{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Chapter 13: Poisson Processes\n", " \n", "This Jupyter notebook is the Python equivalent of the R code in section 13.5 R, pp. 534 - 536, [Introduction to Probability, 2nd 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 matplotlib.pyplot as plt\n", "import numpy as np\n", "\n", "%matplotlib inline" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 1D Poisson process\n", "\n", "In Chapter 5, we discussed how to simulate a specified number of arrivals from a one-dimensional Poisson process by using the fact that the interarrival times are i.i.d. Exponentials. In this chapter, Story 13.2.3 tells us how to simulate a Poisson process within a specified interval $(0, \\left.L\\right]$. We first generate the number of arrivals $N(L)$, which is distributed $Pois(\\lambda L)$. Conditional on $N(L) = n$, the arrival times are distributed as the order statistics of $n$ i.i.d. $Unif(0, L)$ r.v.s. The following code simulates arrivals from a Poisson process with rate 10 in the interval $(0, \\left. 5\\right]$:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "np.random.seed(2971215073)\n", "\n", "from scipy.stats import poisson, uniform\n", "\n", "L = 5 \n", "lambd = 10 \n", "n = poisson.rvs(lambd*L, size=1)[0]\n", "t = np.sort(uniform.rvs(loc=0, scale=(L-0), size=n))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To visualize the Poisson process we have generated, we can plot the cumulative number of arrivals $N(t)$ as a function of $t$:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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rqmrNypUruw5HPQ7adwVHHLRP12FIkiR1psuE+oYkqwDa/xs7jEWSJEmaly7H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\nABNqSZIkaQD/H/5fZfVvPr9+AAAAAElFTkSuQmCC\n", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.figure(figsize=(12, 4))\n", "\n", "nt = np.arange(1, n+1)\n", "plt.plot(t, nt, linestyle='-', drawstyle='steps')\n", "\n", "plt.xlabel(r'$t$')\n", "plt.xlim([0,5])\n", "plt.ylabel(r'$N(t)$')\n", "plt.title(r'Number of arrivals in Poisson process, rate=10, in interval (0,5]')\n", "\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This produces a staircase plot as in Figure 13.8." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Thinning\n", "\n", "The following code starts with an array of arrival times `t` and the corresponding number of arrivals `n`, generated according to the previous section. For each arrival, we flip a coin with probability `p` of Heads; these coin tosses are stored in the array `y`. Finally, the arrivals for which the coin landed Heads are labeled as type-1; the rest are labeled as type-2. The resulting arrays of arrival times $t_1$ and $t_2$ are realizations of independent Poisson processes, by Theorem 13.2.13. To perform this with $p = 0.3$, we can execute" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "np.random.seed(1)\n", "\n", "from scipy.stats import binom\n", "\n", "p = 0.3 \n", "y = binom.rvs(1, p, size=n)\n", "\n", "t1 = t[np.where(y==1)]\n", "t2 = t[np.where(y==0)]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As before, we can plot the number of arrivals in each Poisson process, $N_{1}(t)$ and $N_{2}(t)$, as a function of $t$." ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/png": 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M4xteOo/iLlvHlr/PbXadkqSpNxmfAZLUzcZMqCPiRcDfAa8BNgEuAr4LLCwb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\nRU5ErkJggg==\n", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "_, (ax1, ax2) = plt.subplots(1, 2, sharey=True, figsize=(12, 4))\n", "\n", "# graph for t1\n", "nt1 = np.arange(1, t1.size+1)\n", "ax1.plot(t1, nt1, linestyle='-', drawstyle='steps', color='#ef8a62', label='rate=3')\n", "ax1.set_xlabel(r'$t$')\n", "ax1.set_xlim([0,5])\n", "ax1.set_ylabel(r'$N_{1}(t)$')\n", "ax1.legend()\n", "\n", "# graph for t2\n", "nt2 = np.arange(1, t2.size+1)\n", "ax2.plot(t2, nt2, linestyle='-', drawstyle='steps', color='#67a9cf', label='rate=7')\n", "ax2.set_xlabel(r'$t$')\n", "ax2.set_xlim([0,5])\n", "ax2.set_ylabel(r'$N_{2}(t)$')\n", "ax2.legend()\n", "\n", "plt.suptitle('Number of arrivals in two independent Poisson processes')\n", "\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 2D Poisson process \n", "\n", "Simulating a 2D Poisson process is nearly as straightforward as simulating a 1D Poisson process. In the square $(0, \\left. L\\right] \\times (0, \\left. L \\right]$, the number of arrivals is distributed $Pois(\\lambda L^2)$. Conditional on the number of arrivals, the locations of the arrivals are Uniform over the square, so by Example 7.1.22, their horizontal and vertical coordinates can be generated independently:" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [], "source": [ "np.random.seed(1)\n", "\n", "L = 5 \n", "lambd = 10 \n", "n = poisson.rvs(lambd*(L**2), size=1)[0]\n", "x = uniform.rvs(loc=0, scale=(L-0), size=n)\n", "y = uniform.rvs(loc=0, scale=(L-0), size=n)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can then easily plot the locations of the arrivals as in Figure 13.7:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "image/png": 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wR0A7oZh0EDhY5yFcyiAbNoj5YagnElrDOosXzsSydbvGBKN5eYvYBcQXblj7BG68bwiX\nLToOl59+om1znCDm8CVDPWQCrWue7D8ADtZ5DFMrJ9II6TTX8+b3yauYfPQisUirUGR5QvP7+6Lx\njnymtXc2r38Ke2tIQj2XrNgCALhs0XG4ZdNu3LJpN266YI5ly9yDHn8E+L5KKRkPUytJWRjjjwCm\n/5EYaNTzbw3/19jYx7z+KdHU8yIxfgo/ISQYOLjLwV1CSEQwpJkfCj8pDCfKEBfh2Ed+ohd+ilhx\nOFGmN6qoa6yv7fnogv7MTLUY4vtFiV74KWLF6WXmKAWrmrrG+kqqIHrh5/T33ij6EOwyguVTo9HJ\n1irqGusrqYLohR8oLmKh0YuwNk8Iu/n+J3Hz/cMTPm/+fRnB8snL7WZrL3Wt9frM7+/DghOmRltf\nXcQn5wSg8APg9PeiwtqaPXH56cfjurt3jIl/u9/32sD65OV2s7WXutZ6fW6+fxh3bXsa5wwc7XR9\n9U0My+CTcwKAT+Dik4USGue9dM2Oruef9fSu5RuG9KSr7+34+yLHyMKn571m2VqmrjW+u2TlVp1x\nxSpdvmGo8D5ME9u9VbZ+lwUFnsBlXeSzNpPCb/oRhC5TVlg7/b6sCNi+qYrQztayda1RvktWbu15\nH6bx6bpVgU3nhMJPClP2Bu32+zKi55Pn2GrblXds11nXrB5nay9C7bOA+tRTK4Pta0ThJ4Woyhuv\nS5h96pW12rpxaERnXbNar7xj+9jromXjU8PXim0xNIUL14jCTwpRVlh9EmYblBU/X8vXBTE0hQvX\nqIjwc5E2Qgzgy9OyqlzJlavCmoWLtBHiED6lC1eZlsglFNyFwo+48o2JWXxbMdKnOROkdyj88HDy\nBfEGH1eMjH0me6/45EBS+EEvxzd8usF8DHf4FJpyCZ8cSAp/iikvxyfRchWfbjDf8C005RI+OZAU\n/hRTXg5Fqzw+3WBFse0Y+BiacglfwmQUfpj1ckIWLZP4coMVpdUxuOrOQVyyYss4x6BsQ9CpcfEx\nNOUSvoTJKPww7+WEKlom8eUG60bWksuLF87ERbduxg1rn8CqwWfGfb+KHqKJXqftnosNvAqT5Z3p\nZXILfeZuLNPY6yKkGaHtzmXJym1j69vUUV/qroNVXCMXZsMWwba94JIN7hKSaNnC9g1WNa0ivHzD\n0ARRrmOhs7oXT6tq4T/eK/koIvxBLtng8lRxl20rg+vn5bp9jSUdzhmYhvU7R8dCj5uGR3HJii0A\ngN+fPwO3PfhUJWNCjbDE+W8/prJ9ZlF2qQpTdoZA9Es2uJw5E+rgmctlDrhtX/N4xepHn8XihTMn\n1JGzZh9VWdzYVCy6inEYjofVRN6ugcmtilAP4+jmcb3MXbSvWzijjrCWiVBZVWEaF6+Zq4Ax/oRY\nHgDhEq6XuWv2hTZe0aCK82KMvxhFhD/IGD/A2KANXC9z1+0j43F9XMY1isT4rXv3WVtZj5+egnlc\nL3OT9oXqxceAz9cOBTz+IAd3Oe3cPK6XuUn7ehlIjnHCk4u4nARQJcGGegixSdGwUnOmTSONk8t5\n2MHXkGD06ZyE2KZoGiLXcMpP3b2jGFJIKfyE1EAvOewxCE4V1B2OCWUdqE5Mtm0AIa7SmlXy2fXD\nmHQQsP8AxrJKsrJMWsM08/qn5PLgWwVnXv8Uin8Gzb2jqsMxvV4736DHT0gbWj3LSQcB1929A5PS\nu6adp9nLQLJXKzs6QF29I9eTFKqCg7ttYA4xASYO9C1eOBPL1u0a52kO7tlXuq6EUt9MnYevA7B1\n4uTgrohMEpFtIrLK1DHLEEtaF+lMq2f5h6f1T/A0q6groazhNHv64bjo1s24+f5koLVRFpMOQmWD\nr1X1jmJOoTUZ6vk4gMcNHq8URbMsqqxEMVdI12iNu998//CEgb9GXbno1s344y8/MiEtM6brNr+/\nD5effjyuu3vHWFk0eklVOU1VhWOidu7yzvQqswGYDuA/ACwCsKrb911ajz/v2i5Vzgx1fRZsLLSW\n+/INQzrjilW6fMNQ5ueNh6csWbkt8/OYaJTFbyzb6HQZVLkInO1Zvygwc9dUVs+nAfwpgMPafUFE\nLgZwMQAcc8wxhszqTJEsiyozDerMWiD5afUs9x8APvWBk7D/QPJ5q6e5fucIzhk4GndtexqAjltX\nPyY2DY9i/c4RnDrjDXho9w9xzsA0Z8ugOZR32aLjStnZ6EFkTcJzjrwtRK8bgLMA/J/0/4XwxOPv\n1euucvVH11aSJNlM9Py3pp7/VsuWmadRFo2niC1ZuXVcL8k1ql722eYy0nBsrZ53AvigiOwGsBLA\nIhG5zcBxS9FrSl5VEz9imEQSCs11JfF2R3HOwDSsfvRZp66bibGjwT37xmL6nzlvAP/4WwP41AdO\nwg1rv+NUWQC9DxJ3KkdvJuHlbSGq2OCRx18UxviJ69fNlH22Y9156dXOTuXoi8dvNI9fRBYC+KSq\nntXpey7k8RelyvzlUHK6Y8OH68b892rIKkcAVhfaK5LHzwlchERG2Qegt+JDg1cHreVouxycnMBF\niC04L+JV6hg7CjEfvludySpHnybhUfhJ8IQgTFU0XnWtBxTiktKd6kwI6ypR+EnwhCBMVTRedS5A\n5k02S0461ZkQFnJjjJ9EQ9WxbdO4PDDrsm1l8KnOMMZPSAshzItw1asOIfSRRQh1ph0U/gCIefAy\nz7mHIkyuClEIoY9WQqkz7YhC+EMXxhAGL3slz7m7Ikxl6qHLQuRTNks7Wq9NYwZyo470Wmec1Z68\nM71MblXP3C06Y9GXmYfN2JwxaBtfzr3MzFkf66RP1DWr2eRsbhSYuWtd5LO2OpZsKCIOrk+9b0fM\ni7r5cu6+NFK+UUXDWNe1MXXNiwh/FKEeoNjAmI/pf67Gf03g07m7OkDrO1WEO+u6Ni5e82iEv6g4\nuHix2uFy/LdufDt3nxopn6jCWavr2jh5zfN2DUxutmP8zd/xoUsec/zXp3P3NYRogqquY68hP8b4\nHdiqFv6ilYo3KKmDqhspnxq9blRxz5Vx1jqVZZlyNnmNKPwlCemGIuESmoNSRrirKouse3/5hiE9\n6ep7nS/nIsLPJRsI8ZjQlkrodYmEqpZEbl1Dv/G68VQxl8u5yJINph62TgipgSofFm6b1kHQef1T\ncp9PlrjP7+8rXB7Ng8StIv/8T14JopyBiLJ6CGng7GzKHnAyY6QHXMrOysroC6WcG1D4SXSEssSF\nS2JZFleW1QAmNqY33z8cTDk3YIyfRIlrsfFeYtS2H/UXIlkx/otu3YzLTz8e+w9grLwb5Tx7+uHO\nlDeXZSakC65N0OulFxLC4miukdXz+NyFc8dEv3GN5vf3jb32racI0OMnFeOLF+qax++qTWQ8Ll8j\nevzEGj7Ez12NjbvWCyETCeUaUfhJpfiwwF23gcS6sn667Te0zJEQCeYa5Z3pZXKzPXOXlMeXZZKz\nsLFuS2izcEOk12tkaiUAcFlmYhPfvaK6ei3z+/twxtuOxCUrtozbLwAs37Br3DFanwAF+DvXIBR6\nTTl1MfzJwV1SKe2mvLsW7slDr8sHdGLT8Cj+x60P46WXD+CyRcdh5IWfYtXgM7jpgjlj5bNpeBRf\n274Xax57NohyJGYGhblkA7FGJ6/IJ8Eqs3xANw6elHS0l9+/CweJYNJBMu64DYE/++RpzmaQkGK4\ntrRGEKGekKbg+04IueV1Zf009nvTBXNw8Wkz8dLLB3BAFZe9+7jMsFIoGSTEvfBnEMLvYgyN+Etd\nywc09gtgTAQOnnQQdo28mCnwrolFKJh2FJ1MH847Cmxy6yWrx6cnZpF4ycoMmXXNap11zepxdZdZ\nPvVhumxdzOoJanC3jsE4QqqkdWbzpuFRXLJiC86afRSuP3f2mHd4xtuOxNknT3N+BnQdmJj97fIM\n3F6JcuYuu8XEB1rHQAb37MNNF8zB9efOBvBqWOnYKYd4P1bSKyZCt7GPnwTh8YeUQkiIDVxbY6lu\nj5wefwC4tJY3IT7iWoJEnR65k4OthgnC4yeElMclL7hOW1zr3VRFdB4/8QPOt3AbV+LedXvkIcw1\nKQuFnxjDtXCCr9TVgLqSIMHQbf0w1EOM4lI4wVfqSGZggoT/MNRDnMWVcILP1LF6KL3suOAibcQo\ndS5+FhNVL/qVFd9urBdEwoMef41wMHM8TKOrDlfi8cRPghV+F0SXg5njYTihGupsQF24b0j9BCv8\nLoiuD8+fNQnT6KqhzgbUhfuG1E/QWT2uZJBw8bhihDrBxmWay7xx3yw4YSpWP/p9fO7CuVbuG9aD\nYjCrJ6XqDJJeusGMxRbHttcZY7ijuczn9/dhwQlT8dVtT+N9s95krYdqux6ETNDCX7XoFq2IHMzs\nDdshMl8Ep8oGqrnM//jL23DXtqdxzsDRWL9zxFp9tV0Pgibvwv0mt14exNJKXQ9bKPLAl6IPYDD1\nwAZfWLpmhx57xSpdumaH8WP78GCfOur4kpVb9dgrVumSlVsr22dZbNQDH+9FFHgQS7Aef10DYEXC\nR0UHM33xNE1gO0Tmw0Szqj3iTcOjWP3oszhnYBrW7xwdC/vYzLyyVQ+CvxfzthAmtyo8/rqo2xP0\nwdOsGxceO1j0Otj0EKvwiF0oc9ds8u1eRAGP37rIZ22uCr+pimgzxOECtrvZvVxnWyJVlTjZLvMs\nXLDJp3vRKeEH8BoADwHYDuAxAH/Z7TeuCr+Jiuibl2EKkyLQ67FMXzvbHnHo+HYvuib8AuDQ9P+D\nATwIYF6n37gq/HXDG7k9vpSNSQ/RBY84VHypb80UEX6jE7hE5HUAHgCwWFUfbPe9WJdl5oSVzrgy\nIa8drttH8uPjvVhkApcR4ReRSQC2ADgOwD+p6hWdvh+r8JPuuDoLOu969j4Iig82kok4N3NXVfer\n6ikApgM4VURmtX5HRC4Wkc0isnlkZMSEWcQzbKd4diJv+rAPaYI+2EjKYXytHhG5BsCLqvoP7b5D\nj5+0EtITonwICflgY6+E2qNxyuMXkakickT6/2sBvAfAjrqPS8IipCWdfZkc5rqNvcIejZlQz1EA\nviEigwAeBvB1VV1l4LgE4Sw4FtKSzi6HrBr4YGOvcA0gA8KvqoOqOqCqs1V1lqpeW/cxyavQu3GL\nrIX7Lrp1M26+f3jC92w1zjEsLhhyjyYPwa7VQxJ69W586in4ZGtWyOry04/HDWu/40zjHFJYrR0h\n92hykTfh3+QWwwQu05Nvik4s8mkCi0+2tsO3WaI+k6e++Dg5Dlyd031MhmB68W58ioP6ZGs7Yg89\nmCRPjyb4EGneFsLkFoPHr2rGyyvrDfu0SJVPtrZCj989fLsmiN3j9yXma8LLKxOv9SkO6pOtrcQw\nmOojIffCghR+X7ppJsSq1zRIn8TIJ1uziGEw1Ud8dia6krdrYHKr8tGLrnbTXB+Q9GlwyydbiR+4\nfn9mAVdX58xLVUs2uLqgFxDutHFCQsDH+9O51TmLUoXwN681cvP9T+Ly04/HH57WP+5zly8iIYQU\nwam1emzQGvO9/PTjcd3dO8ZmR7oa8yckFHxJsMhLaOcTpPAv37ALixfOHOum/eFp/Tjv7W/G3937\nhLd53lmEVhljJcTr6EuCRV5CO58ghf/id83EsnW7xl2kex99FmeffFRQqVmhVcZYCfE6hjCprpnQ\nzieKGP9tDz6FxQuTxsDG+uJ1DhSFvG56TIR6HV1OsOgFl88n+hg/MH7yxYITpmLZul3W8rzr9OhC\nnmQSEyFex9Dy4EM6n2CFv/kirRrcizNnHTlugszihTOxfMMuI7bU2U0MqTLGTGjX0fdJda3UcT5W\nx3byJvyb3MpO4GqdbLF8w5DOuGKVLt8wlPm5KapeS8bEJBNOjqofHycLdSO0elPH+VR93VFgApd1\nkc/aygp/1kVavmFIT7r63spn8uatEHXMJDZxc4UoSq4RmkiShDzXtUpdiF7421HH6o15hNF38XR9\n+QviHkUas1Abvrz3fVW6VET4g43xt1JXDDVP/N73RbhCHHgk9VIkocF0Oqup2HoebbA2tpO3hTC5\nVe3xm/C4W1vtkLwYevzVE1L9aEeRemOyjpnugbfz6G3G+KPw+Ov2uLNa7VAm5YSWneEKodSPThTp\nKZrsVZqcjNXJo7caCcjbQpjcfHoCV6dWOwRPOQbP1BYh1I9OuOrxN6j7iW2mexbg4K45ugmjz48D\nJPUTaoiwiOjZSH4w0dCYvpYUJ8RtAAAXeUlEQVQUfkcI3aMj5ciqH75ngDVwOasnlDJupYjwd12r\nR0T+HcAnVHV7/YGnhKoexGKT5tj4/P6+Ca9J3HSqHwCCXLfHFXx8yEoeKn0Qi4j8MoB/APBdAJ9S\n1WfKm9iZEIQ/1MpFqqFb/XB5MTDiJrU8gUtEPgzgzwHcCeDvVfUnvZvYmRCEn5BeCXWlTlIvla/O\nKSIC4AkAywB8DMB3ROSC3k0khGTB9Fligq7CLyIPAHgawD8COBrAhQAWAjhVRJbXaRyZSIhPayKv\n4vssb+IHeTz+jwI4WlXfq6p/pqqrVHVIVT8G4LSa7SMtxDDxJ2Y+uqB/Qlhnfn+fc+NCdED8pqvw\nq+qj2n4g4AMV20O6ENoj4Ei1mBJkOiB+U2rJBlU18yQTMg4umkbaYUqQ6YD4TRRr9YSGjRX92LX3\ng6oEOc/1pgOSDxfvHQq/Z9jK+mDX3h+qEOQ81zu0x0XWhZP3Tt4pviY320s2uLxeik3buASFH1R1\nnTrtJ9RlD+rCxL0DrtVTjjoqtcuNSRG46JzbVF13213vUOqzSeq+d4oIP0M9GdQxcOVkdw/F4o++\ndO1djKmaosp5AJ2uty9pp67g3L2Tt4Uwudn2+BtU3UK7GCrJ6yH61LX3yVZXsV2GofQolq0b0uUb\nhsaV3fINQ3rS1fdWXpagx1+eOlrowT37sOCEqeMG3Wx7onl7Nz7NKGWqYXlsX29Xe8hFmT39cNyw\n9jtYvHDm2P2+bN0uXH768VbvndyLtJnE9iJtdS2pfPP9w7ju7h340MDRWL9zBIsXzsSydbucEKUQ\nV4MM8ZxioLFyKfDq8tS3bNqNs2YfhevPnW3ZuuKYWnSv8kXaYqMOb6fR0n/qAydh/c4RLDihD9fd\nvWPME7CJc/HHCgjxnGKh4e0DGEtLfXn/AZx98jTLlvWGi/MdJts2wEWyBqjm9/eVumDNjcnzP3kF\nN943hHMGpmH/gTKWlqe1NzOvf4r3oZEQzykmGo7WJSu24OX9B/Cagw/CwZP89VFbnZB5/VOs10N/\nS9MzGlkQzZVg/c5R6zFL27HcOvDxnGLORGrHy/sP4KWXD+Di02bipgvmeLk8tavLbEcp/LZuMhcr\nQYhpeT6eUyiDmVXxte17cfCkg8a8ZADjGm9fGsp2TsjyDbvs2p83/cfkVnc6Z2tq2pV3bNdZ16ye\nMDOx6tSxUFLUXCOUcnUx3dcGeVJJbaeblqUO+8GZu91pvslmXbN6nPD7Volix3cRaIYzo/M35L43\nlFXbX0T4o07nbE73awwA8jmnfhLCc2pDOAfT+J6yW6X9TOfMQetIOwDnUq7Iq3SL6bqYMlcEF8d/\nXMf3lF2b9kcp/Fk32SUrtuCWTbu9rUSh023w03cRcCUTyZdBU98bSuv2540JmdzqjvG3xhA3Do3o\nrGtW65V3bB977WPMMHTaxURDivHbxpey9H1Avw77wRh/MRpTxJvDA5uGRzG4Z5/TKYAxkhUT5fWr\nFo41+IlTMX4RebOIfENEHheRx0Tk43Ufsyg+5n3HSLtwDq9ftfg+XtKJMqGsor91OWxmIsb/CoBP\nqOpbAMwD8Eci8lYDxyUBYT0mGhG+j5d0osxEuaK/dXpSXt6YUFUbgP8H4L2dvuPKevzEHXyP6fpC\nlTF+V69Zmfz5or81OdcArq7HLyIzAAwAeNDkcYn/MJxjhiqzi1z1eMuEsor+1tWwmTHhF5FDAdwB\nYImq/nfG5xeLyGYR2TwyMmLKLEJIE1U2sK4+EKdMKKvob50Nm+XtGpTZABwMYA2Ay/N8n6EeQsLB\npWUoyoSyiv7WdGosXAr1iIgA+ByAx1X1hrqPVyUuj8q7AsuIdMI1j7dMKKvob12ZlJdJ3hai1w3A\nrwJQAIMAHkm393f6TVUef9nBJV8ms9iEZUTawbphFrjk8avqA6oqqjpbVU9Jt3vqPi5QfnDJpRil\nq561S2VE3MJpjzdygl6rpwpRcmVU3tUMCcCdMiJuwUwsdwla+IHyouRKjNJlz9qVMiKE5CN44S+b\nuuXSbFGTnnXe0JJrZUSISVwNwXYjaOEvK0quxShNetZ5Q0uulREhJnE5BNuJoFfnDGnVxuZGbH5/\n34TXdR6TqzQS0h5X7hOnVue0SUiDSzY8aw7aEp8xFYbx8T4JWvhDwkYjxkFb4jOmwjA+3ieTbRtA\n3KQ1lNR4GD3DPcQXmjPh6grD+Hqf0OMvia+j+t3goC0JgbrDMFXcJzY0hMJfEl9H9bvRHFpqVMzm\n0FIIjZtNQnUYXKPuMExWCHZwz74J93+na2tDQyj8JXF5YlVVhNq42YRlWj+25pgUvbY2NCTodE6T\nZD0EPCRcSVkLCZZpvdhM5+7l2pbVEKZzGsbHUf2i+Jiy5jos03qxmc5d9Nqa1hAKf0liWbIghsbN\nNCzTcClyba1oSN71m01uPj2Bq3XN/2XrhnT5hqFxa/678IDpMsS4rnrdDwqPsUxjoei1raquwaX1\n+EOntTs5e/rhWLZu19hATh2DdqYzQmJM7ax78DXGMo2FotfWRkiKg7s1UPegnY11e2KEg6/xEMK6\nXhzctUzdg3YxpJC6AAdf7WOqdxtbem2Qwm97ckzVg3ZZ5wMAb3nTYVGLUt3XmYOv9jElyLE5U0EK\nv83Wu44R+qzzuWTFFgw+vS9qUarzOseSreU6JgU5ph5esDF+W/HZumKFzedzy6bdAICbLpgTfYy/\nruscQsw3JExMkPR9TKdIjN966mbWVlU659I1O/TYK1bp0jU7Cv2u7lS+Xmmcz0eWf9NJ+2zR63Um\nftBIh1y6ZkdtKa8hpNeC6Zzl4rMuDvQ0n8+O7z8/4fPW9C/b4xymYBw+bEyF3KJLr83bQpjcynr8\nVbTeJryMorYUOZ8QPJhuxHCOseNq79tFUMDjDzLGX1V81pWF13o9H99jlt1gHJ6QVykS4w9S+Ksg\nFNE00XhRgAmxDydwlcSXVL5ucXxT8W8Xx0QIIe2h8Gfgy0BPJ8E12XjFNvmFEN9hqMdz2oWkbIRf\nXBkTISRGioR6JtdtDKmX5tmGly06bkzos8R9fn9fbV54a1hpXv8UevyEOApDPZ7jQh67L2MiJCxi\nmatSBxR+j3FFcH0ZEykKhcVtmFTQOxR+j3FFcLMeJDG4Z9+EG9A30aSwuI1LSQW+OQkUfo+x+TDp\nboQgmi4Ji0+YFEFXVtT0rb4HK/y+tcChEYpouiIsVRDiQ01Mj3G1K8NG79uX+h6s8PvWAodICKLp\nwuB5VYT2UBMbY1ydytCn+h6s8A/u2YfFC2eOq3yLF870fsDRJ3wXTVcGz6sitIea2Bjj6lSGPtX3\nYIV/9vTDsWzdLiw4YSpuvG8IC06YimXrdln1+It0tX0PVZkUzbrKypXB8yox5ZWaEME8Y1x11I2s\nMvTNSQhW+Of392Hxwpm4a9vTOHXGG3DXtqexeOFMq92vIl1t30NVJkWzrrJyefC8V0wIsksiWEfd\nyCpD75yEvOs3m9yqeAJXY232JSu36rFXrNIlK7c6sVZ7kXX+XXomgOuwrLpj6vkFrq2hX2XdcPkZ\nEOATuF6N8a/fOYrLFh2H9TtHnYjxF+lq+zRYZBuWVXdMeaWu9ZSqrBveefZtCFb4GzH+5u6m7Rg/\nUKyr7dNgkW1YVt1xTZBNwbqRQd6ugcmtilCPa93NxvHzdhNd7lK6BsuKtKPquuFyXUPsj150lSJL\nJfOpVvlhWZF21FE3XH06Hx+9SKKAgk9s4eKzJ/joRTIB3+cFZNGaqnfVnYO4ZMWWceM4vp8jcY8Q\nxgwo/JHg+7yALFpnUa4afGbc5yGcI3ELl+YolIGhnohwNTZZluZu97z+KUGeI3EDl8OLDPXkIMTQ\nRzdCzHVv7XYDCO4ciTuEkhIbrfDbCH3YbmxCiE02k9XtvmTFFtyyaXcw50hIHUQr/DbWi7cZZ/cx\nNtmtoWydRdngrNlHeXOOhNigduEXkc+LyHMi8mjdxyqK6dCHzYeT+DjVvFtD2drtHtyzDzddMAfX\nnzsbgB/nSIgNah/cFZF3AXgBwBdUdVae35ga3C072NnrQI+LOcCuEuqANCFV49TgrqpuAPCDuo9T\nlCpCH72EbkKLs9dNiAPShNjGmRi/iFwsIptFZPPIyEjtx6si9FE0dONjnN02bCiJ69hO2ugFZ4Rf\nVZer6lxVnTt16tTS++t2MapKyyrikfoYZ7fJpuFRXHTrZixeOHNcQ3nz/cNO31QkLnycHOmM8FeN\nqYtRxCMNJQfYFIN79uHy04/HsnW7sGl4dOypajes/Y7TNxWJC5tJG70y2bYBddF8MeoaGGwO3czv\n7xubNer6Rc/CxRmJjeO+bdrh467j5y6c6135krBp7vlftug45+uniXTOLwH4JoATRWSPiFxU9zEb\n1D0wGFLoxuXuKgd4iev4NhZVu8evqh+p+xjtaL0Y8/qnVCoaWZ7w/P6+sWO46EW3w0QPqVfqvo6E\nlMHHnn+wMX4XMmhc9qKzcNGzduE6EtIJH3v+wa7O6Yq37dMEJBdtdeU6EuI6fAKXY/gwU7e1u9r6\nmhDiNk7N3I0dXwZ9fOyu+kLRCT4+TggifkHhrxGf4tOcY1AfRcd6yo4NseEg3aDw1wi9aAIUn+BT\ndkKQb0kFxDyM8RNiiKJjPWXGhlwcqHeJEJMGGONPYZeXuELRsZ6yY0Mupua6ROy9oqCFP/aLWxQ2\nlPVw1Z2DuGTFlgmPiLzqzsHM71cxNuRCUoHL9SlvOM3lcyhD0MLv4+JJNmFD6QZlx4ZcSSpwvT7l\n6RW5fg69EkWM34c8eldgbLgeTJarS/Frl+tTXttcPodmGONvwoUur0t067oyNlwPJsvVpdRcV+tT\nkV6Rq+dQhqCF35Uur0t067qyoayHWMvV1fMuEk5z9RxKoarObXPmzNEqWLZuSDcOjYx7b+PQiC5b\nN1TJ/n1l49CIDly7Vpeu2aED164dK6PG++1ek96ItVxDOG+fzgHAZs2psVHE+MlEssY9XIoNh0Ss\n5Zp13o1MpuvPnT32nstl4dO14yJtDuJSBfJlsIqEBxcDrA8O7jqIK2lhHPcgNmGKtRtQ+A3hSoV3\ndf2gUCfKkImEmCXjGxR+g7hQ4V1K9WvGlR5RXthQ9U6QWTKeQeE3SLsKTxFxp0eUF98aKldgqNEN\nKPyG6FThKSIJLvSI8uJbQ+UKroYaY4NZPYboltUTQ6ZNiGXA5UCIK0SZ1eN6uKRbbN0nb7dXOvVs\nfAwBMFZNfCUY4fc9XBKDiHQKj/gWAvCxoSKkQVChHh9DBUB8k1pCCI+4NCGPECDSUA/gb7jEN2+3\nDKH0bFxNiyUkD5NtG1AlraIyr3+KF+KfJRbz+/u8sL0IrT2Zef1Tgu7ZEOIqwXj8jLm6T0w9m7qp\nKpnB9aQIUg/BCD9FxX0YHqmOqpIZqk6KYEPiB0EN7hISE1UlM1SZFBFbooJLFBncDSrGT0hMNCcz\nXLbouJ6Ftar9NPbVCLP6ll0XE8GEelzFl66vL3aSV6kqQ6rqTCtfs+tigsJfM75MLPPFTpJQVTJD\nHUkRoaTshgxj/AbwZWKZL3aS6iaQVT0RjTF+e/DRiz1S52xMX2ar+mIncRPOaLZHtDN3y1JXuMOX\nrq8vdhJ3YcquHzCrp4k6MhJ8ma3qi52EkPLQ42+h6owEXyaW+WInIaQ8jPG3wAFOQoiPMMbfI1zv\nhxASA4zxN9EIdzTCG63hDmYmEEJCgB5/E42MhObsnubXLkxmCmWGbSjnQYiPUPgz6PSIwCLUIW6h\nzLAN5TxIeMTglFD421BFdk8d4lZVo2SbUM4jC9PCYeJ4Js/JtvDG4JRQ+NtQxWSmusQtlEWwQjmP\nVkwJR0Mgm4+3aXgUV905WPnxTIqhbeEN2SkZQ1Wd2+bMmaM22Tg0ogPXrtWNQyOZr4uydM0OPfaK\nVbp0zY5K7Vu6Zkcpu2wTynlkYeLcmuvlxqERnXXNaj3x6nt01jWraz2eievlQt2o+r6tGwCbNafG\n0uPPoMrJTFUvgxBKymko59EOE72ZZs/0W8P/hZf3H8BLLx/A78+f0fF4vYZSTPbQbPcGQ1++hMKf\nQVXrjdQhbqHMsA3lPNphSjiaBRJAruP1GkoxKYY2hTd0pwQAQz11smzd0IQu6sahEV22bsiSRcQE\nVYcKux2rNcST53hFQymmz8nUsVQn3qfL1g3p8g1D4+7T5vvW1fsaBUI91kU+awtF+EmcmBKGhiBe\necf2CYKf53hFYtgmxc60sBZtaEw3THkpIvxcqydwuD56uJS5tlyTajxFy8PF8nNurR4ReZ+IPCEi\nQyJypYljuo6pXGXbqXGkPnodi4oihl2QooPJtgefy1K78IvIJAD/BOBMAG8F8BEReWvdx3UdU4Ic\nRU4yKUToA+u9UHQw2fusn7wxoV43AO8AsKbp9VUArur0m1hi/CZzlX3LSSbEFDHG+E2Eeo4G8L2m\n13vS98YhIheLyGYR2TwyMmLALPuY6i56750QUiNFe0Ah9JhqH9wVkd8AcIaq/kH6+gIAp6rqx9r9\nJpbBXRMDRK2PVGx9TQgJA9cGd/cAeHPT6+kA9ho4rtOYGmALwTshhFSLCY9/MoCdAN4N4GkADwM4\nT1Ufa/ebGDx+plkSQqqkiMdf+xO4VPUVEbkUwBoAkwB8vpPox0KWuM/v72P4hRBSO0Yevaiq9wC4\nx8SxCCGEdIaLtBFCSGRQ+AkhJDIo/IQQEhkUfkIIiQwKPyGERAaFnxBCIoPCTwghkUHhJ4SQyKDw\nE0JIZDj56EURGQHw3R5/3gcgtnWHec5xwHOOg17P+VhVnZrni04KfxlEZHPehYpCgeccBzznODBx\nzgz1EEJIZFD4CSEkMkIU/uW2DbAAzzkOeM5xUPs5BxfjJ4QQ0pkQPX5CCCEdCEb4ReR9IvKEiAyJ\nyJW27TGBiHxeRJ4TkUdt22ICEXmziHxDRB4XkcdE5OO2baobEXmNiDwkItvTc/5L2zaZQkQmicg2\nEVll2xZTiMhuEflPEXlERGp7/mwQoR4RmYTkub7vRfJw94cBfERVv23VsJoRkXcBeAHAF1R1lm17\n6kZEjgJwlKpuFZHDAGwB8KGQr7OICIBDVPUFETkYwAMAPq6q37JsWu2IyOUA5gJ4vaqeZdseE4jI\nbgBzVbXWuQuhePynAhhS1V2q+jMAKwH8umWbakdVNwD4gW07TKGqz6jq1vT/5wE8DuBou1bViya8\nkL48ON3899a6ICLTAXwAwD/btiVEQhH+owF8r+n1HgQuCLEjIjMADAB40K4l9ZOGPB4B8ByAr6tq\n8OcM4NMA/hTAAduGGEYBrBWRLSJycV0HCUX4JeO94L2iWBGRQwHcAWCJqv63bXvqRlX3q+opAKYD\nOFVEgg7richZAJ5T1S22bbHAO1X1lwGcCeCP0nBu5YQi/HsAvLnp9XQAey3ZQmokjXPfAeCLqnqn\nbXtMoqo/ArAOwPssm1I37wTwwTTevRLAIhG5za5JZlDVvenf5wB8FUkYu3JCEf6HARwvIr8oIj8H\n4LcB/Jtlm0jFpAOdnwPwuKreYNseE4jIVBE5Iv3/tQDeA2CHXavqRVWvUtXpqjoDyb18n6qeb9ms\n2hGRQ9KkBYjIIQBOB1BLxl4Qwq+qrwC4FMAaJAN+/6Kqj9m1qn5E5EsAvgngRBHZIyIX2bapZt4J\n4AIkHuAj6fZ+20bVzFEAviEig0gcnK+rajTpjZFxJIAHRGQ7gIcA3K2qq+s4UBDpnIQQQvIThMdP\nCCEkPxR+QgiJDAo/IYREBoWfEEIig8JPCCGRQeEnhJDIoPATQkhkUPgJyUH6HID3pv//tYjcaNsm\nQnplsm0DCPGEawBcKyK/gGRV0A9atoeQnuHMXUJyIiLrARwKYGH6PABCvIShHkJyICK/hGTdnJ9S\n9InvUPgJ6UL6yMcvInmq24sicoZlkwgpBYWfkA6IyOsA3AngE6r6OIC/AvAXVo0ipCSM8RNCSGTQ\n4yeEkMig8BNCSGRQ+AkhJDIo/IQQEhkUfkIIiQwKPyGERAaFnxBCIoPCTwghkfH/AWoXatTU+ODM\nAAAAAElFTkSuQmCC\n", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.figure(figsize=(6, 6))\n", "\n", "plt.plot(x, y, 'x')\n", "\n", "plt.xlabel(r'$x$')\n", "plt.ylabel(r'$y$')\n", "plt.title(r'Arrivals in simulated $Pois(\\lambda L^2)$, $\\lambda={}$, $(0,L] \\times (0,L]$'.format(lambd))\n", "\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "----\n", "\n", "Joseph K. Blitzstein and Jessica Hwang, Harvard University and Stanford University, © 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": "13. Poisson Processes", "creator": "buruzaemon", "section": "13.5", "title": "Introduction to Probability, 1st Edition" } }, "nbformat": 4, "nbformat_minor": 2 }