{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": [
     "s1",
     "content",
     "l1"
    ]
   },
   "source": [
    "# Discrete Distributions or Probability Mass Functions (PMFs)\n",
    "\n",
    "Probability distribution is a function that generates the probabilities of occurrence of all possible outcomes in an experiment. Consider an experiement of rolling of the die. If the random variable X is used to denote the outcome of the die roll, then the probability distribution of X would take the value $\\frac{1}{6}$ for $X \\to \\{1, 2, 3, 4, 5, 6\\}$.\n",
    "\n",
    "\n",
    "## Bernoulli Distribution\n",
    "\n",
    "Bernoulli distribution is the probability distribution of a random variable that takes a boolean value such as a 1 or 0. The Probability Mass Function of Bernoulli distribution is mathematically defined as:\n",
    "\n",
    "For a possible outcome k, \n",
    "\n",
    "$$f(k,p) = {p^k}({{1-p})^{(1-k)}} $$\n",
    "\n",
    "\n",
    "where p is the probability of outcome 1 and 1-p is the probability of outcome 0.<br>\n",
    "\n",
    "### Examples\n",
    "\n",
    "Result of a coin toss, if patient has disease or not, any experiment with outcome of success or failure.\n",
    "\n",
    "\n",
    "<img src=\"../images/bernoulli.png\" style=\"width: 700px;\">\n",
    "\n",
    "\n",
    "Let us now plot a Bernoulli distribution\n",
    "\n",
    "### Exercise\n",
    "\n",
    "* Increase number of samples to 1000 and use bernoulli.stats() function to determine the mean of the distribution.\n",
    "* Assign it to variable, mu and print it out."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "tags": [
     "s1",
     "ce",
     "l1"
    ]
   },
   "outputs": [
    {
     "data": {
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7qfX8DeA3M/PGIGrthk4fLxERnwT+NzPf2+cSu66D3/FPAI9S+8LBfwJvz8yrg6i1Wzro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\nQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x2090ee12240>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "from scipy.stats import bernoulli\n",
    "import numpy as np\n",
    "import seaborn as sns\n",
    "\n",
    "p = 0.3\n",
    "x = bernoulli.rvs(p, size=1000)\n",
    "sns.distplot(x, kde=False);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": [
     "s1",
     "l1",
     "hint"
    ]
   },
   "source": [
    "Use stats function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "tags": [
     "s1",
     "l1",
     "ans"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.3\n"
     ]
    }
   ],
   "source": [
    "mu = bernoulli.stats(p, moments='m')\n",
    "print(mu)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "tags": [
     "s1",
     "hid",
     "l1"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "continue\n"
     ]
    }
   ],
   "source": [
    "ref_tmp_var = False\n",
    "\n",
    "try:\n",
    "    if mu == 0.3:\n",
    "        ref_assert_var = True\n",
    "        ref_tmp_var = True\n",
    "    else:\n",
    "        ref_assert_var = False\n",
    "        print('Please follow the instructions given and use the same variables provided in the instructions.')\n",
    "except Exception:\n",
    "    print('Please follow the instructions given and use the same variables provided in the instructions.')\n",
    "\n",
    "assert ref_tmp_var"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": [
     "l2",
     "content",
     "s2"
    ]
   },
   "source": [
    "\n",
    "<br/><br/><br/>\n",
    "## Binomial Distribution\n",
    "\n",
    "The number of successes, $x$ in a fixed number of $n$ independent Bernoulli trials with probability of success $p$ or $\\theta$, follows a binomial distribution. $n$ and $p$ are always fixed in a binomial distribution.\n",
    "\n",
    "\n",
    "<img src=\"../images/binomial.png\" style=\"width: 700px;\">\n",
    "\n",
    "\n",
    "### Examples\n",
    "\n",
    "Number of heads after tossing a coin 100 times. Number of defective bulbs after inspecting 1000 bulbs.\n",
    "\n",
    "\n",
    "### Exercise\n",
    "\n",
    "Let us generate numbers that conform to a binomial distribution. Given N = 40, p=0.5 and the numbers distributed according to the binomial pmf, compute the mean and variance and assign it to variables, mu and var."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "tags": [
     "l2",
     "ce",
     "s2"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x2090f0c0390>"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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hyl3VfTftHRyK7AtdziRHxLdhroLdZiPPmT1r8njjTDPPvlbPypJ0cjMTI+o8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\n5IUQwsQk5IUQwsT+P4MWDw0aCyODAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x2090eee7400>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "import numpy as np\n",
    "import seaborn as sns\n",
    "\n",
    "N = 40\n",
    "p = 0.5\n",
    "binomial_x = np.random.binomial(N, p, 10)\n",
    "sns.distplot(binomial_x)\n",
    "\n",
    "# Compute mean and variance"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": [
     "l2",
     "s2",
     "hint"
    ]
   },
   "source": [
    "Use formula for mu & var."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "tags": [
     "l2",
     "s2",
     "ans"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "20.0 10.0\n"
     ]
    }
   ],
   "source": [
    "mu = N*p\n",
    "var = N*p*(1-p)\n",
    "print(mu, var)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "tags": [
     "l2",
     "hid",
     "s2"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "continue\n"
     ]
    }
   ],
   "source": [
    "ref_tmp_var = False\n",
    "\n",
    "try:\n",
    "    if (mu==20.0) and (var==10.0):\n",
    "        ref_assert_var = True\n",
    "        ref_tmp_var = True\n",
    "    else:\n",
    "        ref_assert_var = False\n",
    "        print('Please follow the instructions given and use the same variables provided in the instructions.')\n",
    "except Exception:\n",
    "    print('Please follow the instructions given and use the same variables provided in the instructions.')\n",
    "\n",
    "assert ref_tmp_var"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": [
     "l3",
     "s3",
     "content"
    ]
   },
   "source": [
    "### Coin toss experiment\n",
    "\n",
    "Let us consider an experiment of tossing a biased coin, in which p is the probability of 'heads' and n is the number of tosses. An experiment is defined as a series of trials. In this case, an experiment will consist of n trials/tosses. We will repeat the experiment 1000 times to simulate the binomial distribution. \n",
    "\n",
    "We must check if the experiment follows the assumptions of binomial distribution:\n",
    "\n",
    "1. Each trial is an independent of the other, meaning outcome of one toss does not affect that of others.\n",
    "2. There are only two possible outcomes for each trial, heads or tails.\n",
    "3. The probability of 'success', p is the same across the n trials.\n",
    "3. The number of trials, n is fixed.\n",
    "\n",
    "Consider n tosses of a coin. Since the tosses are independent, we can calculate the probability of an outcome by multiplying the individual probabilities in each toss. Probability of 'heads' is p and that of 'tails' is 1-p. Outcome of one experiment with n tosses, has k heads and n-k tails and has probability **p<sup>k</sup>(1-p)<sup>n-k</sup>**.\n",
    "There are ${n} \\choose {k}$ number of distinct n-toss sequences that contain k heads. This forms the binomial coefficient.\n",
    "\n",
    "Binomial probability mass function (pmf) = ${n} \\choose {k}$ **p<sup>k</sup>(1-p)<sup>n-k</sup>**\n",
    "\n",
    "We can now go ahead and simulate our experiment.\n",
    "\n",
    "### Exercise\n",
    "\n",
    "Assume the experiment consists of n=30 coin tosses, the probability of getting heads, p=0.6 and this experiment is repeated 1000 times. Generate a random distribution, k which holds number of heads of each repetition of the experiment. Calculate pmf for each value of k and plot a graph with k and corresponding pmf values. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": true,
    "tags": [
     "l3",
     "s3",
     "ce"
    ]
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import random\n",
    "from scipy.stats import binom\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "def toss(p,n):\n",
    "    heads = 0\n",
    "    for i in range(n):\n",
    "        if random.random() < p:\n",
    "            heads += 1\n",
    "    return heads #This gives the number of heads in a single experiment of n trials\n",
    "\n",
    "size = range(1,1001)\n",
    "n,p = 30,0.6\n",
    "\n",
    "k = [] #Holds the number of heads from each experiment\n",
    "pmf= [] #Holds the binomial pmf of each experiment\n",
    "\n",
    "random.seed(12345)\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": [
     "l3",
     "s3",
     "hint"
    ]
   },
   "source": [
    "Use toss function to create list of k values, sort them and for each k, calculate pmf using binom.pmf function. Plot k and pmf to get the distribution plot. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "tags": [
     "l3",
     "s3",
     "ans"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x2090eeb7198>]"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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vywBJQKvX906llE1rPTDCtnYgWSn1WaBBa71JKfUNX4LY7XHYTLAAhMMRfHeH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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x2090d943f98>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for i in size:\n",
    "    k.append(toss(p,n))\n",
    "\n",
    "k.sort()\n",
    "\n",
    "for i in k:\n",
    "    pk = binom.pmf(i,n,p)\n",
    "    pmf.append(pk)\n",
    "\n",
    "plt.plot(k,pmf) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "tags": [
     "l3",
     "s3",
     "hid"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "continue\n"
     ]
    }
   ],
   "source": [
    "ref_tmp_var = False\n",
    "\n",
    "try:\n",
    "    head = k[:10]\n",
    "    prob = pmf[:10]\n",
    "    ks = [9, 9, 10, 10, 11, 11, 11, 11, 12, 12]\n",
    "    pmfs = [round(x,4) for x in [0.00063412401653505015, 0.00063412401653505015, 0.0019974906520854141, 0.0019974906520854141, 0.005447701778414739, 0.005447701778414739, 0.005447701778414739, 0.005447701778414739, 0.012938291723735, 0.012938291723735]]\n",
    "    if ks == head and pmfs == [round(a,4) for a in prob]:\n",
    "        ref_assert_var = True\n",
    "        ref_tmp_var = True\n",
    "    else:\n",
    "        ref_assert_var = False\n",
    "        print('Please follow the instructions given and use the same variables provided in the instructions. ')\n",
    "except Exception:\n",
    "    print('Please follow the instructions given and use the same variables provided in the instructions. ')\n",
    "\n",
    "assert ref_tmp_var"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": [
     "l4",
     "s4",
     "content"
    ]
   },
   "source": [
    "## Cumulative Distributions\n",
    "\n",
    "Any random variable, whether discrete or continous, may be defined by its cumulative distribution function. CDF of a random variable $X$ is denoted by $F_X(x)$ and gives the cumulative probability upto the value $x$, which is $P(X \\leq x)$ for all $x$. In other words, CDF is the proportion of the population having values less than or equal to $x$. \n",
    "\n",
    "\n",
    "### Binomial distribution\n",
    "\n",
    "Consider a binomial random variable $X \\sim Binomial(n,\\mathrm p)$.<br><br>\n",
    "Binomial probability mass function, $p_X(x)$ = ${n} \\choose {k}$ p<sup>k</sup>(1-p)<sup>n-k</sup>.\n",
    "$\\quad$We can derive the cumulative distribution function as,<br><br> \n",
    "$F_X(x) \\quad=\\quad P(X \\leq x) \\quad=\\quad \\sum\\limits_{k=0}^{x} p_X(k)$\n",
    "$\\quad=\\quad \\sum\\limits_{k=0}^{x}$ ${n} \\choose {k}$ p<sup>k</sup>(1-p)<sup>n-k</sup>\n",
    "\n",
    "Coin toss is a perfect example of a binomial random variable. Consider tossing a biased coin, in which p is the probability of 'heads'(success), and n is the number of tosses. Let n=10, p=0.8. To find probability of 2 or less successes,<br>\n",
    "$F_X(x=2) = P_X(X \\leq 2) = p_X(0) + p_X(1) + p_X(2)$\n",
    "<br>\n",
    "\n",
    "* Given the cdf plot, calculate the probability of occurence of number of heads greater than 6 and assign it to p_x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "tags": [
     "l4",
     "s4",
     "ce"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Probability of 7 successes = 0.201326592\n",
      "Probability of 7 successes or less = 0.3222004736\n"
     ]
    },
    {
     "data": {
      "image/png": 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XVrVqkDvnMsA9Sx4+UGK5h6pUk0hFCld16mCntBpdECRNYyyRJAQMqUcuLUZBLk1jPDHP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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x2090edd39b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "from scipy.stats import binom\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Plot cdf of a binomial random variable\n",
    "x = np.linspace(0,20,100)\n",
    "cdf = binom.cdf(x, n=50, p=0.2)\n",
    "plt.plot(x,cdf)\n",
    "\n",
    "# Read the following example\n",
    "\n",
    "# Find probability of k = 7 successes, with n=10 and p=0.8\n",
    "pmf7 = binom.pmf(k=7, n=10, p=0.8)\n",
    "print(\"Probability of 7 successes =\", pmf7)\n",
    "\n",
    "# Read the following example\n",
    "\n",
    "# Find probability of k = 7 successes or less, with n=10 and p=0.8\n",
    "cdf7 = binom.cdf(k=7, n=10, p=0.8)\n",
    "print(\"Probability of 7 successes or less =\",cdf7)\n",
    "\n",
    "# Find probability of k > 6 successes, with n=10 and p=0.8\n",
    "# p_x ="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": [
     "s4",
     "l4",
     "hint"
    ]
   },
   "source": [
    "Read the example solution code given for k=7 and k<=7 successes and compute probability of occurence of greater than 6 heads."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "tags": [
     "s4",
     "l4",
     "ans"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Probability of more than 6 successes = 0.8791261184\n"
     ]
    }
   ],
   "source": [
    "p_x = 1 - binom.cdf(k=6, n=10, p=0.8)\n",
    "print(\"Probability of more than 6 successes =\", p_x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "tags": [
     "s4",
     "hid",
     "l4"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "continue\n"
     ]
    }
   ],
   "source": [
    "ref_tmp_var = False\n",
    "\n",
    "try:\n",
    "    if abs(p_x - 0.8791) < 0.01:\n",
    "        ref_assert_var = True\n",
    "        ref_tmp_var = True\n",
    "    else:\n",
    "        ref_assert_var = False\n",
    "        print('Please follow the instructions given and use the same variables provided in the instructions. ')\n",
    "except Exception:\n",
    "    print('Please follow the instructions given and use the same variables provided in the instructions. ')\n",
    "\n",
    "assert ref_tmp_var"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": [
     "l5",
     "content",
     "s5"
    ]
   },
   "source": [
    "\n",
    "<br/><br/><br/>\n",
    "## Geometric Distribution\n",
    "\n",
    "The number of trials until the first success in a series of Bernoulli trials follows a geometric distribution.\n",
    "\n",
    "\n",
    "<img src=\"../images/geometric.png\" style=\"width: 700px;\">\n",
    "\n",
    "\n",
    "### Examples\n",
    "Number of coin tosses before the first time coin shows head. Number of bulbs tested before the first defective bulb is found. Number of patients screened in hospital before first positive case of a disease.\n",
    "\n",
    "### Exercise\n",
    "\n",
    "Let us generate numbers that conform to geometric distribution. Compute mean and variance for the exponential distribution and assign to variables mu and var."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "tags": [
     "l5",
     "ce",
     "s5"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x2090f5d4828>"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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X95yOdEkxIZQg3wbcAmCM2cjIB5pjfRdIAe4YM8USd06e6eFnL1WRkZrIJ29e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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x2090f4ee048>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "N = 40\n",
    "p = 0.7\n",
    "\n",
    "binomial_x = np.random.geometric(0.7, 10)\n",
    "sns.distplot(binomial_x)\n",
    "\n",
    "# Compute mean and variance"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": [
     "l5",
     "s5",
     "hint"
    ]
   },
   "source": [
    "use formula for mu and var"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "tags": [
     "l5",
     "s5",
     "ans"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mean:  0.42857142857142866 Variance:  0.6122448979591838\n"
     ]
    }
   ],
   "source": [
    "mu = (1-p)/p\n",
    "var = (1-p)/p**2\n",
    "\n",
    "print(\"Mean: \", mu, \"Variance: \", var)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "tags": [
     "hid",
     "l5",
     "s5"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "continue\n"
     ]
    }
   ],
   "source": [
    "ref_tmp_var = False\n",
    "\n",
    "try:\n",
    "    if (abs(mu - 0.428) < 0.1) and (abs(var - 0.612) < 0.1):\n",
    "        ref_assert_var = True\n",
    "        ref_tmp_var = True\n",
    "    else:\n",
    "        ref_assert_var = False\n",
    "        print('Please follow the instructions given and use the same variables provided in the instructions.')\n",
    "        \n",
    "except Exception:\n",
    "    print('Please follow the instructions given and use the same variables provided in the instructions.')\n",
    "\n",
    "assert ref_tmp_var"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": [
     "content",
     "l6",
     "s6"
    ]
   },
   "source": [
    "\n",
    "<br><br/><br/>\n",
    "## Poisson Distribution\n",
    "\n",
    "The probability distribution of a Poisson random variable X is the number of events occurring in a given time interval.\n",
    "\n",
    "Poisson distribution of discrete random variables is defined mathematically as\n",
    "\n",
    "$$P(X)=e^{-μ}\\frac{μ^x} {x!}$$\n",
    "\n",
    "\n",
    "<img src=\"../images/poisson.png\" style=\"width: 700px;\">\n",
    "\n",
    "\n",
    "where\n",
    "\n",
    "X = 0, 1, 2, …\n",
    "\n",
    "e = 2.71828\n",
    "\n",
    "μ = mean number of events in the given time interval\n",
    "\n",
    "### Examples\n",
    "\n",
    "Number of cars crossing a traffic signal in 30 minutes. Number of phone calls at a service desk in 5 minutes. Number of patients coming into a hospital every hour.\n",
    "\n",
    "### Exercise\n",
    "\n",
    "Let us now plot a poisson distribution. \n",
    "\n",
    "* Increase number of samples to 1000 and use np.mean function to determine the mean of the distribution.\n",
    "* Assign it to variable, mu and print it out."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "tags": [
     "ce",
     "l6",
     "s6"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x2090d368710>"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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nX8jpWAAAXBOlAgDEuXAkqjerm1RzrlO+5ATdu65Exbk+p2MBmGDeRI/urCrW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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x2090da099e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ax = sns.plt.subplots(figsize=(18, 8))\n",
    "x = np.random.poisson(lam=2, size=100)\n",
    "sns.distplot(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": [
     "l6",
     "s6",
     "hint"
    ]
   },
   "source": [
    "use formula for mu and var."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "tags": [
     "l6",
     "s6",
     "ans"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.018\n"
     ]
    },
    {
     "data": {
      "image/png": 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4HA6tWZKn375dre1mkwpDaUr1e+yOhST1xu7jI35sMOBXuLN3HNO875rlhROy\nHwAAxhLlAZBEzKPMOjgfl9Op/3LzInk9Lr2x67j+54+3acX8HN1y5WzNKUgf9fMdb+7SM69Xquxw\nixyS1pXk686r5yoz6Bv78Ako1e/RKiOkLfsbtXV/o669rFCOKbS2AwAAQDKiPACSyAubqiVJt62b\nbWuOROR0OPSZDy9QaXGWXtpco12HmrXrULNK5mTplitna8GMaRd8jo6ufv327Sq9ubtOsXhcC2dO\n08c/NF+z8oMT8C9ILPOKMlTVEFZtU5eq6sMqnj76EgYAAACJg/IASBKVte0qr2nTktmZmjvFp85f\nLIfDoRXzQ1o+L0cHatr08uZq7atq1b6qVi0oytDNV85W7rQUdfdF1N0bOfV2QN19EZ0M9+vtvXXq\n6YsqLytV9147V8vn5STtb9wdDofWLsnTi5uqta28UQXZqUrx8SUHAABgsuI7OSBJvLB5cK2DW9fN\nsTlJ4nM4HFoyO0tLZmepsrZdL22pVtnhFh18eo/l5wVSPPrUDXN19fLpcrtYjzaY6tWKBSG9W35C\n28pP6Orl0+2OBAAAgItEeQAkgar6Du070ipjxrQRTb/H++YVZeir9yxTTUNYb+6pUywWU6rPoxS/\nW6k+t1KHvC0KBfjt+lkWzpym6vqwahoG/yTjJRwAAABTAd/lAkngxVNrHdzKWgcXbVZ+UPflG3bH\nmHQcDofWlebrxU3VeudAo/KyUuT38qUHAABgsmFeLTDFHW0Ma3dls+YWpmvRrEy74yAJpad5tWJB\njnr7o9q6v1HxeNzuSAAAABglygNgintpc7Uk6dYr5yTt4n2w36JZmcrLTNHRxk5V14ftjgMAAIBR\nojwAprDjzV3aYTZpdn5QpcVZdsdBEnM4HLqyNF9ul0PvlDeqLdxndyQAAACMAuUBMIW9vLlacQ2u\ndcCsA9gtmOrVSiNX/QMx/XRjBZcvAAAATCKUB8AU1dDarXfKGzUjN6Dl83LsjgNIkhbMyFBBdqrK\nDrforbJ6u+MAAABghCgPgCnq5c3ViselW69k1gESh8Ph0JUl+UrxufWrVw+p+WSP3ZEAAAAwApQH\nwBR04mSPtuxv1PScNF1mhOyOA5whLcWj9dfPV19/VE+8Uq4Yly8AAAAkPMoDYAp6ZUuNYvG4blk7\nS05mHSABXVmSr+XzclRx9KRe21FrdxwAAABcAOUBMMW0tPdq09565WWl6vJFeXbHAc7L4XDoLz5i\nKJDi0bO/gqWsAAAgAElEQVRvHFZDa7fdkQAAAGCB8gCYYl7aUq1o7NSsAyezDpC4MgI+feZGQ/2R\nmH704n4NRKJ2RwIAAMAwKA+AKaShtVtv7alXQXaq1ixh1gES3+qFuVpXkq+q+rCe/J3J7RsBAAAS\nFOUBMIU8/+cjisXjuvODxXI5Ob0xOdz3EUNzCtK1ZX+Dfr/tmN1xAAAAcB78dAFMEVX1HdpecUJz\nCtJ12QLusIDJw+N26a/vLFVGwKtn3qhU2eEWuyMBAADgLJQHwBTx3JuHJUl3XzNXDu6wgEkmM+jT\nl+9cKpfTqR++sE/1LV12RwIAAMAQlAfAFLC/ulUHqtu0ZE6WFs3KtDsOcFGKp6fr/psWqqcvqkef\n26uu3gG7IwEAAOAUygNgkovH43r2jVOzDq6ea3Ma4NKsLcnXTVfMVGNrt3742/2KxmJ2RwIAAIAo\nD4BJb7vZpJqGsC5flKtZ+UG74wCX7K6r52rp3Gztq2rVM68ftjvOmIhEY+ro6lckShkCAAAmJ/eF\nHmAYhlPS9yUtk9Qn6QHTNCuHbP+kpK9KikjaK+mvTNPkuyNgAkSiMT3/5mG5nA7d8cFiu+MAY8Lp\ndOjzty7RN3+2XX9495iKQgFdtbTA7lgjEo0NlgQnw/062dmnk52Db8Pd71+CEUjxKD3Nq4w0rzIC\n77/1ey/4JRkAAMA2I/lO5XZJftM01xqGsUbSI5I+JkmGYaRI+kdJpaZpdhuG8StJt0h6YbwCA3jf\n23vr1djWo2tXFCovM9XuOMCYSfW79ZW7luoffrpdT/6uQgORqK69rMjuWOcVjcV07ESXKmtPqr6l\nW/H4mdt9HpfyslIU8HvU1RtRe1ef6pq7VNd85qKQ03PStGJBjrLT/ROYHgAAYGRGUh5cJWmjJJmm\nudUwjFVDtvVJutI0ze4hz9c7thEBnE/fQFQvvF0lr8epW9fNtjsOMObyslL11XuX6bvPlelnfzio\n2uYuffK6+XK7EuOKu7Zwnypr23WkrkN9A1FJUna6T9kZfmUEfMoM+JQR8CrFd+6X2v6BqNq7+tXe\n2a/2rn6daOt5r1CYlR/U8nk5ygh4J/qfBAAAMKyRlAfpktqHvB81DMNtmmbk1OUJjZJkGMaXJQUk\n/dHqyTIzU+V2uy42LzA6lS0KBhLzt3ih0KWtT/Dsa4d0srNf91w3X/Pn5IxRKiXs8ZIu/Zglo+HG\n0+5xHulYhkJBFc/M0j8+8Y5e33lcLR19evgvViuYOn4/WFsdm/6BqA4dO6kDVa060TbYm6f43Fq+\nIKRFs7OUNYpZA9mZaWe8f6wxrK376lXTENbRxrAWzsrS6sV57/1bef2P3mhf5xN1XjCWmEx4vQJn\nSuZzYiTlQYekoUfIaZpm5PQ7p9ZE+DdJCyTdZZrmWRM2z9TW1m21GRhz4c7EnAzT1BS+6M/t6h3Q\nM386qDS/W1eX5l/Sc50tUY+XdGnHLFmdbzyDAb/t4zyasXRK+vonl+tHLx7QrkPN+up/vKGv3LVU\n03PSLvi5F+N8x6aze0DlNW2qrG3XQDQmh6TCUJrmFWaoKDcgl9Mx7OeO1LQ0j268fIaOnejUroPN\nKq9ulXm0TcaMaSqdm83r/yKMZjwm8rxgLDFZhEJBXq/AEMlyTgxXkIykPNgk6VZJT59a82DvWdt/\nqMHLF25noURgYry0uVrdfRHde+08pfo9dscBxp3f69aX7izVhj8f0ctbavTNn23XFz5WotLi7HHb\nZzweV9PJXpVXt+poY6fiklJ8Li2Zk625RRlKG4dzz+FwaGZeUEW5AVXVdWj3oWaV17TpaGNYJXOy\nVBQKjPk+AQAARmIk5cEGSTcYhrFZkkPS/YZhrNfgJQrbJX1O0luSXjMMQ5K+Y5rmhnHKCyS96oYO\n/fHdWuVk+PWhywrtjgNMGKfDobuunqvCnDQ98UqFvv3MHt1zzTxdv6poTNdBiMXiqmkMq7y6Tc3t\ng7+Jzkr3adGsTM0uSH9vlsF4cjocmluYodkFQe070qo9lS3655/v0F/dXqolc7LGff8AAABnu2B5\ncGo2wRfO+nDFkL8nxspVQBKIRGP6ySsVisXj+uxNC+X1sH4Iks+aJfnKzUzVd58r09OvV2rjOzVa\nW5Kvq5ZOV+FFXsoQicZUUdOmnQeb9E55o3r6BhdAnJEb0KLZmcrLTJHDMf6lwdlcTqeWzctReppX\nW/Y16NvP7NFnbjT0wWXTJzwLAABIbtxUGphENr5zVMdOdOoDSwu0eDa/fUTyKp6erv/12dX63Ts1\n2rq/Ub/fdky/33ZMc6en66qlBbp8Ud5573IwVG9/RHuPtGrnwSaVHW5+rzDweVwyZk7TolmZSk9L\njDsezClI17qSAj32/F49+bsKNZ3s0R0fLJbThkIDAAAkJ8oDYJKoa+7SC5uqlBHw6uMfmmd3HMB2\nmUGf1l+/QPdcM097Kpv1Vlm99lW16HBdh3716iGtmB9Smt+tSDSuaDSmSCyuSDSmaDSu3v6IKo93\nKBIdXKonO92vq0qn67IFOTre0pWQP5QvmDFNf/eZlfr2M3v08pYaNZ3s0eduXiQPdzACAAATgPIA\nmARisbh+8rtyRaJx3fdhg0USgSE8bqdWLczVqoW5au3o1aZ9DXq7rE7vHGi0/LzCUJoumx/SZQtC\nmpkXeO+yhPrWxL0rUF5Wqv7uvlX67nNl2lZ+Qq0dffryXaXjeutKAAAAifIAmBRe21mrw8c7tHph\nrlYsCNkdB0hYWel+3XrlbN28dpbqm7sUj0sul0Nul1Nul3Pw706n3C7HpF0zJJDi0X/9xHI98UqF\n3jnQqH/++U49/OnLlE6BAAAAxhHlAZDgmk/26Lk3jyjN79b6GxbYHQeYFJwOhwqn8G0NPW6X/vLW\nxcpI8+oP7x7Td57Zo7/95Ar5vXxZBwAA44M7JQAJLB6P66cbK9Q3ENUnr5+vjARZvA2A/ZwOhz7+\noXlaV5Kvqvqwvrdh33trOAAAAIw1ygMggW3a26D91W0qKc7S2iX5dscBkGAcDof+4qaFWjo3W/ur\nWvXEK+WKxeN2xwIAAFMQ5QGQoNo7+/TUq4fk87r0FzcutOUe8wASn9vl1BdvL9Hc6enaur9RT79W\nqTgFAgAAGGOUB0ACisXi+ulGU919Ed1zzVxlZ/jtjgQggfk8Lj10zzIVZKfqD+8e08ZtR+2OBAAA\nphjKAyDBxONx/erVQ9pd2axFszJ1zYpCuyMBmAQCKR79zb3LlRn06ZnXD2vT3nq7IwEAgCmE8gBI\nMBu3HdWrO2pVmJOmL91RIieXKwAYoewMv/7m3mVK9bn1k1cqVHa42e5IAABgiqA8ABLIlv0Neub1\nw8oM+vS1e5cp1e+xOxKASaYwFNBD9yyVy+XQ93+zTzUNYbsjAQCAKYDyAEgQ+6tb9cTL5Ur1ufW1\ne5cpK511DgBcnPlF0/TgbUs0MBDTo8+Vqb2zz+5IAABgkqM8ABJATUNYjz2/Vw6H9OW7SlUUCtgd\nCcAkd9mCkO68ulht4T49tmGvBiIxuyMBAIBJjPIAsFnzyR59+5k96u+P6i9vXSJjZqbdkQBMER9d\nM0trFufp8PEO/XRjBbdwBAAAF43yALBRZ8+A/uPpPWrv6tcnrp+v1Qtz7Y4EYApxOBz67E0LNacg\nqM37GvT7bcfsjgQAACYpygPAJq0dvfrW03vU0Nqtm66YqRtWzbA7EoApyOtx6a/vXKppAa+eeb2S\nOzAAAICLQnkA2KCqvkP/68fbVFXfoXWl+brrmrl2RwIwhWUGffryXUvldjv1g9/u1/HmLrsjAQCA\nSYbyAJhA/QNRvV1Wr7f21CsSi+mzNy3Uf/noIjkdDrujAZji5hSk6/6bFqq3P6rvPlumzp4BuyMB\nAIBJhPIAmCCNrd16cVO1jtR1KCfDr/99/+X64LLpclAcAJgga5bk6+a1s3TiZI/+32/2KRLlDgwA\nAGBkKA+AcRaNxbXDbNLvtx1Td19ES+dm6yNXzFReVqrd0QAkoTs+WKwV83NUXtOmX/zxIHdgAAAA\nI0J5AIyT/oGoDtW265UtNdpf1apgqkcfuXymls/PkdPJbAMA9nA6HPrLWxdrZm5Ab+6u4w4MAABg\nRNx2BwCmkmgsruNNnaqqD+vYiU7FYoO/0ZtXlKHVC3PlcdPXAbCf3+vWV+5eqn/8z+165vVK5Wam\n6LIFIbtjAQCABEZ5AFyigUhMrR29qqrvUHVDWP0Dg9cQZ6R5VTw9XXMK0hVI9dicEgDOlJXu10N3\nL9M//2KHHn9xvx7+1GWanZ9udywAAJCgKA+AC4hEY+ofiKq7L6pwd7/C3QNnvO3pi7732BSfS4tn\nZ2rO9HRlBX0shgggoc3KD+rBW5fosef36jvPlul/3rdKWel+u2MBAIAERHmApBWLxdXZM6CO7n6F\nuwbf9vRF1NcfVd9AVH0Dg6VBNHb+xcQcktJSPCrI9ik9zasZuQHlZ6dy20UAk8qKBSHd+6F5+vVr\nlfrOs2V6+FOXKcXHtwcAAOBMfHeApBCPx9XS3qujJzrV2tGncHe/OnsGNNwi416PUz6PS6l+n3we\np7wel/xel4IpXgXTPEpP9SotxSMXCx8CmAI+vHqGGtt69Mau4/rhC/v1lbuWsrArAAA4A+UBpqxY\nPK665k5VVLWqpjGs7t7Ie9v8XpdyMvxKT/UqmOZVMPX9QsDrcTJ7AEBScTgcWn/9fDWd7FHZ4RY9\n9dohrb9+gd2xAABAAqE8wJQSj8d18NhJbSs/oZ0Hm9Te1S9J8ridKp6erpl5AeVnpcrrcdmcFAAS\ni9vl1Bc/VqJ/+vkO/Wl7rbKCfn3kipl2xwIAAAmC8gBTxrETnXrq1UMqr2mTJAVSPFo0O0vTs1OV\nn53KJQYAcAGpfre+evdS/dPPd+jp1yvldDr04dUz7I4FAAASAOUBJr2Orn5teOuI/rynTvG4VFKc\npZsun6kFM6dp95E2hTt77Y4IAJNGzrQUfX39ZfrXX+7UU68eksMh3bCKAgEAgGRHeYBJayAS1R+3\n1+qlzdXq7Y+qIDtVn7huvkqLs+2OBgCTWn5Wqr7+yRX6t1/u0q/+dEhOh0PXrSyyOxYAALAR5QEm\nnXg8rh1mk55+vVLN7b0KpHj0qRvm6poV0+VyOu2OBwBTQkF2mr6+foX+9Ze79Is/HpQkCgQAAJIY\n5QEmlYFIVE/+rkJb9jfKdepa3FvXzVaa32N3NACYcgqy007NQNipX/zxoJwO6drLKBAAAEhGlAeY\nNNrCfXrs+b2qqu/QnIJ0ff7WxcrLSrU7FgBMadNz0vS36y/Tv/1yp372h4NyOBy6ZkWh3bEAAMAE\nY443JoXDde36Pz99V1X1HbqyJF8Pf2oFxQEATJDCnDT97SdXKJjq0X/+3tQf3j2meDxudywAADCB\nKA+Q8Dbtrde//mKXOrr69fEPzdPnbl4kj9tldywASCpFocB7BcJTrx7SD367Xz19EbtjAQCACUJ5\ngIQVi8X19GuV+vHL5fK4nfrqPct04+Uz5XA47I4GAEmpKBTQ3392teYVZejdihP6P0++q6ONYbtj\nAQCACUB5gITU3Tugbz+7Rxu3HVV+Vqr+x30ruQUjACSArHS/vv7JFbrpiplqbOvRN3+2Q3/eU8dl\nDAAATHGUB0g43b0RPfLr3dp3pFUlxVn6H/etVEF2mt2xAACnuF1O3XPtPH3lrqXyup168ncV+vHL\n5errj9odDQAAjBPKAySU3v6Ivv3MHlXVh7WuNF9fvXuZUrkNIwAkpOXzc/T3n12tOQVBbd7XoH/4\nz+2qbeq0OxYAABgHlAdIGH0DUT36bJkqj7drzeI83X/TIjmdrG8AAIksZ1qKvvHplbp+ZZHqmrv0\n909s0w9+u4+1EAAAmGLcdgcAJGkgEtVjz5Wp4uhJrTRC+twtFAcAMFm4XU6tv2GBFs/J0m/eOqJt\n5Se0rfyESouzdfPaWVowY5rdEQEAwCWiPIDtItGYvrdhn/ZXt2n5vBw9eNsSuZxMigGAyWb5vBwt\nm5utfVWtenlLjfYeadHeIy2aV5Shm9fM0tK52dwxBwCASYryALaKRGP6wW/3q+xwi0rmZOmLt5fI\n7aI4AIDJyuFwqLQ4W6XF2aqsbdfLW6q153CLvvNsmfIyUzR/xjQVF6SreHq6CkNp41YWD0Si6umL\nKB6X3C6H3C4nM9oAALgElAewTSwW1//30gHtPNikhTOn6a/vLJXHTXEAAFPFvKIMPXTPMh070alX\nttZo16EmvV1Wr7fL6iVJXrdTs/KDKp6erpm5QXk9rvd+0He7HHK5nHI5HXI5Herpi6i7N6Ku3oi6\n+yLq7h0Y/Pvpt+9tH1B3b0QDkdg5eZwOnXruwef3+9zKTvcrO8OvWQUOORVnZgQAAMOgPIAt4vG4\nfrqxQtvKT2h+UYYeunuZvB6X3bEAAONgRm5AD962RNFYTHXN3TpS166q+g4dqetQ5fF2Haptv+R9\nOB0OpfrdSvW7lRX0KdXvUWfPgBySIrG4ItGYIpGYItGYorG4BqIxhdt6dKKtR5L0dlm9PG7ne2VC\naJp/XGdGAAAw2VAewBavbK3RW2X1mpUf1FfvWSafl+IAAKY6l9OpGbkBzcgN6OrlhZIGb9Fb0xBW\nXXOXBqJxRaPv/4AfiQ7+0B+LxeX3uZXqcyvN71aq33PqrVtpfo9S/W75va5zZg28sfu4ZZ5INKbW\njj61tPeqvbtfDS3damgd/CNJPo9LxdPTNb8oQ9OCvvE5KAAATBIXLA8Mw3BK+r6kZZL6JD1gmmbl\nWY9JlfRHSZ8zTbNiPIJi6thecULPvXlEWek+PXT3UqX46LAAIFn5vW4ZMzNlzMyc8H27XU7lZqYo\nNzNFwYBf4c5e9Q9E1dLRq+NNXTpS16HymjaV17QpNM2veUXTNDs/yCV2AICkNJKf2m6X5DdNc61h\nGGskPSLpY6c3GoaxStIPJBWNT0RMJUfqOvSjlw7I53XpobuXaVqA3+QAABKH1+NSQXaaCrLTtGJB\nSLUnOnWotl11zV1qOtmgd8sbNbsgXaXFWQqmeu2OCwDAhBlJeXCVpI2SZJrm1lNlwVA+SXdI+tkY\nZ8MU09zeo0efK1MkGtNDdyzVjNyA3ZEAABiWy+nQrPygZuUH1dkzoMPH21VZO/jnyPF2LZyVqdK5\n2fKxZg8AIAmMpDxIlzR0JaOoYRhu0zQjkmSa5iZJMgxjRDvMzEyV280X2WTT1TOg7z35rjq6+vXg\nHaW6bs2cidlxZYuCAf/E7GuUQqGg3RHOK1GPl5S4xyyRDTeedo9zIo+l3cdmOIl8zBLVaMfS6vHB\ngF8FoaCuXFaow7UntWVvgw5Ut+lwXYdWL8pTydzsES+uyFhiMuH1Cpwpmc+JkZQHHZKGHiHn6eLg\nYrS1dV/sp2KSisZi+s4zZappCOu6lUW6wgipqSk8YfsPd/ZO2L5GYyKPwWgk6vGSEveYJbLzjefp\na7vtlMhjafexGU4iH7NENZqxHM15kZ+ZotvWzVL50ZPae7hFb++p055DTVpphDQjN3DB2z0ylpgs\nQqEgr1dgiGQ5J4YrSEZSkW+S9FFJOrXmwd6xi4WpLh6P65d/PKR9Va1aOjdbn7hunt2RAAC4ZC6X\nUyVzsnTHB+fImDlNnT0DemNXnf6w7Zjawn12xwMAYMyNpDzYIKnXMIzNkr4l6WuGYaw3DOPz4xsN\nU8Gfttfq9V3HVRRK04O3LeF+2QCAKcXvdev/b+/Og+So77uPv7vnnp2d2Xu1h7Q6VrS0QgdYehAR\n5vCBMcGYgJ04hw/iIzxPsB0/lVQlTpyknDiVVMV24vhIbExiOzi2MYFgEQLEoBSHLEAgBDpa565W\nu9Le98zuHN35Y2YvJBYB2u09Pq9SV/9muqfnq5K0mv7M77iiqZqbd6ykvrKIjr4UDz3TzMvHe3Ac\n1+vyRERELprXHbZg27YD3PGqp89ZjtG27WsvUk2ySLxysocfPX6URFGQz35gs5ZkFBGRRSsRC/GO\nt9VzunOY3QfO8uLRblo7h9mxsYZEbOGsyrBrX5vXJZzXtVvqvC5BRGTJ09fAMis6epP84wMH8JkG\nd962kfLE/JyATERE5GKqr4px845VrKoppntglJ3PNHOouQ/XVS8EERFZ2BQeyEWXGsvytfv2kxzL\n8tEb1rGmNuF1SSIiInMmFPTx9s21XLOlFr/P5LnDnTz6bCtDybTXpYmIiLxpCg/konJcl+/87CBn\nepK8e+tydmys8bokERERTzQsK+bmq1ayvCpGR1+Knz3dzNHWfvVCEBGRBUnhgVxUDzx5kn3Humla\nWcqvvmON1+WIiIh4KhLyc+1ltVy1aRmmYbD7QAff+o8DJEczXpcmIiLyhmgGO7lonjvcyc5nmqks\nCXPH+y/VygoiIiKAYRisrk1QVRrlqf1neP5wJ81nBvnUzRtorNPQPhERWRh0dycXxamOIb770EFC\nQR+fuW0TsUjA65JERETmlVgkwPXblnPzjpX0DIzy1//6Ag/tbsbRMAYREVkAFB7IWzaUTPMP971M\nOuPwyZuaqKuMeV2SiIjIvGSaBre8fTV/8OuXkYgFue9/TvDlH+2jf3jM69JERERmpPBA3pJszuFb\nD7xCz+Aot1y1issvqfS6JBERkXlvXUMpf377NrY0VnCopY8/u/tZ9h/v9rosERGR16TwQN4013X5\n4WNHOHyqn7ddUslNO1Z6XZKIiMiCURwN8unbNvIb71pLaizL3927n3seO0I6k/O6NBERkXMoPJA3\n7fEX2ti1r53lVTE+ftN6TMPwuiQREZEFxTAM3rV1OX/yka3UlEf5+d7T/MX3nudUx5DXpYmIiEyj\n8EDelAMne/m3/z5KvCjIZ27bRDiohTtERETerBXVxfzZx7bxzsvraese4S+//zyPPntKkymKiMi8\nofBA3rAzPSN884FXME2489aNlCfCXpckIiKy4AUDPn7z+kv4vQ9uIhry86PHj/GVH++jb0iTKYqI\niPcUHsgbMpzK8LWf7ic1luX2967X+tQiIiIX2aY1FXzx41ewaU05B5v7+NPv7mGv3el1WSIissQp\nPJALNr6yQkdfihu3N3Dlpcu8LklERGRRihcF+ewHNvHh6y8hnXX4xv2v8I37X1YvBBER8YwGqssF\n+7efH+VQSx+Xra3g1mtWe12OiIjIomYYBtddXo+1opR/efgwe+0uDpzs5bZr1nDdZXWYpiYqFhGR\nuaOeB3JBHn/hNE+80EZ9ZYxPvq9JKyuIiIjMkdqKIv7wty7nozdYmIbBPY8d4Us/0IoMIiIytxQe\nyOt65UQPP3zsKPFogM98YKNWVhAREZljpmFwzZY6vvSp7WxvqubkmSG++C/P8+PHjzKaznpdnoiI\nLAEKD2RGJ9oH+cb9r+DzGdx56yYqEhGvSxIREVmyEkVBPnXzBv7/r22mPBHikWdb+cJde3j65TPk\nHMfr8kREZBFTeCCv6UzPCH9370uksznueP8GGuu1soKIiMh8cOmqcv7i41fwy1c20D+c5rsPHeKP\nv72Hp/YrRBARkdmh8EDOq29ojK/8+CWGUxk+esM6Lltb6XVJIiIiMkUw4OO2a9bwN3dcyXWX1dEz\nOMrd/6kQQUREZofCAzlHcjTDV3+yj57BUX7l6tVcvbnW65JERETkNZTFw3z4PVY+RLi8jt6hfIjw\n+W//gif3t5PNKUQQEZG3TjPfyTTpTI6v/XQ/p7tGeOfl9dx0ZYPXJYmIiMgFKIuH+fD1Fr+8vYGH\nftHCky+188//eZifPH6Mbeuq2L5hGY31Ca2YJCIib4rCA5ngOC7/9OABjpweYNu6Kn79XWsx9AFD\nRERkQZkaIjz6XCt7Dnawa187u/a1U5EIs31DNdubllFbUeR1qSIisoAoPBAAXNflB4/avHi0m/UN\npXzipiZMU8GBiIjIQlUWD/Ohd67lg9et4XBLP7sPnGXvkS52PtPCzmdaaKguZnNjOY31CVbXJIiG\n9bFQRERem/6XEFzX5d4njvM/+9pZUR3jzls3EvBrOgwREZHFwGeabFhVxoZVZXw4k2Pf0W52HzjL\ngZO9tHQMAWAAdZVFRMN+KksiVJZEKI4G1ANRREQmKDxY4hzX5Z7HjvDEC20sK4vyuQ9uJhLSXwsR\nEZHFKBTwcUVTNVc0VTOcynCsbYDjbQMcOz3AyTODpLMOR1oHAAj4TIoifmKRAEWRALEpWzTsJxjw\n4VMvRRGRJUN3iUuY47j888OHePrls9RXxvj9D20hXhT0uiwRERGZA7FIgC2NFWxprAAgm3N44KmT\ndPWl6OpPMTCSZjiZoX84/ZrX8JkGwYBJ0O/L7wM+gv7JfSDgI1R4HPCfe1xDJEVEFg6FB0tUNudw\n186DPHuok1U1xXzuV7cQiwS8LktEREQ84veZVCTCVCTCrKcUyA9tTGcdhlMZRlIZhgtbcjRLOuuQ\nyeRIZx1G0zkGk2lc942+pzEtUIiE/MQiforCgWk9HkRExHsKD5agTDbHtx44wL5j3aytT/B7Gqog\nIiIi52EYBqGAj1DAR3k8POO5ruuSzbmksznSGYd0IVgY32cmHjuT5xT2ybHsjD0cHnzqJNVlUeor\nYyyvilFfWUR9VYyisIIFEZG5ojvGJWYsk+Pr9+3nQHMfTStL+fStmwgFfV6XJSIiIgucYRgE/AYB\nv0nRzDnDeTmuy+hYluFUdqKXw8hofu840HJ2iBPtg9NeUxYPsbwyxvLqYhrrEjTWxYkqUBARmRUK\nD5aQ1FiWv7/3JY6cHmDzmnL+369cSsCv4EBERES8ZxoG0XAgf/NfGpl27NotdWRzDmd7krR2DXO6\nc3hi/9LxHl463gNMrhrRWF/C2roEjfUJKhLhJblqxK59bW/5GsWxMEPDoxehmumu3VJ30a8pIrNP\n4cES0dmf4uv3vczprmG2rqviU+9rwu/TcowiIiKyMPh9JvVVMeqrYrBh8vmhZJrms0McPT3AsdP9\nnDRP+C0AAA0LSURBVGgf5HTXCLtezN88l8SCrG8oZX1DGU0rSyl7neEXIiJyfgoPloD9x3v49oMH\nSI5lue6yOn7j3WvxmQoOREREZOErjgbZuLqcjavLgfyk0K2dwxw9PcDR0/0cbe1n94EOdh/oAKC6\nNELTyjLWN5SyrqFUEzKKiFwghQeLmOO67Hymmf948iQ+n8ntN67j7ZtqvS5LREREZNb4fSarauKs\nqolz/bbluK5LW9cIB1v6ONTci93azxMvtvHEi20YwIrqYppWlrJ+ZSlr60sIBTSkU0TkfBQeLFLJ\n0Sx37TzIvmPdlMdD/O6tG1m5LO51WSIiIiJzyjCMieEO129bTjbn0HxmiIMtvRxq7uNY2wAtHUM8\nvOcUfp/BmtoE61eW0rSyjFU1xeqtKSJSoPBgEWrrGubr//4yHX0p1jeU8jvv30A8GvS6LBERERHP\n+X0mjfX5yRRv3rGKsXSOo6f7Cz0T+jjS2o/d2s8DT54kHPRhLS9h/coymhpKqassWpKTL4qIgMKD\nRcV1XfYc6uB7D9uMZXK894oV3HrNaiXmIiIiIq8hFPRx6epyLi3MmTCcynC4pW9imMPU1Rzi0QDr\nC/MlWMtLqCqNKEwQkSVD4cEi0dWf4p7HjrD/eA+hgI//e8ulbFtX5XVZIiIiIgtKLBJg67oqthY+\nR/UOjnKwuY9DLb0cbO5jz8EO9hzMT75YFPazqjbO6po4q2vz8ywUq7eniCxSCg8WuGzO4ZFnT/Gz\np5tJZx3WN5TykfdYVJdFvS5NREREZMEri4e5alMNV22qwXVd2nuSHGru5Xj7ICfaB3jlRC+vnOid\nOL+qJELDsmJqK4qoKY9SW1FEdWmUgF89QUVkYVN4sIAdae3n+4/YtHePEI8G+Nh713FFU7W6z4mI\niIjMAsMwqKsooq6iiHcVnhtMpjnZPsjJM4OcKOyfO9w57XWmYVBZGqG2PMqysihl8TAlsRBl8RAl\nsRCJoiCmqc9vIjK/KTxYgIaSae7ddZyn9p/BAK69rI7brllNUVjrFIuIiIjMpXg0yObGCjY3VgD5\nOah6B8do7xnhTPcI7T3JifaLvcnzXsM0DBKxICWxEEURP9GQn0hocj/eDvhNfKaBz2fgMyfbfl+h\nbRr4Cu2R0QymYeQ308Aw8u9jGOiLJhF5UxQeLCADI2l+vreVx/e2kRzLsrwqxkfeY7GmLuF1aSIi\nIiJC/sa8PBGmPBFmY2ESRsiHCkPJDGd7k/QPj9E3NGUbHqNvcIzWziGyOXf2ayzUaRhgjocOZj6E\nMKe0Q0EfBhAMmAT8PoJ+k0DAJFhoh4I+IkE/4ZAPv0/DMkQWO4UHC0BHX5JHnm3lqf1nyOYcYpEA\nH3pHI+/cWq+VFEREREQWAMMwiBcFiRe99oSKruuSyTokx7KkxrIkRwv7wpbNOuQcN7/lHLK5Qttx\nyL2q3d4zguOC47g4rovr5q8/3nacwt6dvN5YJjfRdt5ghuH3GURCfsJBH+Ggn2g4vxWFAxQV9tGw\nX8MzRBYwhQfz2Mkzgzy85xR77U5cFyoSYW64YgU7NtYQCvi8Lk9ERERELiLDMAgGfAQDPkpiobd0\nrV372t7S6x3XJRIJ0tefJJ11yGQc0lmHdDZXaOcYTee31Fi20M7SPZDBnSF4iIR8PL3/DBUlESpL\nwlQkIlQmwlSWRCiNh/TFmMg8pvBgnhlOZXjxSBe7D5zl8Kl+AFZUx7hxewNvsyr1A1VEREREZp1p\nGAT9PqLhAG9kDS/XdRnL5EiO5ntOjIxmGBlvp/Lt5rNDHG8fPO97lsVDVE4NFkoiVJSEqUxEKI4G\nNF+DiIcUHswDQ8k0Lx7t5rnDnRxu6SNX6Ce2YWUpN2xvoKmhVD8oRURERGTeMwyDcNBPOOinLH7+\nc67eVEvf0BjdAyk6+1N094/SNTC5P9TSx6GWc18XCvgmgoTx/dRwIRRUz1yR2aTwwAOu69LZn//B\n+PzhTg639OMU+nc1LCtm27oqtlqVVJW+kZxXRERERGT+M83JSSWtFaXnHE9ncnQPjNI9kKKrf5Su\n/hRd/Sm6B/Lttq6R8143Hg1QURKhojAMonJKu7Q4pEkdRd4ihQdzIJ3J0Xx2iGNtAxxvG+BY2wBD\nyczE8VU1cbauq2SrVUVlScTDSkVEREREvBUM+KitKKK2ouicY67rMjKanRYmdBfCha6BUVrODnHi\nPEMiAIqjAUpiocKWXxqzpDhEoihILBKgOBqgOBrMT+yoXr8i51B4cBFlsg5d/Sk6epOc7UvS0Zuk\ntXOEUx1DE0MRAMriIf7P+ioa6xJsWVtBRUKBgYiIiIjI6zEMg1gkQCwSYFXNueMiHMedGBIx3muh\neyBF39AY/cNpuvpTtHYOz/gepmEQiwYoLrxPJFRYPSLkP6cdHF+6MpBfvjIYMKe0ffhMQ8OPZdF4\n3fDAsiwT+CawGRgDPmHb9rEpx98H/CmQBe62bfs7s1Srp8bSOQaSaQaH0wyMjDE4kmZgJM3gSJru\ngVHO9ibpGRw9Z3ZZn2mworqYxroEjfUJ1tTGKYuHvflNiIiIiIgsYtOHRJz/nNRYloGRNP1DY/QN\n5z/XDyUzHDndz1hh1YjRdC4/RKL7/EMkLpRh5O8H/D4Tv88stA1M08A0DIzC3jSYeM40DQyDifaK\nqmJ8PgOfWdh8Jv4p7cnnDXymOb3tMwrn5tumYUy/1vjz5uR5ZuH16n0hr3YhPQ9uAcK2bV9pWdZ2\n4MvA+wEsywoAXwW2ASPA05ZlPWjbdsdsFTzXXNflr/51L8fbzt/9aVyiKMjaugTVZVGWlUWpLmxV\nJRECfo2vEhERERGZDyKFXgPLyqbPL3a+5S0d1yUzsVRljnRhn8k6pDMOOcchm3PJ5hxyTmGfc8k6\nLrmcM+X5/OPRdP41OcedcUnLqezCCmxzzYCJEMKcFlDkN9OcDDHMV4cX48+9OuA4T2gx7XmfORFw\n+CeuYU57b8MwMAoFjreNwhOGkW8bGBR+FY4bjGch4z1BTGPyHAxwnfyft+u6OI6L4+bvBR3XxSkc\nKz4zRP9ACsdxJ45lc4W/I1mHTC6/zxbamxvLuXRV+Zz/2c2WCwkPrgL+C8C27V9YlrV1yrH1wDHb\ntvsALMt6CrgauPdiF+qlVcviREJ+EtEg8VjwVfsQZcUhIiGNABERERERWUxMwyAU8BEK+IDARb22\n6+YDBGfiZnXyJnXqDezlayvJOvlQIue4+fBhop0PJaa288GFOxFSTHtd4Tzn1ecUHjvuq6435VjO\ncfKvGw9DMllyOafwmslzZVLPwOiiCg8M93UiL8uy7gLus2374cLjU8Bq27azlmVdBXzatu1fKxz7\nInDKtu27ZrluEREREREREZkjF9KffhAonvoa27azr3GsGPCmX42IiIiIiIiIzIoLCQ+eBm4EKMx5\n8PKUY4eAtZZllVmWFSQ/ZGH3Ra9SRERERERERDxzIcMWxldb2ER+PonbgcuBmG3b356y2oJJfrWF\nb8xuySIiIiIiIiIyl143PBARERERERGRpU1rCIqIiIiIiIjIjBQeiIiIiIiIiMiM/F4XIDIbpszV\nsRkYAz5h2/Yxb6sS8ZZlWQHgbmAlEAL+0rbtBz0tSmQesCyrCtgLvNu27cNe1yPiNcuy/gi4GQgC\n37Rt+7selyTiqcJnqO+R/wyVAz65FP+/UM8DWaxuAcK2bV8J/CHwZY/rEZkPfgvosW377cANwNc9\nrkfEc4UPhP8EpLyuRWQ+sCzrWuCXgB3ANcByTwsSmR9uBPy2bf8S8EXgSx7X4wmFB7JYXQX8F4Bt\n278Atnpbjsi8cC/whULbALIe1iIyX/wt8I9Au9eFiMwT7yG/NPv9wM+And6WIzIvHAH8hd7NcSDj\ncT2eUHggi1UcGJjyOGdZlobpyJJm2/awbdtDlmUVAz8F/sTrmkS8ZFnWx4Au27Yf8boWkXmkgvyX\nLh8E7gDusSzL8LYkEc8Nkx+ycBj4DvA1T6vxiMIDWawGgeIpj03btvUtqyx5lmUtB54AfmDb9g+9\nrkfEY78NvNuyrF3AFuD7lmUt87YkEc/1AI/Ytp22bdsGRoFKj2sS8drnyP+7uIT8nGrfsywr7HFN\nc07fxMpi9TTwPuAnlmVtJ9/9TmRJsyyrGngUuNO27Z97XY+I12zbvnq8XQgQ7rBt+6x3FYnMC08B\nn7Us6ytADVBEPlAQWcr6mByq0AsEAJ935XhD4YEsVveT/zbpGfJju2/3uB6R+eDzQCnwBcuyxuc+\neK9t25ooTkREALBte6dlWVcDz5Lvpfy7tm3nPC5LxGtfBe62LOtJ8quQfN627RGPa5pzhuu6Xtcg\nIiIiIiIiIvOY5jwQERERERERkRkpPBARERERERGRGSk8EBEREREREZEZKTwQERERERERkRkpPBAR\nERERERGRGSk8EBEREREREZEZKTwQERERERERkRkpPBARERERERGRGf0vzHNVNiUaHkQAAAAASUVO\nRK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x2090f627b70>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ax = sns.plt.subplots(figsize=(18, 8))\n",
    "x = np.random.poisson(lam=2, size=1000)\n",
    "sns.distplot(x)\n",
    "\n",
    "mu = np.mean(x)\n",
    "print(mu)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "tags": [
     "hid",
     "s6",
     "l6"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "continue\n"
     ]
    }
   ],
   "source": [
    "ref_tmp_var = False\n",
    "\n",
    "try:\n",
    "    if (abs(mu - 1.921) < 0.25):\n",
    "        ref_assert_var = True\n",
    "        ref_tmp_var = True\n",
    "    else:\n",
    "        ref_assert_var = False\n",
    "        print('Please follow the instructions given and use the same variables provided in the instructions.')        \n",
    "except Exception:\n",
    "    print('Please follow the instructions given and use the same variables provided in the instructions.')   \n",
    "\n",
    "assert ref_tmp_var"
   ]
  }
 ],
 "metadata": {
  "executed_sections": [],
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}