Poisson.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Poisson distribution\n",
    "\n",
    "เรามาทำความเข้าใจกับ Poisson distribution ผ่านทาง binomial distribution กัน\n",
    "\n",
    "ก่อนอื่นเรามาทบทวน binomial distribution กันก่อน  เวลากล่าวถึงตัวแปรสุ่มที่มีการกระจายแบบ binomial distribution เราจะพิจารณาการทดลองลักษณะดังนี้\n",
    "\n",
    "**การทดลองสุ่มที่ 1**  ให้เหรียญที่ออกหัวด้วยความน่าจะเป็น $p$ ทดลองโยนเหรียญ $n$ ครั้ง (แบบอิสระต่อกัน) จากนั้นนับจำนวนครั้งที่เหรียญออกหัว\n",
    "\n",
    "ตัวแปรสุ่มที่แทนจำนวนครั้งที่เหรียญออกหัวจากการทดลองสุ่มที่ 1 จะมีการกระจายแบบ binomial   สาเหตุที่เรียกว่าเป็นการกระจายแบบ binomial เพราะว่าถ้าเราคิดความน่าจะเป็นที่ตัวแปรสุ่มดังกล่าวจะมีค่า $i$ เราจะพบว่ามีค่าเท่ากับ\n",
    "\n",
    "$$ {n \\choose i}\\cdot p^i (1-p)^{n-i} $$\n",
    "\n",
    "สังเกตว่าจากการทดลองทั้งหมด จำนวนรูปแบบที่ได้หัว $i$ ครั้งมี ${n \\choose i}$ แบบ แต่ละแบบเกิดขึ้นไม่พร้อมกัน (disjoint) และมีความน่าจะเป็นที่ได้แต่ละแบบเท่ากันหมด คือ $p^i(1-p)^{n-i}$\n",
    "\n",
    "### มาดูหน้าตากัน\n",
    "\n",
    "เราจะทดลอง plot การกระจายดูสำหรับค่า $n$ และ $k$ บางค่า"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.stats import binom\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "เริ่มจากค่ามาตรฐาน $n=10$ และ $p=0.5$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Container object of 11 artists>"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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RWgn8FPCnVXU58N+MeVpgmLp57e3Mvrn9MPCCJG8bb1Wj131oday3TLYY+r0eDXGhS/L9zAb+R6vqU+OuZ8iuALYlOcHs9NwvJPmr8ZY0dDPATFV9919otzP7JnCxuBr4UlWdrqpvA58CfnbMNY3Kvyf5IYDu+xPjLKbF0O/zWIkLWvdY6w8Bj1TVH4y7nmGrqndX1bqqmmD253dPVV1UZ4lV9RXgZJIf7ZquYvaT7ReLx4DXJXl+9/t6FRfRheo5Bh9TsxP42zHWcn48ZXM5PddjJcZc1rBdAfwy8PkkD3dtv9d9sloXjl8HPtqdnBwHfmXM9QxNVd2f5HbgQWbvNnuI8+yTq+ciyceAnwdWJ5kBbgb2ArcluQH4MvBL46vQT+RKUlNanN6RpGYZ+pLUEENfkhpi6EtSQwx9SWqIoS9JDTH0Jakhhr4kNeR/AD4CFnFzEVBSAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fb3ebf00ba8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n, p = 10, 0.5\n",
    "\n",
    "x = np.arange(0, n+1, 1)\n",
    "y = [binom.pmf(i, n, p) for i in x]\n",
    "\n",
    "plt.bar(x,y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "ลองเพิ่ม $n=40$ และปรับ $p$ ให้ลดลงเป็น $0.1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Container object of 41 artists>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fb421079320>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n, p = 40, 0.1\n",
    "\n",
    "x = np.arange(0, n+1, 1)\n",
    "y = [binom.pmf(i, n, p) for i in x]\n",
    "\n",
    "plt.bar(x,y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "สังเกตว่าการกระจายจะไปสูงที่ใกล้ ๆ 4 ซึ่งจริง ๆ แล้วน่าจะเป็นจุดที่มีค่าสูงที่สุดและเป็นค่าเฉลี่ยของตัวแปรสุ่มนี้ด้วย\n",
    "\n",
    "สำหรับตัวแปรสุ่มแบบ binomial ค่าเฉลี่ย (หรือค่าคาดหวัง expectation) จะมีค่าเท่ากับ $np$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### ทดลองปรับค่าเล่น (เพิ่ม $n$ ลด $p$)\n",
    "\n",
    "ทีนี้ ถ้าเราลองเพิ่ม $n$ แต่พยายามให้ค่าเฉลี่ยเท่าเดิม ลองพิจารณารูปการกระจายด้านล่าง\n",
    "\n",
    "เพื่อให้เปรียบเทียบไม่ยาก จะขอวาดการกระจายแค่ที่ตัวแปรสุ่มมีค่าระหว่าง 0 - 40 เท่านั้น\n",
    "\n",
    "**กรณีแรก ให้ $n=100, p=0.04$**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Container object of 41 artists>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fb3e9e41748>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# ให้ n = 100, p = 0.04, np = 4\n",
    "n, p = 100, 0.04\n",
    "\n",
    "x = np.arange(0, 41, 1)\n",
    "y = [binom.pmf(i, n, p) for i in x]\n",
    "\n",
    "plt.bar(x,y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**กรณีสอง ให้ $n=200, p=0.02$**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Container object of 41 artists>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fb3e9da30b8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# ให้ n = 100, p = 0.04, np = 4\n",
    "n, p = 200, 0.02\n",
    "\n",
    "x = np.arange(0, 41, 1)\n",
    "y = [binom.pmf(i, n, p) for i in x]\n",
    "\n",
    "plt.bar(x,y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**กรณีที่สาม ให้ $n=1000, p=0.004$**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Container object of 41 artists>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fb3e9d27780>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# ให้ n = 100, p = 0.04, np = 4\n",
    "n, p = 1000, 0.004\n",
    "\n",
    "x = np.arange(0, 41, 1)\n",
    "y = [binom.pmf(i, n, p) for i in x]\n",
    "\n",
    "plt.bar(x,y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "สังเกตว่ากราฟหน้าตาแทบไม่ต่างกันเลย จริง ๆ ดูด้วยตาไม่น่าออก"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### มาวิเคราะห์การกระจายกัน\n",
    "\n",
    "กลับมาดูค่าความน่าจะเป็นที่ตัวแปรสุ่มจะมีค่าเท่ากับ $i$ อีกครั้ง (จะเขียนแทนด้วย $f(i)$) เราจะลองพิจารณาค่าดังกล่าว ถ้าปรับค่า $n$ ให้เข้าใกล้ $\\infty$ แต่ยังคงค่าเฉลี่ย $np$ ให้เท่าเดิม\n",
    "\n",
    "เพื่อความสะดวกเรากำหนดให้ $\\lambda = np$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "จากนิยาม เราทราบว่า\n",
    "\n",
    "$$f(i) = {n\\choose i}p^i(1-p)^{n-i}$$\n",
    "\n",
    "พิจารณาเทอม ${n\\choose i}$  สังเกตว่า\n",
    "\n",
    "$${n\\choose i}=\\frac{n!}{(n-i)!\\cdot i!} = \\frac{n(n-1)(n-2)\\cdots (n-i+1)}{i!}$$\n",
    "\n",
    "เมื่อ $n\\rightarrow\\infty$ ค่า $n-i+1$ จะใกล้กับ $n$ มาก เราจึงเขียนได้ว่า\n",
    "\n",
    "$${n\\choose i} = \\frac{n(n-1)(n-2)\\cdots (n-i+1)}{i!} \\approx \\frac{n^i}{i!}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "เราจะพิจารณาส่วนถัดไป เนื่องจาก $\\lambda=np$ เราจะเขียน $p = \\lambda/n$  ดังนั้นจะได้ว่า\n",
    "\n",
    "$$p^i(1-p)^{n-i}=\\left(\\frac{\\lambda}{n}\\right)^i \\left(1-\\frac{\\lambda}{n}\\right)^{n-i}$$\n",
    "\n",
    "เนื่องจาก $n-i$ ใกล้กับ $n$ มากเมื่อ $n\\rightarrow\\infty$ เราจึงเขียน\n",
    "\n",
    "$$p^i(1-p)^{n-i}=\\left(\\frac{\\lambda}{n}\\right)^i \\left(1-\\frac{\\lambda}{n}\\right)^{n-i} \\approx \\left(\\frac{\\lambda}{n}\\right)^i \\left(1-\\frac{\\lambda}{n}\\right)^{n}$$\n",
    "\n",
    "อย่างไรก็ตาม เมื่อ $n\\rightarrow\\infty$ เราทราบว่า\n",
    "\n",
    "$$\\left(1-\\frac{\\lambda}{n}\\right)^{n}\\approx e^{-\\lambda}$$\n",
    "\n",
    "ดังนั้น\n",
    "\n",
    "$$p^i(1-p)^{n-i}=\\approx \\left(\\frac{\\lambda}{n}\\right)^i \\left(1-\\frac{\\lambda}{n}\\right)^{n} \\approx \\left(\\frac{\\lambda}{n}\\right)^i e^{-\\lambda}$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "รวมสองส่วนเข้าด้วยกันเราจะได้ว่า\n",
    "\n",
    "$$f(i)\\approx \\frac{n^i}{i!}\\cdot \\left(\\frac{\\lambda}{n}\\right)^i e^{-\\lambda} = \\frac{\\lambda^i e^{-\\lambda}}{i!}$$\n",
    "\n",
    "ซึ่งตรงกับนิยามของการกระจายแบบ Poisson ([ดูที่วิกิพีเดีย](https://en.wikipedia.org/wiki/Poisson_distribution))\n",
    "\n",
    "ลองมาพลอตการกระจายคู่กันเมื่อ $n$ มีค่าต่าง ๆ และ $\\lambda=np=4$ อีกครั้ง เพื่อความชัดเจนจะพลอตแค่ $x$ มีค่าไม่เกิน 15 เท่านั้น"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math\n",
    "def faprox(lmda,i):\n",
    "    return (lmda**i) * math.exp(-lmda) / math.factorial(i)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### ทดลองที่ n = 10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fb3e9c7f278>]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fb3e9c7fcf8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n, p = 10, 0.4\n",
    "\n",
    "x = np.arange(0, 16, 1)\n",
    "y = [binom.pmf(i, n, p) for i in x]\n",
    "yy = [faprox(4,i) for i in x]\n",
    "\n",
    "plt.bar(x,y)\n",
    "plt.plot(x,yy,'ro-')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "สังเกตว่าลักษณะการเปลี่ยนแปลงค่อนข้างตรง แต่ค่ายังต่างกันอยู่มาก\n",
    "\n",
    "#### ทดลองที่ n = 40"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fb3e9c9f278>]"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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CEGlU6bdlFZ+mnUFJvQTXaZza2LI9K9ucxtXWbWRMnbCCEGHSvt1OauFOPj45fscPAr3afRBdd34Fq1a5jmJMzLOCEGH6+eMHn1hBAODtzhdwuF4ivPii6yjGxDwrCBGm35ZVfNO4JZta2O0hAPY2OpEFp2TASy9512YYY2qNFYQIIlrCuVtW8Un6md4VygbwlrJgxw6YN891FGNimhWECNK5YDMtvy/kkzTrLgq0oGMfaNECpk51HcWYmGYFIYKct7n0+gMrCIEOJ9b37if9xhuwb5/rOMbELCsIEaTfllVsbNGO7U1buY4SeUaPhoMHYfZs10mMiVlWECJE4pFi+ubl2Oyi8vTpA6efbt1GxtSisAqCiAwWkQ0ikisi40K8319EVohIsYgMD2jvKSKfikiOiKwWkesC3nteRL4SkZX+I37XaQDO/ObfND78vXUXlUfEO0tYtAi++sp1GmNiUoUFQUQSgMeBS4GuwEgR6Rq02VbgJmB6UHsRMFpVuwGDgUdEpFnA+79T1Z7+Y2UVv0NM6LdlFSUIn6ad4TpK5LrxRu/nSy+5zWFMjArnDKEvkKuqm1T1MDATGBa4gapuVtXVQElQ+79V9Uv/+TZgJ5BSI8ljTL8tq8g5qSP7GjZxHSVypaXBwIFet5Gq6zTGxJxwCkI74OuA13l+W6WISF8gCdgY0PyA35X0sIgkV/aYsaLh4YP0yv/CuovCMXo05ObCp5+6TmJMzKmTQWURaQO8CPxcVUvPIv4AdAb6AC2A35ez71gRWSYiywoKCuoibp3LyF9HUkmxDSiH4+qrvTvJ2eCyMTUunIKQD7QPeJ3qt4VFRJoC7wDjVXVJabuqfqOeQ8BzeF1TZajqZFXNUNWMlJTY7G3qt2UVh+slsjS1m+soka9JE7jqKpg1y5uGaoypMeEUhKVAJxHpICJJwAggK5yD+9u/DkxV1dlB77XxfwpwBbC2MsFjyXlbVvF5u858n9TAdZToMGYMfPstvP226yTGxJQKC4KqFgN3AnOB9cDLqpojIhNFJBNARPqISB5wDfCUiOT4u18L9AduCjG9dJqIrAHWAK2A+2v0m0WLPXvovn2jjR9UxqBB0LatdRsZU8MSw9lIVecAc4LaJgQ8X4rXlRS830tAyDmCqjqoUklj1cKF1EP55GSbbhq2hARvCurf/w47d0Lr1q4TGRMT7Epl17KzOVC/AavanOY6SXQZNcpbDnvmTNdJjIkZVhBcy87mX+278UNCfddJokv37nDWWdZtZEwNsoLgUn4+bNhg4wdVNXo0LF8OOTkVb2uMqZAVBJeyswH45OS4Xsap6kaO9MYT7CzBmBphBcGl7Gxo1Yr1rdNdJ4lOrVvDpZd6axsdOeI6jTFRzwqCK6peQRg4EBX7Y6iyMWNg2zaYP991EmOinv0mcuXf//bGEC680HWS6DZ0KDRrZt1GxtQAKwiu+OMHVhCqqUEDuO46eO012L/fdRpjopoVBFeys73lnE85xXWS6Dd6NBQVeUXBGFNlVhBcKCmBBQu8swMR12mi37nneoXVuo2MqRYrCC6sXAl791p3UU0pvb3mggWwdavrNMZELSsILpSOHwwc6DZHLLnxRm/mlt1e05gqs4LgQnY2dOnirdhpakbHjnDBBXZ7TWOqIazVTk0NOnwYPvoIbr7ZdZKolj7unTJtf9nXkJEbNlBSrx7bmqbwUP/RZHUrexa2edKQuohoTNSxM4S6tmSJNyPGxg9qVGbOAq5YtxDw/qdOLSxg0nuPkZmzwGkuY6KJFYS6lp0N9erBT3/qOklMuWfRVBoWHz6mrVHxIe5ZZDOPjAmXFYS6lp0NvXt7V9eaGtO2cFel2o0xZYVVEERksIhsEJFcERkX4v3+IrJCRIpFZHjQe2NE5Ev/MSagvbeIrPGP+ah/b+XY9t138Nln1l1UC7Y1bVWpdmNMWRUWBBFJAB4HLgW6AiNFpGvQZluBm4DpQfu2AO4Dzgb6AveJSHP/7SeAXwCd/MfgKn+LaLFokXeXLysINe6h/qMpSkw+pu1gQn0e6j/aUSJjok84Zwh9gVxV3aSqh4GZwLDADVR1s6quBkqC9r0EmKeqe1R1LzAPGCwibYCmqrpEVRWYClxR3S8T8bKzITkZ+vVznSTmZHUbyLjBd5LXNIUShBKEtSedEnKWkTEmtHCmnbYDvg54nYf3L/5whNq3nf/IC9FehoiMBcYCpKWlhfmxESo7G847Dxo2dJ0kJmV1G/hjAbg3+2lGr3ib1vt3s7NJS8fJjIkOET+orKqTVTVDVTNSUlJcx6m6ggJYtcq6i+rI1LOGkFBSwg0r33MdxZioEU5ByAfaB7xO9dvCUd6++f7zqhwzOi3w58NbQagTW5q3Zf4pGVy/8l2Sin9wHceYqBBOQVgKdBKRDiKSBIwAssI8/lzgYhFp7g8mXwzMVdVvgEIROcefXTQaeLMK+aPH/PnQpAlkZLhOEjee751JStG3DPniI9dRjIkKFRYEVS0G7sT75b4eeFlVc0RkoohkAohIHxHJA64BnhKRHH/fPcB/4xWVpcBEvw3gl8AUIBfYCLxbo98s0mRnw4ABkGirhdSVxek9+bJle25a/patb2RMGML67aSqc4A5QW0TAp4v5dguoMDtngWeDdG+DOhembBRa+tWyM2FO+5wnSS+iPDCWUO5f94T9Nq2gc/bdXadyJiIFvGDyjHBbpfpzGvdB1GY1Mg7SzDGHJcVhLqQnQ2tW0P3+DghiiRFSQ15+YyLuGzDYlrv3+06jjERzQpCbVP1CsKgQXa7TEemnjXUn4Ia28NUxlSXFYTatn49bN9u3UUObW3exp+C+p5NQTXmOKwg1DYbP4gINgXVmIpZQaht2dnQoYP3MM6UTkH9+fIsm4JqTDmsINSm4mJYuNDODiKBCC/0vpwztud6d60zxpRhBaE2rVgB+/ZZQYgQr3UbSGHyCfDoo66jGBORrCDUptLxg0GD3OYwgD8FtcfPYPZs2LbNdRxjIo4VhNqUnQ09enjXIJiIMPWsoXDkCDz5pOsoxkQcKwi15eBB+PhjOzuIMFubt4GhQ72CcOiQ6zjGRBQrCLXl00+9omDjB5Hnrru8+1PMmuU6iTERxQpCbcnOhoQEb4VTE1kuvBC6dPEGl20KqjE/soJQW7KzoU8faNrUdRITTAR+9StYvtw7kzPGAFYQakdhISxdat1FkWzUKDjxRPjnP10nMSZiWEGoDR9+6M1ksYIQuRo3hltu8aag5sf23VuNCVdYBUFEBovIBhHJFZFxId5PFpFZ/vufiUi6336DiKwMeJSISE//vYX+MUvfi525mdnZ0KABnHuu6yTmeO64w6agGhOgwoIgIgnA48ClQFdgpIh0DdrsFmCvqp4KPAw8CKCq01S1p6r2BEYBX6nqyoD9bih9X1V31sD3iQzZ2XD++V5RMJGrY0dvCupTT3kzwoyJc+GcIfQFclV1k6oeBmYCw4K2GQa84D+fDVwoUmbx/5H+vrFr2jRo3x7WroVly7zXJrLZFFRjfhROQWgHfB3wOs9vC7mNqhYD+4CWQdtcB8wIanvO7y66N0QBAUBExorIMhFZVlBQEEZcR6ZNg7FjIS/Pe/3tt95rKwqRrXQK6j//aVNQTdyrk0FlETkbKFLVtQHNN6hqD+AC/zEq1L6qOllVM1Q1IyUlpQ7SVtH48VBUdGxbUZHXbiKXiHeWYFNQjQmrIOQD7QNep/ptIbcRkUTgRCDwBrYjCDo7UNV8/+d+YDpe11T02rq1cu0mcpROQbVVUE2cC6cgLAU6iUgHEUnC++WeFbRNFjDGfz4cmK/qnX+LSD3gWgLGD0QkUURa+c/rA0OBtUSztLTKtZvIccIJNgXVGMIoCP6YwJ3AXGA98LKq5ojIRBHJ9Dd7BmgpIrnAb4DAqan9ga9VdVNAWzIwV0RWAyvxzjCerva3cemBB7ylKgI1auS1m8h3xx1QUmJTUE1cSwxnI1WdA8wJapsQ8PwgcE05+y4EzglqOwD0rmTWyHb11XDrrd5U06Ii78zggQfghhtcJzPh6NgRLr/cm4I6frxNGTZxya5UrinvvuvNZX/1Ve9fmps3WzGINjYF1cQ5Kwg1ZcYMSEmx5Sqi2aBB0LUr/OMfNgXVxCUrCDWhsBDeeguuvRYSw+qFM5GodBXUzz+HTz5xncaYOmcFoSa8+abXXTRypOskprpsFVQTx+yfs1WQPu6dY14/98o/6NQ0hQuy9qBvvVPOXkdtnjSktqKZ6jrhBG9ywCOPeFedp6a6TmRMnbEzhGpqXrSP8zev5O0u/VGx/5wxwaagmjhlv8Gq6bINH1O/5AhvdrVbZcaMDh2OTkG1VVBNHLGCUE2Z6z7ky5btWZ/SwXUUU5Puugt27fK6jOrVg/R0W6jQxDwrCNXQprCAs/NyyOrS35uhYmLH9u3en+nu3d4U1C1bbPVaE/OsIFTD0PUfAZBl3UWxZ/z4stci2Oq1JsZZQaiGzPUfsrJNJ7Y0b+s6iqlptnqtiUNWEKqo4+48euzYyFtd7OwgJtnqtSYO2XUIVZS5/kNKEN7qfIHrKKaagq8rAcjsMZxJ+Y/RqPjQj20HE+pzT4/hZAVsb9eUmFhiBaEqVLl8/SI+S+vOzibBdwo1sSCr20AA7lk0lbaFBagI2xu35O0u/R0nM6b2WJdRFXTbsZFT9uTzpnUXxbSsbgM5//bn6Pj7t7l76G9J37edm5a/7TqWMbXGCkIVDFv3IYfrJfLu6f1cRzF1JKvLABZ07M3/+2gqqft2uI5jTK0IqyCIyGAR2SAiuSIyLsT7ySIyy3//MxFJ99vTReR7EVnpP54M2Ke3iKzx93lUJEom8peUMPSLj1jUoRf7GjZxncbUFRHGX3IHJVKP/3nvMVse28SkCguCiCQAjwOXAl2BkSLSNWizW4C9qnoq8DDwYMB7G1W1p/+4LaD9CeAXQCf/MbjqX6MOLV5M2/277NqDOLStaWse6j+a/ps/56qc+a7jGFPjwjlD6AvkquomVT0MzASGBW0zDHjBfz4buPB4/+IXkTZAU1VdoqoKTAWuqHR6F2bM4PvEZD449WzXSYwDL541hOVtO3Nv9hRaHvjWdRxjalQ4BaEd8HXA6zy/LeQ2qloM7ANKp990EJHPReRDEbkgYPu8Co4ZeX74AV55hXmdzqYoqaHrNMYBlXr8/tK7aPTD99yXPdl1HGNqVG0PKn8DpKlqL+A3wHQRaVqZA4jIWBFZJiLLCgoKaiVk2ObNg927ybLZRXEtt1Uaj597HZnrF8HbNuvIxI5wCkI+0D7gdarfFnIbEUkETgR2q+ohVd0NoKrLgY3Aaf72gXceCXVM/P0mq2qGqmakpKSEEbcWzZgBzZqxqMNZbnMY5544ZzgbWqXB7bd7t1A1JgaEUxCWAp1EpIOIJAEjgKygbbKAMf7z4cB8VVURSfEHpRGRjniDx5tU9RugUETO8ccaRgNv1sD3qT1FRfDGG3D11RxOrO86jXHsh4T6jBt8F+Tnwx/+4DqOMTWiwoLgjwncCcwF1gMvq2qOiEwUkUx/s2eAliKSi9c1VDo1tT+wWkRW4g0236aqe/z3fglMAXLxzhzeraHvVDveeQe++w6uv951EhMhPm/X2btvwv/9Hyxe7DqOMdUW1tIVqjoHmBPUNiHg+UHgmhD7vQq8Ws4xlwHdKxPWqenToU0bGDAA3n/PdRoTKe6/3ztzvPVWWLkSGjRwnciYKrMrlcPx7bcwZw5cey0kJLhOYyJJ48berTY3bIAHHnCdxphqsYIQjtdfh8OHYeRI10lMJLrkEhg1CiZNgtWrXacxpsqsIIRjxgzo2BH69nWdxESqhx+G5s29rqMjR1ynMaZKrCBUZMcOyM72zg6iZLkl40DLlvDoo7B0qffTmChkBaEiL78MJSXWXWQqdt11MGQI/OlP8NVXrtMYU2lWECoyYwb06AHdurlOYiKdCDzxhDfxYOxYWxHVRB0rCMezeTN8+qmdHZjwtW/vDS5/8AFMneo6jTGVYgXheGbO9H6OGOE2h4kut90G/frB3Xd7Y1DGRAkrCMczfTqcey506OA6iYkm9erBlClw4IB3JbMxUcIKQnlycmDNGusuMlXTuTO5d8JHAAANfUlEQVTce683KSEreOkvYyKTFYTyzJjh/Uvv2mtdJzHR6p57vAkJt98O+/a5TmNMhawghKLqFYRBg+Ckk1ynMdEqKcnrOtq+HcaVuRW5MRHHCkIoS5fCpk22sqmpvr594de/hiefhJ/8xDvrTE+HadNcJzOmjLBWO40706d7/7q78krXSUws6N7du0ahdMbRli3edQoAN9zgLpcxQewMIdiRIzBrFlx2GTRr5jqNiQUTJ5a9SK2oCMaPd5PHmHJYQQj24Yden6/NLjI1ZevWyrUb40hYBUFEBovIBhHJFZEyo2Mikiwis/z3PxORdL/9IhFZLiJr/J+DAvZZ6B9zpf9oXVNfqlpmzPDWuB861HUSEyvS0kK3t21btzmMqUCFBcG/J/LjwKVAV2CkiHQN2uwWYK+qngo8DDzot+8CLlfVHnj3XH4xaL8bVLWn/9hZje9RMw4dgtmz4YoroFEj12lMrHjggdD/PxUVwapVdZ/HmHKEM6jcF8hV1U0AIjITGAasC9hmGPBn//ls4DEREVX9PGCbHKChiCSr6qFqJ68Nc+d6d0ez7iJTRenj3gnR2ozMQbdzz6KptC3cxbamrZjWczCjV8yhSd9zuWPYOD7s2BuAzZOG1G1gYwKEUxDaAV8HvM4Dzi5vG1UtFpF9QEu8M4RSVwMrgorBcyJyBO++y/erOl4ecsYMb137iy5yGsPEnqxuA8nqNvCYtte7Xcizr/4Xz8z+LyZcfDvTe17qKJ0xnjoZVBaRbnjdSP8R0HyD35V0gf8YVc6+Y0VkmYgsKygoqL2QBw54Swxccw3Ur197n2OMb3vTVlxz/YN81KEX/zP3ccYtfM6794YxjoRTEPKB9gGvU/22kNuISCJwIrDbf50KvA6MVtWNpTuoar7/cz8wHa9rqgxVnayqGaqakZKSEs53qpqsLK9P17qLTB06kNyIW6+ewIu9LuO2z171brLz/feuY5k4FU5BWAp0EpEOIpIEjACCV+vKwhs0BhgOzFdVFZFmwDvAOFX9uHRjEUkUkVb+8/rAUGBt9b5KNU2fDqmpcP75TmOY+HOkXgL3XnQ79w+8GV591VsypTbPho0pR4UFQVWLgTuBucB64GVVzRGRiSKS6W/2DNBSRHKB3wClU1PvBE4FJgRNL00G5orIamAl3hnG0zX5xcI2bZp3U5O33/YWIJsxw0kME+dEmNL3KnjlFVi5Es45B774wnUqE2fCWrpCVecAc4LaJgQ8PwhcE2K/+4H7yzls7/Bj1pJp07wlBIqKvNf799uSAsatq6+Gdu0gMxPOOw9efx0GDHCdysSJ+L5Sefz4o8WglC0pYFw75xxYssRbafeii+Cll1wnMnEivguCLSlgIlXHjvDJJ96tOEeNCr0ekjE1LL4LQnlLCpTXbkxdat7cu1hy1Ci47z74+c/h8GHXqUwMi++CEGpJgUaNvHZjIkFSErzwAvz5z97PwYNh8mTvngp2bwVTw+L7fgilA8fjx3vdRGlpXjGwAWUTSUS8M4SOHeGmm2DhwqPdR3ZvBVOD4qYghF5jBqAZjHj86Ms1QNC2tr6MiQijRsHvfnf0RjulSidCWEEw1RTfXUbGRJud5SwKbBMhTA2wgmBMNClvwoMITJgAeXl1m8fEFCsIxkSTUBMhkpPhjDPg/vu9QearroLsbJumaiotbsYQjIk24d5b4aH+o8nqNpDUs7dzw8r3uO6992nx+utw+ulw++0wZozdH9yExQqCMVEm1L0VAPKa/YQHf3oTj5x/PZdu+JhHdn8C//mf8Mc/wvXXwx13QM+eDhKbaGFdRsbEmEOJSbzRbaB3pfOKFd7so+nToVcvb32kl16Cgwe9jadNs2sazI+sIBgTy3r18i5ky8+HRx6B3bu96avt28Pll8MvfuFdy6B69JoGKwpxywqCMfGgWTP49a+9JbU/+AD69/eWfA++GY8t7hjXbAzBmHgiAhde6D3q1Qs9E2nLFm/soUcP79GtG5xwQt1nNXXOCoIx8SotzfvlHywpCZ5++ujS8CLeshmlBaL0ceqpkBj0K2TaNFsKJoqFVRBEZDDwDyABmKKqk4LeTwam4t30Zjdwnapu9t/7A3ALcAS4S1XnhnNMY0zNCp7GmtljOJPyH6NR8aEf24oSkxl38Z281XUA7b/dQeeCzZxesJnftjkMa9Z49x4vKfE2Tk6GLl2OFoiCAnjssaPdUNVdZ8mKS52rsCCISALwOHARkAcsFZEsVV0XsNktwF5VPVVERgAPAteJSFe8ezB3A9oCH4jIaf4+FR3TGFOLSqeuhrqmAWBr8zZsbd6G9087l9+Wrud18CCsX+8Vh9JHdja8+GLoDykqgv/4D/jXv7zlvEsfLVoc+7p5c2jQ4Oh+wXczjKTiUtOFKoIKXzhnCH2BXFXdBCAiM4FhQOAv72HAn/3ns4HHRET89pmqegj4yr/ncl9/u4qOaYypZeVd01CuBg28mUu9eh3bvns3pKSEHpM4cACefx4KC49/7IYNjxaH3Fw4dOjY94uKvGspNm70urXCfcyfD3/5y9Gptlu2eLOrCgrgyiu9sZTyHiLHvn75ZS9DqEJ1/fXe9pVR04WvmsIpCO2ArwNe5wFnl7eNqhaLyD6gpd++JGjfdv7zio5pjIlQoa6iXtykFamFBWXa85qmcP7tz5FQcoQmhw6w8ld9YO9e77Fnz9HngY+cnNAfvG+ftxR4dX3/Pdx9t/eorqIiuPFG7xGotDiE+ln6/IcfQh/P0eq1ohWsdyIiw4HBqnqr/3oUcLaq3hmwzVp/mzz/9Ua8X/B/Bpao6kt++zPAu/5uxz1mwLHHAn7J5HRgQ9W+aoVaAbtq6djVEam5IHKzRWouiNxs1c7VClqkwckSMJ1doWQrbNkFeypzrDOgR31ICm7/AQ6v9hapD1tvb2wzpOWw3NWxauN4AYL/PE9W1ZSKdgrnDCEfaB/wOtVvC7VNnogkAifiDS4fb9+KjgmAqk4GJoeRs1pEZJmqZtT251RWpOaCyM0WqbkgcrNFai6I3GyRmguqni2cC9OWAp1EpIOIJOENEmcFbZMFjPGfDwfmq3fqkQWMEJFkEekAdAL+FeYxjTHG1KEKzxD8MYE7gbl4U0SfVdUcEZkILFPVLOAZ4EV/0HgP3i94/O1exhssLgbuUNUjAKGOWfNfzxhjTLjCug5BVecAc4LaJgQ8PwhcU86+DwBl7lof6piO1Xq3VBVFai6I3GyRmgsiN1uk5oLIzRapuaCK2SocVDbGGBMfbHE7Y4wxgBUERGSwiGwQkVwRGec6TykRaS8iC0RknYjkiMivXWcKJCIJIvK5iLztOksgEWkmIrNF5AsRWS8i57rOBCAid/t/jmtFZIaINKh4r1rL8qyI7PSni5e2tRCReSLypf+zeQRl+6v/57laRF4XkTq//VuoXAHv/VZEVERa1XWu42UTkV/5/91yROShcI4V1wUhYFmOS4GuwEh/uY1IUAz8VlW7AucAd0RQNoBfA+tdhwjhH8B7qtoZOJMIyCgi7YC7gAxV7Y43kWKEw0jPA4OD2sYB2araCcj2X7vwPGWzzQO6q+oZwL+BP9R1KELnQkTaAxcDW+s6UIDnCcomIgPxVn84U1W7AX8L50BxXRAIWJZDVQ8DpUtoOKeq36jqCv/5frxfbO2Ov1fdEJFUYAgwxXWWQCJyItAfb9YbqnpYVb91m+pHiUBD/zqdRsA2V0FUdRFlLxQbBrzgP38BuKJOQ/lCZVPV91W12H+5BO+6Jee5fA8D9wDOBmPLyXY7MMlfNghV3RnOseK9IIRaliMifukGEpF0oBfwmdskP3oE7y9BiesgQToABcBzfnfWFBFxvpC/qubj/QttK/ANsE9V33ebqoyTVPUb//l24CSXYY7jZo6uduCUiAwD8lV1lessIZwGXCAin4nIhyLSJ5yd4r0gRDwRaQy8CvynqlawOlid5BkK7FTV6lxWX1sSgbOAJ1S1F3AAd10fP/L744fhFay2wAkicuPx93LHv6g04qYfish4vK5U5/f4FJFGwB+BCRVt60gi0AKvu/l3wMv+gqPHFe8FIZxlOZwRkfp4xWCaqr7mOo+vH5ApIpvxutgGichLbiP9KA/IU9XSM6nZeAXCtZ8BX6lqgar+ALwGnOc4U7AdItIGwP8ZVhdDXRGRm4ChwA0aGXPlT8Er8Kv8vwupwAoR+YnTVEflAa+p5194Z/MVDnrHe0GI2CU0/Gr+DLBeVf/uOk8pVf2Dqqaqajref6/5qhoR/9pV1e3A1yJyut90IZGxpPpW4BwRaeT/uV5IBAx2BwlcfmYM8KbDLMcQ72Za9wCZqlrkOg+Aqq5R1daqmu7/XcgDzvL/H4wEbwADAcS7B00SYSxeGNcFwR+oKl1CYz3wcgQtodEPGIX3L/CV/uMy16GiwK+AaSKyGugJ/I/jPPhnLLOBFXgrddbD4VWuIjID+BQ4XUTyROQWYBJwkYh8iXdG4+QOhuVkewxoAszz/x48GSG5IkI52Z4FOvpTUWcCY8I5s7IrlY0xxgBxfoZgjDHmKCsIxhhjACsIxhhjfFYQjDHGAFYQjDHG+KwgGGOMAawgGGOM8VlBMMYYA8D/B7jH/PYxyj7AAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fb3e9c9f9e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n, p = 40, 0.1\n",
    "\n",
    "x = np.arange(0, 16, 1)\n",
    "y = [binom.pmf(i, n, p) for i in x]\n",
    "yy = [faprox(4,i) for i in x]\n",
    "\n",
    "plt.bar(x,y)\n",
    "plt.plot(x,yy,'ro-')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "เริ่มใกล้กันมากขึ้น\n",
    "\n",
    "#### ทดลองที่ n = 100"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fb3e9c20c18>]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fb3e9cdeef0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n, p = 100, 0.04\n",
    "\n",
    "x = np.arange(0, 16, 1)\n",
    "y = [binom.pmf(i, n, p) for i in x]\n",
    "yy = [faprox(4,i) for i in x]\n",
    "\n",
    "plt.bar(x,y)\n",
    "plt.plot(x,yy,'ro-')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "สังเกตว่าแทบมองไม่เห็นความแตกต่าง"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### แล้วจะเข้าใจ Poisson distribution ได้อย่างไร\n",
    "\n",
    "จากการอธิบายข้างต้น เราอาจจะพิจารณา Poisson distribution ว่าเป็นเหมือนการทดลองที่มีการทดลองย่อยทำซ้ำจำนวนมาก ๆ (โยนเหรียญจำนวนมากๆ) โดยที่แต่ละการทดลองย่อยมีโอกาสสำเร็จน้อยมาก แต่เราทราบว่าโดยเฉลี่ยแล้วจะมีจำนวนการสำเร็จเท่ากับ $\\lambda$ ครั้ง (มักนิยมเรียกว่าเป็น rate ของ Poisson distribution)\n",
    "\n",
    "ยกตัวอย่างเช่น มีคนกล่าวว่าในการขุดบิตคอยน์นั้น จะได้บล็อคใหม่โดยเฉลี่ยทุก ๆ สิบนาที และเวลาในการขุดเป็น Poisson distribution  ถ้ามองจากมุมที่เราได้พิจารณามาก็สอดคล้องกับสภาพความเห็นจริงที่ในการสร้างบล็อคนั้น ต้องมีการพยายามทำให้ค่า hash มีคุณสมบัติบางอย่างที่ในทางปฏิบัติต้องหาโดยการทดลองจนกว่าจะเจอ (trial-and-error) และมีผู้พยายามหาจำนวนมาก แต่ละคนมีโอกาสสำเร็จน้อยมาก ๆ แต่มีอัตราการสำเร็จอยู่ที่ 0.1 ครั้งต่อนาที   (ดังนั้นในการคิดก็คือจะใช้ $\\lambda=0.1$ เป็นต้น)"
   ]
  }
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