Ch5.ipynb

{
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   "cell_type": "markdown",
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   "source": [
    "# Chapter 5: Continuous Random Variables\n",
    " \n",
    "This Jupyter notebook is the Python equivalent of the R code in section 5.9 R, pp. 253 - 255, [Introduction to Probability, Second Edition](https://www.crcpress.com/Introduction-to-Probability-Second-Edition/Blitzstein-Hwang/p/book/9781138369917), Blitzstein & Hwang.\n",
    "\n",
    "----"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Python, SciPy and Matplotlib\n",
    "\n",
    "In this section we will introduce continuous distributions in Python and SciPy, learn how to make basic plots, demonstrate the universality of the Uniform by simulation, and simulate arrival times in a Poisson process."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Uniform, Normal, and Exponential distributions \n",
    "\n",
    "For [continuous distributions in `scipy.stats`](https://docs.scipy.org/doc/scipy/reference/tutorial/stats/continuous.html), the `pdf` function gives the PDF, the `cdf` function gives the CDF, and the `rvs` function generates random numbers from the continuous distribution. This is in keeping with the application programming interface of the [discrete statistical distributions in `scipy.stats`](https://docs.scipy.org/doc/scipy/reference/tutorial/stats/discrete.html). Thus, we have the following functions: \n",
    "\n",
    "### Uniform\n",
    "[`scipy.stats.uniform`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.uniform.html#scipy.stats.uniform) provides functions for Uniform continuous random variables. \n",
    "* To evaluate the $Unif(a, b)$ PDF at $x$, we use `uniform.pdf(x, a, b)`. \n",
    "* For the CDF, we use `uniform.cdf(x, a, b)`. \n",
    "* To generate $n$ realizations from the $Unif(a, b)$ distribution, we use `uniform.rvs(a, b, size=n)`. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "PDF of Unif(0, 4) evaluated at 3 is 0.25\n",
      "\n",
      "CDF of Unif(0, 4) evaluated at 3 is 0.75\n",
      "\n",
      "Generating 10 realizations from Unif(0, 4):\n",
      "[3.24085002 2.48793265 0.69337744 3.98212677 3.53906749 0.19890551\n",
      " 3.30252222 2.37991343 1.42342695 1.92222338]\n"
     ]
    }
   ],
   "source": [
    "# seed the random number generator\n",
    "np.random.seed(1597)\n",
    "\n",
    "from scipy.stats import uniform\n",
    "\n",
    "# to learn more about scipy.stats,uniform un-comment ouf the following line\n",
    "#print(uniform.__doc__)\n",
    "\n",
    "a = 0\n",
    "b = 4\n",
    "x = 3\n",
    "n = 10\n",
    "\n",
    "print('PDF of Unif({}, {}) evaluated at {} is {}\\n'.format(a, b, x, uniform.pdf(x, a, b)))\n",
    "\n",
    "print('CDF of Unif({}, {}) evaluated at {} is {}\\n'.format(a, b, x, uniform.cdf(x, a, b)))\n",
    "\n",
    "print('Generating {} realizations from Unif({}, {}):\\n{}'.format(n, a, b, uniform.rvs(a, b, size=n)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Normal\n",
    "[`scipy.stats.norm`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.norm.html#scipy.stats.norm) provides functions for normal continuous random variables. \n",
    "\n",
    "* To evaluate the $N(\\mu, \\sigma^2)$ PDF at $x$, we use `norm.pdf(x, loc, scale)`, where the parameter `loc` corresponds to $\\mu$ and `scale` corresponds to standard deviation $\\sigma$ (and **not** variance $\\sigma^2$).\n",
    "* For the CDF, we use `norm.cdf(x, loc, scale)`. \n",
    "* To generate $n$ realizations from the $N(\\mu, \\sigma^2)$ distribution, we use `norm.rvs(loc, scale, size=n)`. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "PDF of N(0.0, 2.0) evaluated at 1.5 is 0.15056871607740221\n",
      "\n",
      "CDF of N(0.0, 2.0) evaluated at 1.5 is 0.7733726476231317\n",
      "\n",
      "Generating 10 realizations from N(0.0, 2.0):\n",
      "[-4.56890553  1.4163793  -0.72812069  1.80610792 -1.5542524   0.4551897\n",
      " -1.82794816  1.58959742 -3.26902153 -2.71002864]\n"
     ]
    }
   ],
   "source": [
    "np.random.seed(2584)\n",
    "\n",
    "from scipy.stats import norm\n",
    "\n",
    "# to learn more about scipy.stats,norm un-comment ouf the following line\n",
    "#print(norm.__doc__)\n",
    "\n",
    "mu = 0.0\n",
    "sigma = 2.0\n",
    "x = 1.5\n",
    "\n",
    "print('PDF of N({}, {}) evaluated at {} is {}\\n'.format(mu, sigma, x, norm.pdf(x, mu, sigma)))\n",
    "\n",
    "print('CDF of N({}, {}) evaluated at {} is {}\\n'.format(mu, sigma, x, norm.cdf(x, mu, sigma)))\n",
    "\n",
    "print('Generating {} realizations from N({}, {}):\\n{}'.format(n, mu, sigma, norm.rvs(mu, sigma, size=n)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "&#x2623; 5.9.1 (Normal parameters in `scipy.stats.norm`). Note that we have to input the standard deviation for the `scale` parameter value, not the variance! For example, to get the $N(10, 3)$ CDF at 12, we use`norm.cdf(12, 10, np.sqrt(3))`. Ignoring this is a common, disastrous coding error."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "N(10, 1.7320508075688772) CDF at 12 is 0.8758934605050381\n"
     ]
    }
   ],
   "source": [
    "loc = 10\n",
    "var = 3\n",
    "scale = np.sqrt(var)\n",
    "x = 12\n",
    "\n",
    "ans = norm.cdf(x, loc, scale)\n",
    "\n",
    "print('N({}, {}) CDF at {} is {}'.format(loc, scale, x, ans))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exponential\n",
    "[`scipy.stats.expon`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.expon.html#scipy.stats.expon) provides functions for exponential continuous random variables. The probability density function for $Expo(\\lambda)$ is:\n",
    "\n",
    "\\begin{align}\n",
    "  f(x) &= e^{-x}\n",
    "\\end{align}\n",
    "\n",
    "\n",
    "The probability density is defined in the “standardized” form. To shift and/or scale the distribution use the `loc` and `scale` parameters. Specifically, `expon.pdf(x, loc, scale)` is identically equivalent to `expon.pdf(y) / scale` with `y = (x - loc) / scale`.\n",
    "\n",
    "* To evaluate the $Expo(\\lambda)$ PDF at $x$, we use `expon.pdf(x, scale=1/lambda)`, where the $\\lambda$ corresponds to `scale=1/lambd`.\n",
    "* For the CDF, we use `expon.cdf(x, scale=1/lambd)`. \n",
    "* To generate $n$ realizations from the $Expo(\\lambda)$ distribution, we use `expon.rvs(scale=1/lambd, size=n)`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "PDF of Expo(2.1) evaluated at 5 is 5.782654363446904e-05\n",
      "\n",
      "CDF of Expo(2.1) evaluated at 5 is 0.9999724635506503\n",
      "\n",
      "Generating 10 realizations from Expo(2.1):\n",
      "[0.44907946 0.77554149 1.469014   0.15728342 0.92258566 0.54294386\n",
      " 0.80347768 0.23157762 0.34759354 1.03891708]\n",
      "\n"
     ]
    }
   ],
   "source": [
    "np.random.seed(4181)\n",
    "\n",
    "from scipy.stats import expon\n",
    "\n",
    "# to learn more about scipy.stats,expon un-comment ouf the following line\n",
    "#print(expon.__doc__)\n",
    "\n",
    "lambd = 2.1\n",
    "x = 5\n",
    "\n",
    "expon.pdf(x, scale=1/lambd)\n",
    "\n",
    "print('PDF of Expo({}) evaluated at {} is {}\\n'.format(lambd, x, expon.pdf(x, scale=1/lambd)))\n",
    "\n",
    "print('CDF of Expo({}) evaluated at {} is {}\\n'.format(lambd, x, expon.cdf(x, scale=1/lambd)))\n",
    "\n",
    "print('Generating {} realizations from Expo({}):\\n{}\\n'.format(n, lambd, expon.rvs(scale=1/lambd, size=n)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Due to the importance of location-scale transformations for continuous distributions, SciPy has default parameter settings for each of these three families. The default for the Uniform [`scipy.stats.uniform`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.uniform.html#scipy.stats.uniform) is `loc=0, scale=1`, the default for the Normal [`scipy.stats.norm`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.norm.html#scipy.stats.norm) is `loc=0, scale=1`, and the default for the Exponential [`scipy.stats.expon`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.expon.html#scipy.stats.expon) is `loc=0, scale=1`.\n",
    "\n",
    "For example, `uniform.pdf(0.5)`, with no additional inputs, evaluates the $Unif(0, 1)$ PDF at 0.5."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "uniform.pdf(0.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`norm.rvs(size=10)`, with no additional inputs, generates 10 realizations from the $N (0, 1)$ distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.79839927, -0.69769073,  0.34055804, -0.22043838,  0.79782274,\n",
       "       -2.60292799, -1.53866926,  0.93084378,  0.81590117, -0.42043989])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "norm.rvs(size=10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This means there are two ways to generate a $N(\\mu, \\sigma^2)$ random variable in SciPy. After choosing our values of $\\mu$ and $\\sigma$,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "mu = 1\n",
    "sigma = 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "we can do either of the following:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "norm.rvs(mu, sigma, size=1) = [1.99342831]\n",
      "mu + sigma * norm.rvs(size=1) = [1.99342831]\n"
     ]
    }
   ],
   "source": [
    "ans1 = norm.rvs(mu, sigma, size=1, random_state=42)\n",
    "ans2 = mu + sigma * norm.rvs(size=1, random_state=42)\n",
    "\n",
    "print('norm.rvs(mu, sigma, size=1) = {}'.format(ans1))\n",
    "print('mu + sigma * norm.rvs(size=1) = {}'.format(ans2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Either way, we end up generating a draw from the $N(\\mu, \\sigma^2)$ distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plots in Matplotlib\n",
    "\n",
    "Plotting in Python within a Jupyter notebook is done in [Matplotlib](https://matplotlib.org/). \n",
    "\n",
    "First, we import `matplotlib.pyplot` module. To cut down on typing, we can alias that imported `matplotlib.pyplot` module reference as `plt` for short. We then invoke the `%matplotlib inline%` Jupyter notebook magic command in order to create Matplotlib graphs right in your notebook."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here is how we can create a plot of the standard Normal PDF from −3 to 3 using Matplotlib in Jupyter notebook.\n",
    "\n",
    "[`numpy.linspace`](https://docs.scipy.org/doc/numpy-1.15.0/reference/generated/numpy.linspace.html) creates an evenly-spaced numerical sequence, ranging from `start` to `stop`. The third function parameter `num` lets you specify how many numbers in the sequence to generate. \n",
    "\n",
    "We use `numpy.linspace(-3, 3, 1000)` to generate 1000 numbers ranging from -3 to 3 as our `x` values in the graph; with 1000 values in the sequence, the curve looks very smooth. If we were to choose `num=20`, the piecewise linearity would become very apparent."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.linspace(-3, 3, 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can then call `norm.pdf(x)` to generate the `y` values in the graph, and pass both the `x` and `y` values to [`matplotlib.pyplot.plot`](https://matplotlib.org/api/_as_gen/matplotlib.pyplot.plot.html). This will render a graph of $N(0,1)$ from -3 to 3. `pyplot.show` will display the graph."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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p+yFV5en30+gdGWZP35o2ISL8x9QLyD9TwUufH3I6jqnHSt8Pfbgnlx3Zp3jg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\nQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, norm.pdf(x))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Alternately, we could \"freeze\" an instance of the `scipy.stats.norm` distribution object to fix the shape, location and scale parameters. This might be useful if you are working with multiple distributions with differing shape, location and scale, and wanted to use these distributions throughout your notebook."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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z5vIZZTqav1Mrgr61tTXp6ek68FcjUkrS09Or1O4ZfzaL3w6fYlwXH5xsFDqO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\nAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rv1 = norm(0, 1)\n",
    "rv2 = norm(0.5, 1)\n",
    "rv3 = norm(1, 1)\n",
    "\n",
    "plt.plot(x, rv1.pdf(x), label='norm(0, 1)')\n",
    "plt.plot(x, rv2.pdf(x), label='norm(0.5, 1)')\n",
    "plt.plot(x, rv3.pdf(x), label='norm(1, 1)')\n",
    "\n",
    "plt.legend()\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also set the axis labels and plot title with [`pyplot.xlabel`](https://matplotlib.org/api/_as_gen/matplotlib.pyplot.xlabel.html), [`pyplot.ylabel`](https://matplotlib.org/api/_as_gen/matplotlib.pyplot.ylabel.html), and [`pyplot.title`](https://matplotlib.org/api/_as_gen/matplotlib.pyplot.title.html) functions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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1kKWqI4BXgUedfbsAD+GZM2Ms8JCIWA9omDhcWcOrKwu4eERvu1w2jFw0ohfd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\nGGOMT/8fkODAHeiH40wAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rv = norm()\n",
    "plt.plot(x, rv.pdf(x))\n",
    "\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('norm.pdf(x)')\n",
    "plt.title('Standard Normal PDF')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The axis limits can be set manually with [`pyplot.xlim`](https://matplotlib.org/api/_as_gen/matplotlib.pyplot.xlim.html) and [`pyplot.ylim`](https://matplotlib.org/api/_as_gen/matplotlib.pyplot.ylim.html) If, for example, you wanted the vertical axis to range from 0 to 1, you would follow the call to `pyplot.plot` with `pyplot.ylim([0, 1])`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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Vch8B/LHy+zeBJ83MXC3M+XRIqcf3+7Wq+NcwsfIrIZbkxBgaJcbSMCGGBgkx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\nRUR8SOUuIuJDKncRER9SuYuI+ND/A6RjAUp9jdDwAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, rv.pdf(x))\n",
    "plt.ylim([0, 1])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, to change the color of the plot, add `color=\"orange\"` or `color=\"green\"`, or whatever your favorite color is! (see [Specifying Colors](https://matplotlib.org/users/colors.html) in the Matplotlib documentation)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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wG768Ctr2h8F3+U5jTHDJqXDcc25pRevmaXRW9OPFwtthzzeuWye5ue80xtRN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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, rv.pdf(x), color='orange')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Universality with Logistic\n",
    "\n",
    "We proved in Example 5.3.4 that for $U \\sim Unif(0, 1)$, the r.v.  $log\\left(\\frac{U}{1−U}\\right)$ follows a Logistic distribution. In SciPy and `uniform.rvs`, we can simply generate a large number of $Unif(0, 1)$ realizations and transform them."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 2.0197625 , -0.72108035, -1.86938026, ..., -1.2638552 ,\n",
       "        1.8532956 ,  1.46624836])"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u = uniform.rvs(size=10**4)\n",
    "x = np.log(u/(1-u))\n",
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now `x` contains 10<sup>4</sup> realizations from the distribution of $log\\left(\\frac{U}{1−U}\\right)$. We can visualize them with a histogram, using [`matplotlib.hist`](https://matplotlib.org/api/_as_gen/matplotlib.pyplot.hist.html). The histogram resembles a Logistic PDF, which is reassuring. To control how fine-grained the histogram is, we can set the number of bins in the histogram via the `bins` parameter (the 2nd parameter passed into `pyplot.hist`: `hist(x, 100)` produces a finer histogram, while `hist(x, 10)` produces a coarser histogram.\n",
    "\n",
    "To illustrate, we will generate two graphs side-by-side with [`pyplot.figure`](https://matplotlib.org/api/_as_gen/matplotlib.pyplot.figure.html), and use a [`pyplot.subplot`](https://matplotlib.org/api/_as_gen/matplotlib.pyplot.subplot.html) for each graph."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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4myRJkqROM/RIkiRJ6jQPb9OCm3uJY0mSJGmSnOmRJEmS1GmGHkmSJEmd5uFt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\n",
      "text/plain": [
       "<Figure size 1008x432 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(14, 6))\n",
    "\n",
    "ax1 = fig.add_subplot(121)\n",
    "ax1.hist(x, 100)\n",
    "ax1.set_title('bins=100')\n",
    "\n",
    "ax2 = fig.add_subplot(122)\n",
    "ax2.hist(x, 10)\n",
    "ax2.set_title('bins=10')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Poisson process simulation\n",
    "\n",
    "To simulate $n$ arrivals in a Poisson process with rate $\\lambda$, we first generate the interarrival times as i.i.d. Exponentials and store them in variable `x`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.17627964 0.01902565 0.02146015 0.19802154 0.16173354 0.02476036\n",
      " 0.02394206 0.11207592 0.00051518 0.09787187 0.06122226 0.07313865\n",
      " 0.07720919 0.15936267 0.00195037 0.01965782 0.00122744 0.03186262\n",
      " 0.13671518 0.29913587 0.05325137 0.1329256  0.18825129 0.31339715\n",
      " 0.07390052 0.2792474  0.19905106 0.1237123  0.12739159 0.05063309\n",
      " 0.00787248 0.20327465 0.28319828 0.0039541  0.0084741  0.01842843\n",
      " 0.00354277 0.08189181 0.07338873 0.03824727 0.02011739 0.0211869\n",
      " 0.02824106 0.07150573 0.24463567 0.35755971 0.01220348 0.00555063\n",
      " 0.05732313 0.20553653]\n"
     ]
    }
   ],
   "source": [
    "n = 50\n",
    "lambd = 10\n",
    "x = expon.rvs(scale=1/lambd, size=n)\n",
    "print(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then we convert the interarrival times into arrival times using the [`numpy.cumsum`](https://docs.scipy.org/doc/numpy-1.15.0/reference/generated/numpy.cumsum.html) function, which stands for \"cumulative sum\"."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.17627964 0.19530529 0.21676545 0.41478698 0.57652053 0.60128088\n",
      " 0.62522295 0.73729887 0.73781405 0.83568592 0.89690818 0.97004683\n",
      " 1.04725602 1.20661869 1.20856907 1.22822689 1.22945433 1.26131695\n",
      " 1.39803213 1.697168   1.75041936 1.88334496 2.07159625 2.38499339\n",
      " 2.45889391 2.73814132 2.93719238 3.06090468 3.18829627 3.23892936\n",
      " 3.24680184 3.4500765  3.73327478 3.73722888 3.74570298 3.76413141\n",
      " 3.76767418 3.849566   3.92295473 3.961202   3.98131938 4.00250628\n",
      " 4.03074735 4.10225307 4.34688874 4.70444845 4.71665193 4.72220256\n",
      " 4.77952569 4.98506222]\n"
     ]
    }
   ],
   "source": [
    "t = np.cumsum(x)\n",
    "print(t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The array `t` now contains all the simulated arrival times."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----\n",
    "\n",
    "Joseph K. Blitzstein and Jessica Hwang, Harvard University and Stanford University, &copy; 2019 by Taylor and Francis Group, LLC"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.3"
  },
  "notebook_info": {
   "author": "Joseph K. Blitzstein, Jessica Hwang",
   "chapter": "5. Continuous Random Variables",
   "creator": "buruzaemon",
   "section": "5.9",
   "title": "Introduction to Probability, 1st Edition"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}