Sample Code/Day_27_SampleCode.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 今天的範例，帶大家運用python 裡面的套件\n",
    "* 如何模擬這些分配的樣本點\n",
    "* 進行一些機率的運算\n",
    "\n",
    "包含以下離散型分配        \n",
    "\n",
    "1. 離散均勻分配( Discrete Uniform Distribution )     \n",
    "2. 伯努利分配( Bernoulli Distribution )    \n",
    "3. 二項分配(Binomial Distribution)   \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# library\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "from scipy import stats\n",
    "import math\n",
    "import statistics"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1. 離散均勻分配( Discrete Uniform Distribution )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1 2 3 4 5 6]\n",
      "[0.16666667 0.16666667 0.16666667 0.16666667 0.16666667 0.16666667]\n"
     ]
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "'''\n",
    "# 離散均勻分配 (Discrete Uniform Distribution)\n",
    "# 前提：其中有限個數值擁有相同的機率。\n",
    "'''\n",
    "# 1.定義離散均勻分配的基本資訊\n",
    "\n",
    "low=1 \n",
    "high=7\n",
    "r = np.arange(low,high)\n",
    "# 2.計算離散均勻分配的概率質量分佈 (probability mass function)\n",
    "# 之所以稱為質量，是因為離散的點\n",
    "# 產生 x 軸的點\n",
    "#r = np.arange(stats.randint.ppf(0.01, low, high),\n",
    "#              stats.randint.ppf(0.99, low, high),1)\n",
    "print(r)\n",
    "# P(X=x) --> 是機率\n",
    "probs = stats.randint.pmf(r,low,high)\n",
    "print(probs)\n",
    "plt.bar(r, probs)\n",
    "plt.ylabel('P(X=x)')\n",
    "plt.xlabel('x')\n",
    "plt.title('pmf of DU(1,6)')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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Kr+DuXb/92sz+Dsw4sAQkfO7OjPQN3DE9i135hfzu1J78+uQe1Kml8RAi8SCw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\nRK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# 3.計算離散均勻分配的累積機率 (cumulative density function)，pmf 的累加\n",
    "# P(X=x) --> 是機率\n",
    "cumsum_probs = stats.randint.cdf(r, low,high)\n",
    "plt.ylabel('P(X<=x)')\n",
    "plt.xlabel('x')\n",
    "plt.title(' cdf of DU(1,6)')\n",
    "plt.plot(r, cumsum_probs)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1. 2. 3. 4. 5. 6.]\n"
     ]
    }
   ],
   "source": [
    "# 4.透過 cdf ，給定一個 機率值，反推出對應到的 x\n",
    "k = stats.randint.ppf(cumsum_probs , low, high)\n",
    "print(k)\n",
    "#看上圖看結果"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[5 4 5 4 6 2 2 5 3 5 4 3 4 4 1 2 4 4 3 2]\n"
     ]
    },
    {
     "data": {
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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# 5.產生符合隨機樣本點 (random sample)\n",
    "X = stats.randint.rvs(low,high,size=20)\n",
    "print(X)\n",
    "plt.hist(X,bins=13)\n",
    "plt.show()\n",
    "#試試看，，每一次的結果一樣嗎?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(array(3.5), array(2.91666667), array(0.), array(-1.26857143))\n",
      "<class 'tuple'>\n",
      "randint mean= 3.5\n",
      "randint variance= 2.9166666666666665\n",
      "randint kurtosis= 0.0\n",
      "randint skew= -1.2685714285714285\n"
     ]
    }
   ],
   "source": [
    "#6.計算固定參數下，隨機變數的平均數、變異數、偏度和峰度。\n",
    "stat_randint=stats.randint.stats(low,high,moments='mvks')\n",
    "print(stat_randint)\n",
    "print(type(stat_randint))\n",
    "print(\"randint mean=\",float(stat_randint[0]))\n",
    "print(\"randint variance=\",float(stat_randint[1]))\n",
    "print(\"randint kurtosis=\",float(stat_randint[2]))\n",
    "print(\"randint skew=\",float(stat_randint[3]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2. 伯努利分配( Bernoulli Distribution )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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IJKlxvQZBkvVJrk2yO8mWMdOfm+Sa7nV5klP6rEeSdG+9BUGSZcB5wAZgLXBWkrUj3f4e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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "'''\n",
    "# 伯努利分配( Bernoulli Distribution )\n",
    "# 前提：是只有兩種可能結果（成功或失敗）的單次隨機試驗，成功的機率為p\n",
    "'''\n",
    "# ①定義伯努利分配的基本資訊\n",
    "\n",
    "# ①定義伯努利分配基本資訊\n",
    "p = 0.4 # 事件A 機率 0.4\n",
    "r = np.arange(0,2) # 可以出現的範圍為 0、1,2種可能出現的結果\n",
    "\n",
    "# ②計算二項分佈的概率質量分佈 (probability mass function)\n",
    "# 之所以稱為質量，是因為離散的點\n",
    "# P(X=x) --> 是機率\n",
    "probs = stats.bernoulli.pmf(r,p)\n",
    "#array([ 0.07776, 0.2592 , 0.3456 , 0.2304 , 0.0768 , 0.01024])\n",
    "plt.bar(r, probs)\n",
    "plt.ylabel('P(X=x)')\n",
    "plt.xlabel('x')\n",
    "plt.title('Bernoulli(p=0.4)')\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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QCyw0s7nunlXqsR8BHgmPvxD4T3ffXuphTjt0C1KRIyksLuHpT1bxlw+yaVS/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\nRK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# ③計算伯努利分配的累積機率 (cumulative density function)，pmf 的累加\n",
    "# P(X=x) --> 是機率\n",
    "cumsum_probs = stats.bernoulli.cdf(r,p)\n",
    "plt.ylabel('P(X<=x)')\n",
    "plt.xlabel('x')\n",
    "plt.title('bernoulli(p=0.4)')\n",
    "plt.plot(r, cumsum_probs)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0. 1.]\n"
     ]
    }
   ],
   "source": [
    "# ④ 透過 cdf ，給定一個 機率值，反推出對應到的 x\n",
    "p_loc = stats.bernoulli.ppf(cumsum_probs, p)\n",
    "print(p_loc)\n",
    "#看上圖看結果"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 1 0 1 1 0]\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXAAAAD4CAYAAAD1jb0+AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# ⑤產生符合伯努利分配的隨機樣本點 (random sample)\n",
    "X = stats.bernoulli.rvs(p,size=20)\n",
    "print(X)\n",
    "plt.hist(X)\n",
    "plt.show()\n",
    "#試試看，每一次的結果一樣嗎?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(array(0.4), array(0.24), array(0.40824829), array(-1.83333333))\n",
      "<class 'tuple'>\n",
      "bernoulli mean= 0.4\n",
      "bernoulli variance= 0.24\n",
      "bernoulli kurtosis= 0.40824829046386296\n",
      "bernoulli skew= -1.8333333333333337\n"
     ]
    }
   ],
   "source": [
    "#6.計算固定參數下，隨機變數的平均數、變異數、偏度和峰度。\n",
    "stat_ber=stats.bernoulli.stats(p,moments='mvks')\n",
    "print(stat_ber)\n",
    "print(type(stat_ber))\n",
    "#E(X)\n",
    "print(\"bernoulli mean=\",float(stat_ber[0]))\n",
    "print(\"bernoulli variance=\",float(stat_ber[1]))\n",
    "print(\"bernoulli kurtosis=\",float(stat_ber[2]))\n",
    "print(\"bernoulli skew=\",float(stat_ber[3]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3. 二項分配(Binomial Distribution)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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JkuMPvGZwqe0Xx+krSZobD/o9iKr6J+CdwDuTPBQ4CfhuVX3zQfrtT3IRcBWwCLikqrYn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\nRK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "'''\n",
    "# 二項分佈 (binomial distribution)\n",
    "# 前提：獨立重複試驗、有放回、只有兩個結果\n",
    "# 二項分佈指出，隨機一次試驗出現事件A的機率如果為p，那麼在重複 n 次試驗中出現 x 次事件A的機率為：\n",
    "# f(n,x,p) = choose(n, x) * p**x * (1-p)**(n-x)\n",
    "'''\n",
    "# 1.定義二項分佈的基本資訊\n",
    "p = 0.5 # 事件A 機率 0.4\n",
    "n = 5  # 重複實驗5次,\n",
    "r = np.arange(0,6) # 可以出現的範圍為 0,1,2,...,5-->6種可能出現的結果\n",
    "#print(type(k))\n",
    "\n",
    "# 2.計算二項分佈的概率質量分佈 (probability mass function)\n",
    "# 之所以稱為質量，是因為離散的點\n",
    "# P(X=x) --> 是機率\n",
    "probs = stats.binom.pmf(r, n, p)\n",
    "#array([ 0.07776, 0.2592 , 0.3456 , 0.2304 , 0.0768 , 0.01024])\n",
    "plt.bar(r, probs)\n",
    "plt.ylabel('P(X=x)')\n",
    "plt.xlabel('x')\n",
    "plt.title('binomial(n=5,p=0.5)')\n",
    "plt.show()\n",
    "\n",
    "#學生額外小練習: 可以調整 p 的不同值，p接近於1 時，p=0.5, p 接近於 0時，看 pmf 的變化。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# 3.計算二項分佈的累積機率 (cumulative density function)，pmf 的累加\n",
    "# 之所以稱為質量，是因為離散的點，預設體積（即寬度）為1\n",
    "# P(X=x) --> 是機率\n",
    "cumsum_probs = stats.binom.cdf(r, n, p)\n",
    "#array([ 0.07776, 0.2592 , 0.3456 , 0.2304 , 0.0768 , 0.01024])\n",
    "plt.show()\n",
    "plt.ylabel('P(X<=x)')\n",
    "plt.xlabel('x')\n",
    "plt.title('binomial(n=5,p=0.4)')\n",
    "plt.plot(r, cumsum_probs)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0. 1. 2. 3. 4. 5.]\n"
     ]
    }
   ],
   "source": [
    "# 4.透過 cdf ，給定一個 機率值，反推出對應到的 x\n",
    "p_loc= stats.binom.ppf(cumsum_probs, n, p)\n",
    "print(p_loc)\n",
    "#看上圖看結果"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1 5 4 3 3 3 3 2 2 4 3 3 2 2 3 4 1 0 3 3]\n"
     ]
    },
    {
     "data": {
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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# 5.產生符合二項分佈的隨機樣本點 (random sample)\n",
    "X = stats.binom.rvs(n,p,size=20)\n",
    "#array([2, 3, 1, 2, 2, 2, 1, 2, 2, 3, 3, 0, 1, 1, 1, 2, 3, 4, 0, 3])\n",
    "print(X)\n",
    "plt.hist(X)\n",
    "plt.show()\n",
    "#試試看，，每一次的結果一樣嗎?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(array(2.5), array(1.25), array(0.), array(-0.4))\n",
      "<class 'tuple'>\n",
      "binomial mean= 2.5\n",
      "binomial variance= 1.25\n",
      "binomial kurtosis= 0.0\n",
      "binomial skew= -0.4\n"
     ]
    }
   ],
   "source": [
    "#6.計算固定參數下，隨機變數的平均數、變異數、偏度和峰度。\n",
    "stat_bin=stats.binom.stats(n,p,moments='mvks')\n",
    "print(stat_bin)\n",
    "print(type(stat_bin))\n",
    "#E(X)\n",
    "print(\"binomial mean=\",float(stat_bin[0]))\n",
    "print(\"binomial variance=\",float(stat_bin[1]))\n",
    "print(\"binomial kurtosis=\",float(stat_bin[2]))\n",
    "print(\"binomial skew=\",float(stat_bin[3]))"
   ]
  }
 ],
 "metadata": {
  "colab": {
   "name": "Day33.YOLO 細節理解 - 網路輸出的後處理_HW.ipynb",
   "provenance": []
  },
  "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": "3.7.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}