "       [4, 4]], dtype=int64)"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "72a7783f-3fd9-4f49-8156-157fe82f7714",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5b61a71e-be78-46f2-87d6-a6f23cc0dd97",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "id": "9751f073-9818-4e02-a175-e47b46623444",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[3, 3],\n",
       "       [4, 0],\n",
       "       [4, 2]], dtype=int64)"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Shuffling on the x-axis\n",
    "rng.permuted(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e99f9b99-05a9-465f-b2d6-6bc58177db6d",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "256ce5db-b2a0-4f39-9002-770ce6e89d41",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "id": "07532b5a-93e8-4c81-9a2a-fdd953ac4af7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0, 3],\n",
       "       [2, 3],\n",
       "       [4, 4]], dtype=int64)"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Shuffling on the y-axis\n",
    "rng.permuted(x,axis=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5949543e-0088-41fa-9fb8-8100f28f07ce",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "42fd15e5-b070-4a8e-8323-cd97d5c1361f",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bc085657-38a6-419f-a516-e4f8aa64d7ef",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "***\n",
    "\n",
    "## Distributions\n",
    "\n",
    "***"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "30ea79e2-a4c7-494d-b7d1-c409d5735cbf",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "81202d68-d99f-4443-87a3-f3de2d4e819a",
   "metadata": {},
   "source": [
    "The distribution of data obtained from a sample is pivitol in understanding how to analyse it. Probability distributions one of the corner stones of statistics. Understanding which probability distribution function fits your data is important for analysis. There are different probability distributions for different types of data. \n",
    "At the highest level, data can be broadly classified into 2 groups[11]:\n",
    "\n",
    "- Qualitative/Categorical data: This is data which answers questions such as \"what type?\". It deals with traits, characteristics, judgements and descriptions. e.g. male or female. Qualitative data contains subtypes. Nominal data, this is data that doesn't follow a pattern or sequence e.g. hair colour. The other subtype is ordinal data. This data can be ranked in decending or ascending order e.g. tooth pain on a scale of pain. This can be refered to binary data if only two cateogries are present e.g. yes/no. If there is more than two categories it is non binary data e.g. yes/no/maybe. [11].\n",
    "\n",
    "- Quantitative/Numerical data: is data which deals with objective measurements. e.g. height. It is data which can be counted. Quantitative data can be broken down into discrete data. This is data which is represented in whole numbers e.g. how many people are in the crowd. Continuous data is data which can take any value within a range e.g. a person height can be any value within the range of human heights [11]\n",
    "\n",
    "\n",
    "The 5 distributions we will look deeper at are: \n",
    "\n",
    "- Normal Distribution\n",
    "- Poisson Distribution\n",
    "- Binomial Distribution\n",
    "- Logistic Distribution\n",
    "- Uniform Distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3499f15b-c5ad-40d3-b53f-8734f0de7966",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a7b14321-ac03-4715-86de-8d07eb79de66",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8ac507bb-703e-4b66-8543-7599ee540500",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "391398f1-d5f7-4785-a68c-e7d277e26f9f",
   "metadata": {},
   "source": [
    "***\n",
    "## Poisson Distribution\n",
    "***\n",
    "Named after after French mathematician Siméon Denis Poisson, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.[12]  When the total number of occurrences of the event is unknown, we can think of it as a random variable. This random variable follows the Poisson Distribution. The Poisson distribution is a limiting case of the Binomial distribution when the number of trials becomes very large and the probability of success is small [13]. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.\n",
    "\n",
    "Features of a Poisson Process:\n",
    "- The important point is we know the average time between events but they are randomly spaced (stochastic). We might have back-to-back failures, but we could also go years between failures due to the randomness of the process.\n",
    "- A Poisson Process meets the following criteria (in reality many phenomena modeled as Poisson processes don’t meet these exactly):\n",
    "- Events are independent of each other. The occurrence of one event does not affect the probability another event will occur.\n",
    "- The average rate (events per time period) is constant.\n",
    "- Two events cannot occur at the same time [13]\n",
    "\n",
    "\n",
    "Examples:The number of patients arriving in an emergency room between 10 and 11 pm, Vistors to a website, calling a helpline.\n",
    "\n",
    "We need to apply the Poisson Distribution to a Poisson Process to do interesting things like finding the probability of a number of events in a time period or finding the probability of waiting some time until the next event.\n",
    "\n",
    "![img](images/poisson.PNG)\n",
    "\n",
    "\n",
    "- k: The number of occurrences of an event for which poisson distribution has to be found\n",
    "- λ: The expected number of event occurrences in the given interval\n",
    "- e: It is the Euler’s constant whose approximate value is 2.72.\n",
    "\n",
    "Parameters:\n",
    "\n",
    "- lam: It is a float value or an array of float values. The lam corresponds to the λ value in the poisson distribution formula. It denotes the expected number of event occurrences for a given time interval. Its value should be greater than or equal to zero since no event can occur a negative amount of times.\n",
    "\n",
    "- size: It is an optional parameter whose default value is ‘None’. For None, it will return a single value as a sample. If you pass any (x,y) value, then x * y samples will be drawn from the poisson distribution. If any other value n is passed, then n samples would be drawn from it. [14]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bd6c7cfb-0add-4eb4-abe7-11623a7a7422",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "id": "dd8de7bf-00ee-4ad4-8711-0d478d22259c",
   "metadata": {},
   "outputs": [],
   "source": [
    "# The frequency an event will occur 5 times.\n",
    "x = rng.poisson(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "id": "a2fc0bd8-94d8-4bdb-a14c-e6c941066a31",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e8db1093-df7a-49b7-a3dc-5684e560d830",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "id": "9134dd67-f594-4c0a-914b-967afefdf41d",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'Frequency')"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x720 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plotting x\n",
    "plt.hist(x, alpha=0.8)\n",
    "plt.ylabel(\"Frequency\") # Label the y axis"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "55ee6914-edd3-4bbf-8ecb-c43bedbe75ce",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "af90e770-9696-472d-b512-940ddbe30636",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "id": "c8077909-8a38-4d45-93cf-70c2fc0180a3",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Adding a size of 20. This means 20 samples will be taken.\n",
    "y = rng.poisson(5, 100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "id": "5a46d7dc-0639-421a-b695-78e50fde330b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 8,  7,  8,  4,  2,  2,  3,  4, 10,  7,  6,  9,  5, 11,  9,  1,  3,\n",
       "        4,  7,  0,  5,  6, 11,  5,  6,  0,  2,  9,  9,  2,  8,  7, 10,  5,\n",
       "        2,  3,  4, 10,  7,  3,  3,  5,  1,  5, 11,  4,  2,  6,  3,  7,  8,\n",
       "        2,  1,  2,  9,  4,  7,  5,  1,  1, 10,  5,  5,  6,  5,  4,  5,  6,\n",
       "        4,  3,  6,  4,  8,  4, 14,  7,  3,  3,  1,  8,  6,  5,  4,  5,  2,\n",
       "        5,  8,  7,  2,  2,  1,  6,  1,  3,  6,  4,  4,  4,  2,  2],\n",
       "      dtype=int64)"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1f4faa8c-7488-421a-958f-8fcfe50cbfc4",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6ac955c5-e9ce-4fa0-a9de-882988f5122c",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "id": "6b0b509f-4ad6-49f6-89d4-8dd921ec9ffd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'Frequency')"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x720 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plotting y\n",
    "plt.hist(y, alpha=0.8)\n",
    "plt.ylabel(\"Frequency\") # Label the y axis"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "43de4d4e-5241-43de-9b7d-e78b1fd794d9",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "febeaff5-c630-4578-a240-9d43671c063e",
   "metadata": {},
   "source": [
    "<br>\n",
    "The same method can be applied to multidimensional arrays."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a64c1471-a314-44c1-9303-ae5e98a7f7a6",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dca09a6f-a571-4953-a4b3-657e5a933637",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "45634f19-54d5-41ac-9177-7d7599f41e57",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Generating a multi dimeensional array with 16 samples to draw\n",
    "z = rng.poisson(2, (4, 4))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "bcb8c935-d4c2-4530-8c13-7b24730cf7a7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 1, 1, 2],\n",
       "       [0, 3, 2, 0],\n",
       "       [2, 3, 1, 0],\n",
       "       [4, 3, 3, 4]], dtype=int64)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fa459cea-b196-4eb5-9b80-59c86faa3196",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "bccf3e51-077f-4d62-8a3a-e689c55e3fba",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'Frequency')"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x720 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plotting Z\n",
    "plt.hist(z, bins=10)\n",
    "plt.title('Poisson Distribution')\n",
    "plt.ylabel(\"Frequency\") # Label the y axis"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "baa9f73e-3196-4318-9dc8-1185653dfa7f",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6bc3d6d5-1d7b-4404-aad9-4e8dc7564819",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "e3f8c91b-c661-4b17-9c22-6122f42d18ca",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.875"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Mean of samples drawn\n",
    "z.mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "811e59f4-f6ab-4465-a2fb-7524702792a0",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2c7fbc43-a9ae-4142-af8c-d40c997fe62b",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "id": "1173562a-a920-4b1e-9897-16a54909e74a",
   "metadata": {},
   "outputs": [],
   "source": [
    "q = rng.poisson(5, 10000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "id": "17d59518-f2c4-44b6-90d6-09fc86c8936e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([4, 8, 5, ..., 8, 7, 7], dtype=int64)"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "382e3d98-135e-4c42-9dbe-ccf9bdd0fda4",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2a623580-9759-48c1-8ec4-40252b1c42fc",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "id": "4fc9878a-2d96-4e9d-9cdb-d9a25acbb883",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Poisson Distribution')"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x720 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(q, alpha=0.8)\n",
    "plt.title('Poisson Distribution')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f3e8aa60-68b3-40a8-9b03-c05a9d76cae5",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "11f0f3d5-7bce-406b-b316-7abddbb08c5b",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e23d8cad-d1a8-43ab-a4cc-8acba9addd7a",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "id": "5bcbf6bf-fbe3-41d2-bdc4-d5f2356de2a0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.9918"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q.mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "73d68fc8-64cc-444f-ae8d-3c94a3a818ad",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cf603e10-d0c9-4c64-84e8-d81276634d1c",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b7130801-490d-4de8-b149-f086c6f0c524",
   "metadata": {},
   "source": [
    "We can also draw an equal amount of samples for two different values of lam, simultaneously. In this example, we shall be drawing 20 different sample values for two values of lam: λ = 10 and λ = 100. We will be passing lam argument in the form of (10,100) [14]."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "71a96b1c-0bac-46c0-8d6a-e6ca772e673a",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "14a9f49c-7452-4b4a-b6aa-f6233ad5ba73",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "id": "01fa7e34-1454-4b47-ad02-22219a80b6fd",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Code taken from pythonpool\n",
    "r = rng.poisson((10, 100), (20,2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "id": "066ac0dd-4c60-4732-bc68-2aa1a2df8804",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  6,  96],\n",
       "       [  9,  91],\n",
       "       [ 10, 107],\n",
       "       [ 10,  93],\n",
       "       [ 13, 122],\n",
       "       [ 15, 102],\n",
       "       [ 11, 107],\n",
       "       [ 13, 100],\n",
       "       [ 16,  85],\n",
       "       [ 11, 100],\n",
       "       [  8,  89],\n",
       "       [  9,  93],\n",
       "       [ 13,  94],\n",
       "       [ 13, 110],\n",
       "       [  9, 118],\n",
       "       [  9, 101],\n",
       "       [  9,  86],\n",
       "       [ 10,  85],\n",
       "       [ 10, 104],\n",
       "       [ 10, 111]], dtype=int64)"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ad5981f6-97df-46a8-a2f5-d9ad6a14b393",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "262f6bd1-77ad-41bb-9df4-ffdb87b58ff8",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ebb6446d-7d91-4c65-b927-33fc81fb1ad2",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "id": "7410f77e-3d21-4e4c-b2b4-5ef236ca41c9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Poisson Distribution')"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x720 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(r)\n",
    "plt.title('Poisson Distribution')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2607124c-ae6e-49f0-a48e-9b457c2f3301",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d323a3b2-78a3-4447-91cd-9d1149f8cf1d",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c54cbee1-1e61-423e-b9c7-4de1470ccb3f",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7c85e895-335d-4b24-9140-326b28985647",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "69520478-27fa-4939-99ad-d3ce36101887",
   "metadata": {
    "tags": []
   },
   "source": [
    "***\n",
    "## Binomial Distribution\n",
    "\n",
    "A binomial distribution derived from Bernoulli Process, can be thought of as probability of a success or failure outcome in an experiment which is repeated numerous times. The binomial distribution assumes that each experinment/event is independant of each other and that only two outcomes came occur, hence the word bi [15]. An example would be flipping a coin. The result can only be heads or tails, the chance of either appear is even. \n",
    "\n",
    "![img](images/binomial.PNG)\n",
    "\n",
    "\n",
    "- x: total number of successes\n",
    "- P: probability of a success in an individual trail\n",
    "- n: number of trails\n",
    "\n",
    "\n",
    "Parameters:\n",
    "\n",
    "- n: number of trials.\n",
    "- p: probability of occurence of each trial (e.g. for toss of a coin 0.5 each).\n",
    "- size: - The shape of the returned array. If size is None (default), a single value is returned if n and p are both scalars.\n",
    "\n",
    "Each result has a predefined probability of success. For a large number of repetition (size) and equal probability of winning (p=0.5) - e.g. coin toss - the result resembles a normal distribution.\n",
    "\n",
    "Binomial distributions must also meet the following criteria:\n",
    "\n",
    "- The number of observations or trials is fixed. In other words, you can only figure out the probability of something happening if you do it a certain number of times. This is common sense—if you toss a coin once, your probability of getting a tails is 50%. If you toss a coin a 20 times, your probability of getting a tails is very, very close to 100%.\n",
    "- Each observation or trial is independent. In other words, none of your trials have an effect on the probability of the next trial.\n",
    "- The probability of success (tails, heads, fail or pass) is exactly the same from one trial to another [16].\n",
    " \n",
    "\n",
    " \n",
    "Once you know that your distribution is binomial, you can apply the binomial distribution formula to calculate the probability "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5444735f-649a-4bc1-8159-e894e60c3791",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7accb991-6e6e-496e-92ec-9081e8c838c1",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "id": "09c0307e-171a-4c44-bdc9-167c6d0b98e6",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Results of flipping a coin 10 trails, with an even probabilty, tested 10000 times\n",
    "coin_toss = rng.binomial(n=10, p=0.5, size=100000)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b2304a98-e571-4b58-abb4-9355cd6d9334",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d5aa9d2c-a117-419c-ae45-a52a10465036",
   "metadata": {},
   "source": [
    "![img](images/coin.jpg)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2bc2255a-9545-434f-a343-1d820f114909",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "id": "2b84eaa2-ea51-47f4-9af8-c08dbfdae947",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2, 5, 4, ..., 4, 4, 7], dtype=int64)"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "coin_toss"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fcac6341-d1f3-48d2-9efc-67a55277b761",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "id": "6fe032c4-1d2d-41ec-b2e3-1407daeb7f09",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 0, '# of Trials')"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x720 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plotting the coin toss results\n",
    "plt.hist(coin_toss)\n",
    "\n",
    "plt.title('Binomial Distribution') # Adding a title\n",
    "plt.ylabel(\"Probability of Success\") #  y axis label\n",
    "plt.xlabel('# of Trials') # x axis label"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7afaeae4-7fa2-4c98-af67-53a4589a398e",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "096d9fab-ed77-416f-8391-f900082699b0",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8fadaf77-31a1-408d-a6db-7a62701c11cf",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "id": "b24c4868-fe48-4924-a7ba-9ad11e5456e7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.00018"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Mean of the coin toss experiment \n",
    "coin_toss.mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "78ac9eb9-8a48-483e-8bb9-2979228c7a45",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f2f55958-b8c1-410f-ab6d-bceacbc81c68",
   "metadata": {},
   "source": [
    "As expected, around under half the time you choose heads or tails you will be correct. This is to be expected where probaility=0.5. When success and failure are equally likely, the binomial distribution is a normal distribution. Hence, changing the value of p to 0.5, we obtain this graph, which is identical to a normal distribution plot [16].\n",
    "\n",
    "If the probability of success is less than 0.5, the distribution is positively skewed i.e. the probabilities for x is greater for values below the expected value than above it. \n",
    "\n",
    "And if the probability of success is greater than 0.5, the distribution is negatively skewed — i.e. probabilities for X are greater for values above the expected value than below it [17].For example, if we had to choose between 10 boxes but we knew 8 of these boxes contained a prize. The probability of choosing a prize is greater\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "10a33713-fdc9-438c-abe7-d0bc4acfd303",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1ea3e30b-5579-400d-a1bc-19f6c15f85e1",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "id": "4f39c44c-5ba8-4a7e-a672-dff0c0948b22",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Results of flipping a coin 10 trails, with a skewed probabilty, tested 10000 times\n",
    "pos = rng.binomial(n=10, p=0.2, size=100000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "id": "bc27bc90-789c-4c35-8dc4-7be371833e4f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([3, 2, 2, ..., 1, 2, 2], dtype=int64)"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pos"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4300a2d8-530c-4856-a76e-251dea19936c",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ba20019e-6e91-4cde-93cf-c63c02ed4067",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "id": "7af33ba5-28a6-4757-b0bd-0bb1403ec43c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 0, '# of Trials')"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x720 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plotting the coin toss results\n",
    "plt.hist(pos)\n",
    "\n",
    "plt.title('Binomial Distribution') # Adding a title\n",
    "plt.ylabel(\"Probability of Success\") #  y axis label\n",
    "plt.xlabel('# of Trials') # x axis label"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f87c5ed9-8fb4-4cc3-94e3-706e2bbdaa44",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "55201676-500d-44bf-9d3f-576ed87b201a",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "id": "106c13a5-30d0-467f-bf21-8220645836e8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.0038"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Mean of pos\n",
    "pos.mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0d9dc927-8f7b-4a86-855f-6c43b0be272a",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "87b9889a-beca-481d-bd4e-922eb5381f8c",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "id": "3065b7dc-ea93-481b-90ec-5ba1bb91cf50",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Results of flipping a coin 10 trails, with a skewed probabilty, tested 10000 times\n",
    "neg = rng.binomial(n=10, p=0.8, size=100000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "id": "f1e4ff36-dbcb-48ab-a6f4-0eed914f6fbb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 9,  7,  8, ...,  8, 10,  8], dtype=int64)"
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "neg"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "275a5b28-5454-4ea8-9126-8b3ff3b87f3f",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2f203c5c-7647-45bb-89ae-e70fb6215fec",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "id": "a50798cb-074c-4d70-ac6e-0be925a96496",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 0, '# of Trials')"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x720 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plotting the coin toss results\n",
    "plt.hist(neg)\n",
    "\n",
    "plt.title('Binomial Distribution') # Adding a title\n",
    "plt.ylabel(\"Probability of Success\") #  y axis label\n",
    "plt.xlabel('# of Trials') # x axis label"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6d3c03fe-ea3d-4694-ba77-7c6516973a11",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e0591389-7763-45fd-9704-b0b503d5d935",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "id": "4925e53a-143c-40b1-8ce5-6e5e086816ae",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8.00144"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "neg.mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "32bc58dc-6ded-4184-9fdc-ad4b645becc5",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ce6f469a-f335-4d31-a707-dce1167476b1",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "275916b9-54d7-4c1f-84c1-3de65030e276",
   "metadata": {},
   "source": [
    "The differences between binomial and poisson distribution can be drawn clearly on the following grounds:\n",
    "\n",
    "- The binomial distribution is one in which the probability of repeated number of trials is studied. A probability distribution that gives the count of a number of independent events occur randomly within a given period, is called probability distribution.\n",
    "- Binomial Distribution is biparametric, i.e. it is featured by two parameters n and p whereas Poisson distribution is uniparametric, i.e. characterised by a single parameter m.\n",
    "- There are a fixed number of attempts in the binomial distribution. On the other hand, an unlimited number of trials are there in a poisson distribution.\n",
    "- The success probability is constant in binomial distribution but in poisson distribution, there are an extremely small number of success chances.\n",
    "- In a binomial distribution, there are only two possible outcomes, i.e. success or failure. Conversely, there are an unlimited number of possible outcomes in the case of poisson distribution.\n",
    "- In binomial distribution Mean > Variance while in poisson distribution mean = variance [19]."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "09520127-97f0-47ca-ac02-37481b571294",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "47361549-fc01-4bbd-92d1-8352a38b53e6",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f56abd29-49f8-49a0-9a56-79ae9fc5b86c",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6a36fff1-3f07-4a34-b00e-e36d86e05b3b",
   "metadata": {},
   "source": [
    "***\n",
    "\n",
    "## Uniform Distribution\n",
    "\n",
    "\n",
    "A Uniform distribution refers to a type of probability distribution in which all outcomes are equally likely [19] e.g. rolling a dice, you have an equal chance of rolling 1 through 6. likely draw a heart, spade, diamond, clubs from a pack of cards.The concept of a uniform distribution, as well as the random variables it describes make up the skeleton of statistical analysis and probability theory.\n",
    "\n",
    "![img](images/dice.png)\n",
    "\n",
    "\n",
    "With the uniform distribution you provide the lower  and upper  bounds of the interval to specify the minimum (a) and maximum (b) possible values in the range. The smaller number provided will be taken as the lower bounds and the bigger number as the upper bound. Without providing an interval bounds, the function assumes a standard uniform distribution [20].\n",
    "\n",
    "There are two types of uniform distributions: \n",
    "\n",
    "- Discrete: Discrete uniform distributions have a finite number of outcomes. i.e. each has outcome has the same probability of occuring. e.g. heads or tails, the finate is 2 outcomes on a coin, rolling a 6 sided dice - The possible values would be 1, 2, 3, 4, 5, or 6. In this case, each of the six numbers has an equal chance of appearing. Therefore, each time the 6-sided die is thrown, each side has a chance of 1/6. It is impossible to get a value of 1.3, 4.2, or 5.7 when rolling a fair die. However, if another die is added and they are both thrown, the distribution that results is no longer uniform because the probability of the sums is not equal [21].\n",
    "\n",
    "![img](images/uniform_discrete.jpg)\n",
    "\n",
    "- Continuous: A Continuous uniform/rectangular distribution is a statistical distribution with an infinite number of equally likely measurable values. Unlike discrete random variables, a continuous random variable can take any real value within a specified range. A good example of a continuous uniform distribution is an idealized random number generator. With continuous uniform distribution, just like discrete uniform distribution, every variable has an equal chance of happening. However, there is an infinite number of points that can exist [21].\n",
    "\n",
    "\n",
    "\n",
    "![img](images/uniform_continuous.jpg)\n",
    "\n",
    "The uniform distribution is defined by two parameters, a and b:\n",
    "\n",
    "- Low: is the minimum.\n",
    "- High: is the maximum. \n",
    "- Size: if size is None (default), a single value is returned if low and high are both scalars\n",
    "\n",
    "When high == low, values of low will be returned. If high < low, the results are officially undefined and may eventually raise an error [22].\n",
    "\n",
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "47255103-31a4-4228-adb1-b25333e2978d",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "id": "3ea0f190-3df4-462b-89d5-21d5ab5a3e1e",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.2911425359844062"
      ]
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# If the range is not defined, by default is [0,1]\n",
    "# A single random value in the interval [0,1]\n",
    "np.random.uniform() "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c6e281df-0554-454b-9e2b-0461ed63c515",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f2c313b6-cfed-4bfd-aee8-25e97aaf2110",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "id": "c4dfef18-b6fc-45cc-b1bb-82cff7afb413",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.775439860509405"
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Defining a range\n",
    "# A single random value in the interval [-10,10]\n",
    "rng.uniform(-10, 10) "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6b422b94-1072-437b-b175-877d20b67e3a",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2b4c3d44-dca7-4ac4-a45e-1645b83402c9",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "id": "511ea812-ac92-468a-92ec-1a9524b3b947",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([9.60632831, 8.69407901, 6.58618303, ..., 5.90064674, 5.1812084 ,\n",
       "       6.76019069])"
      ]
     },
     "execution_count": 94,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# An array of 20 values in the interval [5,10] \n",
    "rng.uniform(5,10,2000) "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5a5b40e8-804d-492f-b4ca-d0b7a8ffe93e",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7de76780-248c-41ea-bf57-5be446134626",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "id": "b90479d8-09cf-453e-8b92-c0ed3bd64b98",
   "metadata": {},
   "outputs": [],
   "source": [
    "# The probility of rolling a number on a 6 sided dice, if done 100000 times.\n",
    "x = rng.uniform(1,7, 100000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "id": "222c8f1e-0758-47e3-bfad-7e0866a679c1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([5.89731471, 6.52247851, 3.32734975, ..., 1.16973186, 3.17101087,\n",
       "       5.88332086])"
      ]
     },
     "execution_count": 96,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9187b36e-6c92-4876-896e-34c8221e39b7",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1dd9534a-39ad-4e30-8d7f-265af754e394",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "id": "b7b742bd-8c34-46d1-8587-2af2f217deef",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Uniform Distribution')"
      ]
     },
     "execution_count": 97,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x720 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plotting\n",
    "plt.hist(x, alpha=0.8)\n",
    "plt.title('Uniform Distribution')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d9bd4f32-14a0-4f20-aa82-1942d2f62b8c",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "65053870-e47e-4a6c-942a-676bc20b58a0",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e573db1f-6989-4b39-86cb-be1abc1ad6d2",
   "metadata": {
    "tags": []
   },
   "source": [
    "<br>\n",
    "As you can see from the above plot, it mimics a rectangular shape. This is the calling card of a A  uniform distribution. We are almost as equally likely to roll 1, 2, 3, 4, 5, 6 on the dice. The more samples, the more equally the histogram. In the below plot we'll use a smaller sample size to show the difference.\n",
    "\n",
    "<br>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "755076b7-9e14-418c-90fb-dfc85d9e185a",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "106d4202-6721-48a0-b662-3c202ab4e962",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "id": "25545748-f54b-44c5-b24b-d879ec0c0c0e",
   "metadata": {},
   "outputs": [],
   "source": [
    "# The probility of rolling a number on a 6 sided dice, if done 10 times.\n",
    "r = rng.uniform(1,7, 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "id": "20ba77d2-c563-4c49-be43-a04b53f5ff16",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([3.86658962, 4.63592918, 2.76744482, 2.25044787, 2.30868698,\n",
       "       4.78900739, 2.00286077, 3.77494324, 2.23768925, 3.82715032])"
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "da052bee-7dbc-4b0b-9db3-34fb7240c519",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "id": "eaf4e78e-c8ca-4e9d-8d19-821f853919f1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Uniform Distribution')"
      ]
     },
     "execution_count": 100,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x720 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(r, alpha=0.8)\n",
    "plt.title('Uniform Distribution')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e2a0de5f-4ade-411f-bf5e-3d4fb45ed416",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "97d8b133-1062-460c-b1df-e375c7393db1",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "30be5470-7a38-42b0-9320-6e2573afdbb2",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bb6a9a32-a377-4caa-8f2d-218d4ad3b372",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6d105fbc-c3b9-4383-bc13-0ab0048af835",
   "metadata": {
    "tags": []
   },
   "source": [
    "***\n",
    "\n",
    "## Normal Distribution\n",
    "\n",
    "Normal distribution, also known as the Gaussian distribution,is a probability function used in statistics that tells about how the data values are distributed. It usually resembles a bell shaped curve. It is the most important probability distribution function used in statistics because of its advantages in real case scenarios. The normal distribution describes a common occurring distribution of samples influenced by a large or tiny, random distribution or which occurs often in nature [23]. For example, the height of the population. The normal distribution is symmetric about the mean i.e. showing that data near the mean are more frequent in occurrence than data far from the mean e.g. people are more likely to be in and around the average height than they are to be on the smaller or taller side.\n",
    "\n",
    "\n",
    "![img](images/normal_distribution.PNG)\n",
    "\n",
    "\n",
    "For a normal distribution, 68% of the observations are within +/- one standard deviation of the mean, 95% are within +/- two standard deviations, and 99.7% are within +- three standard deviations [24].\n",
    "\n",
    " ![img](images/normal_distribution_std.png)\n",
    " \n",
    "\n",
    "Characteristics of a normal distribution:\n",
    " - mean = median = mode\n",
    " - The normal distribution is the only distribution whose cumulants beyond the first two (i.e., other than the mean and variance) are 0\n",
    " - The mean is 0 and the standard deviation is 1 and has 0 skew.\n",
    " - 50% of values less than the mean and 50% greater than the mean\n",
    " - While normal distributions are symmetrical, not all symmetrical distributions are normal [25].\n",
    "\n",
    " \n",
    "The standard normal distribution has three parameters (none are mandatory): \n",
    "\n",
    "- loc: the mean\n",
    "- scale: the standard deviation\n",
    "- size: Output shape. If the given shape, if size is None (default), a single value is returned if loc and scale are both scalars\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "90b0f951-6b6f-432f-be7b-05abbcab4abe",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ec87b52f-0f53-4bbd-99b5-023b8ec61093",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "id": "0400ff35-bb46-40a1-b972-772c1687b370",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Creating an array of heights within normal distribution from 100,000 samples\n",
    "height = rng.normal(1.75, 0.15, 100000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "id": "9c2fc74b-27a2-4560-92de-d1f38be6dab3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1.61153485, 1.79811581, 1.73030151, ..., 1.775226  , 1.64169269,\n",
       "       1.96306212])"
      ]
     },
     "execution_count": 102,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "height"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1653ab5b-e793-42b6-bd95-efd8b38eec70",
   "metadata": {
    "tags": []
   },
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cdf5eff1-b817-419f-8aed-22c2125046dd",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "id": "e19a13ee-4ad1-4d4f-ae94-b849fd4b7361",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.1109016657670912"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Smallest persons height\n",
    "height.min()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e14f0887-4c19-47b9-b2ec-0c0ab6d53bd2",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "602b8eff-c390-4000-ae33-aa60df36fb9e",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "id": "33597790-d48e-452d-8815-bc13c8f5597d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.3838128137043193"
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# The tallest persons height\n",
    "height.max()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b15d5f51-6e8e-4cc6-bc96-91d02550bbfc",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "892e75aa-8805-47d6-8d9c-5d66137d63a5",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "id": "8a62c8ee-4e96-49e1-b0fd-7810373ee1b3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 0, 'Height')"
      ]
     },
     "execution_count": 105,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x720 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plotting the height sample\n",
    "plt.hist(height, bins=20)\n",
    "plt.title('Normal Distribution')\n",
    "plt.xlabel('Height')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "86e65198-8da7-43bc-ac5b-d74b477111e5",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f2feb546-1e53-40b2-94d1-d9b8d4ad0119",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "52788a0f-804c-4eec-abf8-976c2268f0fb",
   "metadata": {},
   "source": [
    "As we can see from the histogram above, it creates a bell shape. 50% of values are either side of the 1.75 mean. And the majority of those sampled fall within the average height, 1.75m."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ed7443e3-dbbc-410c-98d2-e3ce628115e0",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "317c291a-3f02-4dd0-918a-aad70600ce6f",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2715afdf-1d1f-4170-8802-a896c02ab1f9",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "23b5ca99-1bc6-405d-81a5-b24c65da9c49",
   "metadata": {
    "tags": []
   },
   "source": [
    "***\n",
    "\n",
    "## Logistic Distribution\n",
    "\n",
    "Logistic distribution is a continuous probability distribution. It's cumulative distribution function is the logistic function which appears in logistic regression and feedforward neural networks. It is used for modeling growth, and also used in logistic regression. It is a symmetrical distribution, unimodal (it has one peak) and is similar in shape to the normal distribution. Logistic distribution is used for sports and recently finanical modelling [26]. \n",
    "\n",
    "Some argue that the logistic distribution isn't the right choice for modeling lifetime data as the left-hand limit of the distribution extends into negative infinity. This could result in modeling negative times-to-failure. However, provided that the distribution in  has a relatively high mean and a relatively small location parameter, the issue of negative failure times should not be a problem [27].\n",
    "\n",
    "![img](images/logistic_distribution.jpg)\n",
    "\n",
    "Parameters of Logistic distribution:\n",
    "\n",
    "- The location parameter (μ) tells you where it’s centered on the x-axis.\n",
    "- The scale parameter (σ) tells you what the spread is. In the above equation, s is a scale parameter proportional to the standard deviation [28].\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7b5b75a7-a83f-4a40-8553-1872530237fc",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1ed8b63d-4015-4149-b2c4-3418762aaac0",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "id": "58c4d769-02e5-482b-aba7-e5f26d80b9da",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Generating a array of random numbers in the logistic distribution\n",
    "x = rng.logistic(20, 20 ,size=10000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "id": "2e6d1702-4aa4-4671-a93e-c24cc38be938",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 17.18011301,  68.50132817,  51.02639207, ...,  21.0692812 ,\n",
       "       -23.920347  ,  79.97691251])"
      ]
     },
     "execution_count": 107,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b766774c-909c-421d-89e1-0153b9d18aae",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bad4eb97-fa3f-4194-a55d-eb86e0f7522b",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "id": "a451f7e8-3034-43db-ab82-439c951f39d1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Logistic Distribution')"
      ]
     },
     "execution_count": 108,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x720 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# PLotting the array\n",
    "plt.hist(x, bins =20)\n",
    "plt.title('Logistic Distribution')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "43cd7ed9-060b-45ac-bc4c-28955cd4c62c",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8b2e6af4-abf2-4fe5-b6ad-9b76af188fa2",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8b93665b-aa52-49b5-a50a-d7883a98ce18",
   "metadata": {},
   "source": [
    "<br>\n",
    "Logistic distributiont is symmetrical, unimodal i.e. has one peak and is very similar in shape to the the normal distribution. The major distingushing characteristic between the two distributions is that, the logistic distribution has wider tails than a normal distribution so it is more consistent with the underlying data and provides better insight into the likelihood of extreme events [29]. Let's visualise the difference below. \n",
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b21bd908-6dd2-4faf-8f0b-80a449418bc5",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1576fac8-9905-413c-88e8-5a080e9c9164",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "id": "51ad7056-3aac-40c3-98ec-fe910115cf04",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Normal Distribution')"
      ]
     },
     "execution_count": 109,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Following two pieces of code are taken from W3 schools\n",
    "# Normal distribution \n",
    "\n",
    "a = sns.displot(rng.normal(scale=2, size=1000), label='normal', kde=True)\n",
    "plt.title('Normal Distribution')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "84b0b8d4-3b80-41b9-9e5d-8a07c872d09d",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d91fb0cc-caa2-4930-8d3c-104f8e0d4a10",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "id": "97c18098-4859-4aa9-b7d7-1417c821b2c0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Logistic Distribution')"
      ]
     },
     "execution_count": 110,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Logistic distribution\n",
    "b = sns.displot(rng.logistic(size=1000), label='logistic', kde= True)\n",
    "plt.title('Logistic Distribution')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2a2e1a9c-28af-4d96-a579-536d3f738030",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dbdc19a6-5d35-4c75-818e-d83407cc6669",
   "metadata": {},
   "source": [
    "As we can see from the above, both distributions are almost identical, but the logistic distribution has more area under the tails. This means that it can show more possibility of an occurence of an events further away from mean.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cdedd626-ed38-4f12-be99-b8a296c20a2f",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2fe3e699-927d-4653-820d-5ab7882760bf",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d606f964-69ff-42fc-bd56-6c5a4985c119",
   "metadata": {
    "tags": []
   },
   "source": [
    "***\n",
    "\n",
    "## Seed\n",
    "\n",
    "***"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6bcd3380-c1da-409f-8dd5-118ece28c29a",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2fede291-a6a7-484e-bf93-d9bc10278f01",
   "metadata": {},
   "source": [
    "The methods commonly used to generate random numbers is not a truly random processes. True randomness is hard to achieve, so we employ pseudorandomness instead. This numbers are refered to as psuedo-random numbers. While pseudorandom numbers appear to be genuinely random, they are  generated by a deterministic process i.e. a system in which no randomness is involved. This is because computers are deterministic and follow instructions to the low. If you give a certain input, it will percisely follow instruction. Hence the use of psuedo-random number gerenators [30]. \n",
    "\n",
    "A random number generator is an algorithm that, based on an initial seed, produces a sequence of seemlying random numbers[31].\n",
    "\n",
    "The generated sequence is not truly random, because it is determined by the initial seed value. The numbers they produce have properties that approximate the properties of random numbers. Although, these numbers are decided by an algorithm, for the most part there is no pattern in their choosing, hence making them appear truly random.\n",
    "\n",
    "Psuedo-Random numbers (PRNG) and random  bit  generators(RGB) are fundamental in many areas. In  stochastic simulation, PRNGs  are employed to copy the behavior of a random  variable  with a  given probability distribution [32]. PRNGs are algorithms that use mathematical formulae or simply precalculated tables to produce sequences of numbers that appear random. \n",
    "\n",
    "Characteristics of PRNG:\n",
    "\n",
    "- Efficient: PRNG can produce many numbers in a short time and is advantageous for applications that need many numbers\n",
    "- Deterministic: A given sequence of numbers can be reproduced at a later date if the starting point in the sequence is known.Determinism is handy if you need to replay the same sequence of numbers again at a later stage.\n",
    "- Periodic: PRNGs are periodic, which means that the sequence will eventually repeat itself. While periodicity is hardly ever a desirable characteristic, modern PRNGs have a period that is so long that it can be ignored for most practical purposes [30].\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e4f8f313-0ddd-4ac1-9fe6-5a64920a38a1",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2915b5c9-6f75-4128-a72e-a655cbebaf82",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6ce612b0-3245-4846-9257-abecf4b9ab6c",
   "metadata": {
    "tags": []
   },
   "source": [
    "#### (PCG-64) Seed Examples\n",
    "[PCG-64 Seed Documentation]('https://numpy.org/doc/stable/reference/random/generator.html')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "id": "45338dda-b644-4739-9bd7-3fa6763cb52c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Generator(PCG64) at 0x1DC170B8D60"
      ]
     },
     "execution_count": 111,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Randomly setting the seed\n",
    "rng = np.random.default_rng(seed=0)\n",
    "rng"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9c0f717d-8d93-482b-92ba-65fa4e0dd3a9",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "id": "174ae05c-84c7-420f-9589-d33b32a6ad8b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([4, 3, 2, 1, 1, 0, 0, 0, 0, 4], dtype=int64)"
      ]
     },
     "execution_count": 112,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Randomly generating 10 numbers between 0 - 4\n",
    "rng.integers(0,5,10)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "62d349cf-023e-4aed-be60-7d9472e16623",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "id": "e51a003f-c9cb-4d9a-8525-d2e1d3e5c5c9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([3, 4, 2, 3, 4, 3, 3, 2, 2, 4], dtype=int64)"
      ]
     },
     "execution_count": 113,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Repeating the above generation with a randomly set seed\n",
    "rng.integers(0,5,10)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1ee1b0d4-5730-4997-80d4-278390451643",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0a8ce784-5aa5-43ab-8610-539bd05f03d4",
   "metadata": {},
   "source": [
    "As we can see from the above there is seemingly no pattern in both arrays. This is a result of leaving the seed blank. \n",
    "\n",
    "But pseudo-random numbers can be replicated exactly. PRNG's are repeatable because of their deterministic nature. This is often used when testing or sharing code [30].\n",
    "\n",
    "np.random.default_rng allows us to determine the vaule of the seed which generates these pseudo-random numbers. Let's reapeat the above code but, this time assigning a vaule to the seed"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "893e8c86-9720-45e3-835b-bf05ac5ad288",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "id": "f21a4eb7-9213-45cc-9190-770622a7df46",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Setting the seed as 5\n",
    "rng = np.random.default_rng(seed=5)\n",
    "\n",
    "# Randomly generating 10 numbers between 0 - 4\n",
    "a = rng.integers(0,5,10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "id": "65b6a155-6ea0-4365-8279-ca3b6974c7e8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([3, 4, 0, 4, 2, 2, 3, 1, 4, 0], dtype=int64)"
      ]
     },
     "execution_count": 115,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3aef09a4-5ca1-4be5-9aba-af5d5503b7ff",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "id": "26903d8d-ca3a-4512-ae93-0f5b7a36e0cf",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Repeating the above generations \n",
    "rng = np.random.default_rng(seed=5)\n",
    "b = rng.integers(0,5,10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "id": "4e23df4b-608f-4af2-ba5d-934804bac826",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([3, 4, 0, 4, 2, 2, 3, 1, 4, 0], dtype=int64)"
      ]
     },
     "execution_count": 117,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "662cb247-e52d-45cb-b3d4-c848712d86aa",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2f9ebcd3-504d-46e7-9c08-0bffb728b622",
   "metadata": {},
   "source": [
    "By specifying a seed in the above examples, it enables us to replicate the results."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "49c23b5d-ffc7-4e75-adc3-73afa802ff4c",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "22bec5e3-6979-4361-aeeb-b40aa9333b42",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "990d56a0-bbf5-4604-8a64-bd73d067ecfb",
   "metadata": {
    "tags": []
   },
   "source": [
    "#### Legacy (MT19937) Seed Examples\n",
    "\n",
    "[MT19937 Seed Documentation]('https://numpy.org/doc/stable/reference/random/generator.html')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c2b79052-7b3e-4eb2-9502-a9471cac6ba9",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "id": "509d0752-a286-459d-a92c-7c9c2be8cd1e",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Randomly setting the seed\n",
    "np.random.seed()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "id": "8bb840ac-3dde-4aea-988d-9f0f7185bf79",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 3, 0, 3, 4, 4, 2, 0, 3, 3])"
      ]
     },
     "execution_count": 119,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Randomly generating 10 numbers between 0 - 4\n",
    "np.random.randint(0,5,10)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "babb9145-9673-485e-b257-ed02a48065f9",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "id": "e2aa342c-70a8-4f46-bf8e-fd47098ac187",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 0, 0, 3, 4, 2, 2, 1, 1, 4])"
      ]
     },
     "execution_count": 120,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Repeating the above generation\n",
    "# Randomly setting the seed\n",
    "np.random.seed()\n",
    "\n",
    "# Randomly generating 10 numbers between 0 - 4\n",
    "np.random.randint(0,5,10)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6af85214-d53b-41a0-b231-02392523f251",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "id": "7869ae36-c7cf-461d-a910-147a0cae73ea",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([3, 0, 1, 0, 4, 3, 0, 0, 4, 1])"
      ]
     },
     "execution_count": 121,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Setting the seed as 5\n",
    "np.random.seed(5)\n",
    "# \"Randomly\" generating 10 numbers between 0 - 4\n",
    "np.random.randint(0,5,10)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4bb46696-aeff-4da0-8a93-8fd47a35553f",
   "metadata": {},
   "source": [
    "  <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "id": "690a0376-74a7-4e89-bd3a-2c11ff48a77d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([3, 0, 1, 0, 4, 3, 0, 0, 4, 1])"
      ]
     },
     "execution_count": 122,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Repeating the above generation\n",
    "np.random.seed(5)\n",
    "np.random.randint(0,5,10)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "49aa6ef1-7092-4c78-95ba-2fe726e92dee",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "eaaa6d7a-0489-4a61-ad36-b857c2bc5dcd",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "131b6743-7cc6-451f-a50d-f9cff59d0a0d",
   "metadata": {
    "tags": []
   },
   "source": [
    "***\n",
    "\n",
    "## References\n",
    "\n",
    "***"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b20dd204-c277-4e74-9666-3d45aebccc5e",
   "metadata": {
    "tags": []
   },
   "source": [
    "[1].\tWhat is NumPy? — NumPy v1.21 Manual [Internet]. Numpy.org. [cited 2021 Oct 19]. Available from: https://numpy.org/doc/stable/user/whatisnumpy.html\n",
    " \t \n",
    "[2]\tPajankar A. Introduction to NumPy. In: Raspberry Pi Supercomputing and Scientific Programming. Berkeley, CA: Apress; 2017. p. 109–28.\n",
    " \t \n",
    "[3].\tBisong E. NumPy. In: Building Machine Learning and Deep Learning Models on Google Cloud Platform. Berkeley, CA: Apress; 2019. p. 91–113.\n",
    " \t \n",
    "[4]\tPython R. NumPy tutorial: Your first steps into data science in Python [Internet]. Realpython.com. Real Python; 2021 [cited 2021 Oct 19]. Available from: https://realpython.com/numpy-tutorial/\n",
    " \t \n",
    "[5]\tWikipedia contributors. Applications of randomness [Internet]. Wikipedia, The Free Encyclopedia. 2021 [cited 2021 Oct 19]. Available from: https://en.wikipedia.org/w/index.php?title=Applications_of_randomness&oldid=1016893073\n",
    " \t \n",
    "[6]\tRandom Numbers in NumPy [Internet]. W3schools.com. [cited 2021 Oct 19]. Available from: https://www.w3schools.com/python/numpy/numpy_random.asp\n",
    " \t \n",
    "[7]\tMcKinney W. Python for Data Analysis: Data wrangling with pandas, NumPy, and IPython. Sebastopol, CA: O’Reilly Media; 2012.\n",
    " \t \n",
    "[8]\tDifferences between numpy.random.rand vs numpy.random.randn in Python [Internet]. Stackoverflow.com. [cited 2021 Oct 19]. Available from: https://stackoverflow.com/questions/47240308/differences-between-numpy-random-rand-vs-numpy-random-randn-in-python\n",
    " \t \n",
    "[9]\tLohana A. NumPy Random Choice: How to use it? [Internet]. Codeleaks.io. 2021 [cited 2021 Oct 19]. Available from: https://www.codeleaks.io/numpy-random-choice-python/\n",
    " \t \n",
    "[10]\tWikipedia contributors. Permutation [Internet]. Wikipedia, The Free Encyclopedia. 2021 [cited 2021 Oct 19]. Available from: https://en.wikipedia.org/w/index.php?title=Permutation&oldid=1050569920\n",
    " \t \n",
    "[11]\tRanganathan P, Gogtay NJ. An Introduction to Statistics - Data Types, distributions and Summarizing Data. Indian J Crit Care Med. 2019;23(Suppl 2):S169–70.\n",
    " \t \n",
    "[12]\tM SL, T TM, F HS, editors. Poisson Distribution. Betascript Publishing; 2013.\n",
    " \t \n",
    "[13]\tKoehrsen W. The Poisson distribution and Poisson process explained [Internet]. Towards Data Science. 2019 [cited 2021 Oct 19]. Available from: https://towardsdatascience.com/the-poisson-distribution-and-poisson-process-explained-4e2cb17d459\n",
    " \t \n",
    "[14]\tDiving into python’s numpy random Poisson [Internet]. Pythonpool.com. 2021 [cited 2021 Oct 19]. Available from: https://www.pythonpool.com/numpy-random-poisson/\n",
    " \t \n",
    "[15]\tBinomial Distribution: Formula, What it is, How to use it [Internet]. Statisticshowto.com. 2018 [cited 2021 Oct 19]. Available from: https://www.statisticshowto.com/probability-and-statistics/binomial-theorem/binomial-distribution-formula/\n",
    " \t \n",
    "[16]\tPython - Binomial Distribution - GeeksforGeeks [Internet]. Geeksforgeeks.org. 2020 [cited 2021 Oct 19]. Available from: https://www.geeksforgeeks.org/python-binomial-distribution/\n",
    " \t \n",
    "[17]\tHow to Graph the Binomial Distribution - dummies [Internet]. Dummies.com. 2016 [cited 2021 Oct 19]. Available from: https://www.dummies.com/education/math/business-statistics/how-to-graph-the-binomial-distribution/\n",
    " \t \n",
    "[18]\tSurbhi S. Difference between binomial and Poisson distribution [Internet]. Keydifferences.com. 2016 [cited 2021 Oct 19]. Available from: https://keydifferences.com/difference-between-binomial-and-poisson-distribution.html\n",
    " \t \n",
    "[19]\tChen J. Uniform Distribution [Internet]. Investopedia.com. 2021 [cited 2021 Oct 19]. Available from: https://www.investopedia.com/terms/u/uniform-distribution.asp\n",
    " \t \n",
    "[20]\tParametric continuous distributions [Internet]. Analytica.com. [cited 2021 Oct 19]. Available from: http://wiki.analytica.com/Parametric_continuous_distributions\n",
    " \t \n",
    "[21]\tUniform distribution [Internet]. Corporatefinanceinstitute.com. 2020 [cited 2021 Oct 19]. Available from: https://corporatefinanceinstitute.com/resources/knowledge/other/uniform-distribution/\n",
    " \t \n",
    "[22]\tnumpy.random.uniform — NumPy v1.16 Manual [Internet]. Numpy.org. [cited 2021 Oct 19]. Available from: https://numpy.org/doc/1.16/reference/generated/numpy.random.uniform.html\n",
    " \t \n",
    "[23]\tWikipedia contributors. Normal distribution [Internet]. Wikipedia, The Free Encyclopedia. 2021 [cited 2021 Oct 19]. Available from: https://en.wikipedia.org/w/index.php?title=Normal_distribution&oldid=1049556144\n",
    " \t \n",
    "[24]\tChen J. Normal Distribution [Internet]. Investopedia.com. 2021 [cited 2021 Oct 19]. Available from: https://www.investopedia.com/terms/n/normaldistribution.asp\n",
    " \t \n",
    "[25]\tPatel JK, B. C. Handbook of the normal distribution, second edition. 2nd ed. Boca Raton, FL: CRC Press; 1996.\n",
    " \t \n",
    "[26]\tKissell R, Poserina J. Advanced Math and Statistics. In: Optimal Sports Math, Statistics, and Fantasy. Elsevier; 2017. p. 103–35.\n",
    " \t \n",
    "[27]\tReliaSoft Publishing. Overview of the Gumbel, Logistic, Loglogistic and Gumbel distributions [Internet]. Weibull.com. [cited 2021 Oct 19]. Available from: https://www.weibull.com/hotwire/issue56/relbasics56.htm\n",
    " \t \n",
    "[28]\tStephanie. Logistic distribution [Internet]. Statisticshowto.com. 2016 [cited 2021 Oct 19]. Available from: https://www.statisticshowto.com/logistic-distribution/\n",
    " \t \n",
    "[29]\tLogistic Distribution [Internet]. W3schools.com. [cited 2021 Oct 19]. Available from: https://www.w3schools.com/python/numpy/numpy_random_logistic.asp\n",
    " \t \n",
    "[30]\tEbner J. NumPy random seed, explained [Internet]. Sharpsightlabs.com. 2019 [cited 2021 Oct 19]. Available from: https://www.sharpsightlabs.com/blog/numpy-random-seed/\n",
    " \t \n",
    "[31]\tPseudo random number generator (PRNG) [Internet]. Geeksforgeeks.org. 2017 [cited 2021 Oct 19]. Available from: https://www.geeksforgeeks.org/pseudo-random-number-generator-prng/\n",
    " \t \n",
    "[32]\tVajargah BF, Asghari R. A novel pseudo-random number generator for cryptographic applications. Indian J Sci Technol. 2016;9(6):1–5.\n",
    " \t  \n",
    "[33]\tPermuted Congruential Generator (64-bit, PCG64) — NumPy v1.21 Manual [Internet]. Numpy.org. [cited 2021 Oct 19]. Available from: https://numpy.org/doc/stable/reference/random/bit_generators/pcg64.html\n",
    " \t \n",
    "[34]\tWikipedia contributors. Permuted congruential generator [Internet]. Wikipedia, The Free Encyclopedia. 2021 [cited 2021 Oct 19]. Available from: https://en.wikipedia.org/w/index.php?title=Permuted_congruential_generator&oldid=999160939\n",
    " \t \n",
    "[35]\tShema M. Seven deadliest web application attacks [Internet]. Syngress Publishing; 2010 [cited 2021 Oct 20]. Available from: https://www.sciencedirect.com/book/9781597495431/seven-deadliest-web-application-attacks\n",
    " \t \n",
    "[36]\tWikipedia contributors. Mersenne Twister [Internet]. Wikipedia, The Free Encyclopedia. 2021 [cited 2021 Oct 20]. Available from: https://en.wikipedia.org/w/index.php?title=Mersenne_Twister&oldid=1049043226\n",
    " \t \n",
    "[37]\tRandom sampling (numpy.random) — NumPy v1.18 Manual [Internet]. Numpy.org. [cited 2021 Oct 20]. Available from: https://numpy.org/doc/1.18/reference/random/index.html\n",
    " \t \n",
    "[38]\tRandom Generator — NumPy v1.18 Manual [Internet]. Numpy.org. [cited 2021 Oct 20]. Available from: https://numpy.org/doc/1.18/reference/random/generator.html\n",
    " \t \n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "17ba469b-a971-468d-8e7d-b0a768e3f1c7",
   "metadata": {},
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    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "73436d2f-18c0-43cc-b4c3-1b1b12190142",
   "metadata": {},
   "source": [
    "#### Images\n",
    "\n",
    "- [Numpy Logo](https://www.google.com/imgres?imgurl=https%3A%2F%2Ftwilio-cms-prod.s3.amazonaws.com%2Fimages%2Fjupyter_python_numpy.width-808.png&imgrefurl=https%3A%2F%2Fwww.twilio.com%2Fblog%2F2017%2F10%2Fbasic-statistics-python-numpy-jupyter-notebook.html&tbnid=J-fIWc88mja7tM&vet=12ahUKEwjhxO29p9fzAhUSWhUIHVGKDuQQMygJegUIARDCAQ..i&docid=VNux2Dk9hmRbWM&w=800&h=200&itg=1&q=numpy%20logo&ved=2ahUKEwjhxO29p9fzAhUSWhUIHVGKDuQQMygJegUIARDCAQ)\n",
    " \t\n",
    "- [PCG vs MT19937 ](https://www.pcg-random.org/index.htmlhttps://www.pcg-random.org/index.html)\n",
    " \t\n",
    "- [Permutations](https://www.w3resource.com/w3r_images/csharp-recursion-image-exercise-11.png)\n",
    " \t\n",
    "- [Matrix](https://3xedigital.com/wp-content/uploads/2016/03/Red-Pill-Vs-Blue-Pill.png)\n",
    " \t\n",
    "- [Coin](https://www.google.com/url?sa=i&url=https%3A%2F%2Ffree3d.com%2F3d-model%2F2-euro-coin-italy-4150.html&psig=AOvVaw3914ZDmMJJRuMCZL_bZdBl&ust=1634761631572000&source=images&cd=vfe&ved=0CAsQjRxqFwoTCPjQmeen1_MCFQAAAAAdAAAAABAL)\n",
    " \t\n",
    "- [Dice](https://static.wikia.nocookie.net/game-of-dice/images/c/cb/White_Dice.png/revision/latest?cb=20160113233423)\n",
    " \t\n",
    "- [Std Rule](https://blogs.sas.com/content/iml/files/2019/07/rule6895.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "934221fa-3442-40ef-b013-d8e8cfc2de97",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bf53021f-fda0-4914-bb49-04b0f25fbe02",
   "metadata": {},
   "source": [
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8171bf7a-abee-49cf-aaeb-88129061f3ae",
   "metadata": {},
   "source": [
    "***\n",
    "\n",
    "# End\n",
    "\n",
    "***"
   ]
  }
 ],
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