概率统计上篇.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 事件"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "样本空间(再看抛硬币、丢色子)\n",
    "\n",
    "* 随机试验\n",
    "\n",
    "    * $E_1$：抛一枚硬币，分别用$H$和$T$表示出正面和反面\n",
    "    * $E_2$：将一枚硬币连抛三次，考虑正反面出现的情况\n",
    "    * $E_3$：掷一颗骰子，可能出现的点数\n",
    "    \n",
    "* 样本空间：一个试验所有可能的结果；样本空间的元素为样本点\n",
    "\n",
    "    * $S_1 = \\{H,T\\}$\n",
    "    * $S_2 = \\{HHH,HHT,HTH,THH,HTT,THT,TTH,TTT\\}$\n",
    "    * $S_3 = \\{1,2,3,4,5,6\\}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "事件\n",
    "\n",
    "* 事件：样本空间的子集，满足某种条件的样本点组成的集合\n",
    "\n",
    "    * $E_2$中的事件$A_1$：第一次出现的是$H$，则$A_1 = \\{HHH,HHT,HTT\\}$；$A_2 = \\{HHH,TTT\\}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "事件关系、频率、概率\n",
    "\n",
    "* 事件关系\n",
    "\n",
    "    * 和事件，$A∪B$\n",
    "    * 积事件，$A∩B$，通常就记做$AB$，上页中$A_1∩A_2 = \\{HHH\\}$\n",
    "    * 对立事件，$\\bar{A}$\n",
    "    \n",
    "* 频率：事件发生的频繁程度，频率稳定于概率(伯努利大数定理)\n",
    "\n",
    "* 概率：表征事件发生可能性大小\n",
    "\n",
    "    * $0 \\le P(A) \\le 1$\n",
    "    * $P(S) = 1$\n",
    "    * 可列可加：$AB = ∅$，则$P(A∪B) = P(A) + P(B)$\n",
    "\n",
    "* 加法公式：任意事件$A$和$B$，则有$P(A∪B) = P(A) + P(B) - P(AB)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "古典概率计算\n",
    "\n",
    "* 抛硬币$E_2$事件，事件$A_1$为恰好有一次为正面，则$A_1 = \\{HHH,HHT,HTT\\}$，则$P(A_1) = \\frac{3}{8}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 说明：古典概率假定所发生的事件都是等可能事件，比如抛硬币、掷色子"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "非常简单的思考题：同时丢一个红色子和绿色子\n",
    "\n",
    "* 计算点数和为3的概率：点数和为3的就两种情况，故而有$\\frac{2}{6\\times6} = \\frac{1}{18}$\n",
    "\n",
    "\n",
    "* 计算点数和为6的概率：点数和为3的就六种情况，故而有$\\frac{6}{6\\times6} = \\frac{1}{6}$\n",
    "\n",
    "* 红色子或绿色子出现点数为1的概率，尝试利用$P(A∪B) = P(A) + P(B) - P(AB)$：$\\frac{1}{6} + \\frac{1}{6} - \\frac{1}{6\\times6} = \\frac{11}{36}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* 会计算更复杂的排列组合问题。经典问题：取球，放球，两人生日相同，次品率，分配学生等等，举一反三"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "条件概率：走向贝叶斯\n",
    "\n",
    "* 将一枚硬币连抛两次，设事件$A$为至少有一次是$H$，事件$B$是两次投出同一面。求已知事件$A$发生的条件下事件$B$发生的概率\n",
    "\n",
    "* 样本空间$S = \\{HH,HT,TH,TT\\}$，$A = \\{HH,HT,TH\\}$，$B = \\{HH,TT\\}$\n",
    "\n",
    "* 则有$P(B|A) = \\frac{1}{3}$，显然这里$P(B) = \\frac{2}{4} ≠ P(B \\mid A)$\n",
    "\n",
    "* 另外$P(A) = \\frac{3}{4}$，$P(AB) = \\frac{1}{4}$，$P(B \\mid A) = \\frac{1}{3} = \\frac{P(AB)}{P(A)}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* $P(B|A) = \\frac{P(AB)}{P(A)}$\n",
    "\n",
    "* 与前面和公式类似，对于任意事件$B_1$，$B_2$，有$P(B_1∪B_2 \\mid A) = P(B_1 \\mid A) + P(B_2 \\mid A) - P(B_1B_2 \\mid A)$\n",
    "\n",
    "* $P(AB) = P(B \\mid A)P(A)$\n",
    "\n",
    "* 推广：$P(ABC) = P(C \\mid AB)P(AB) = P(C \\mid AB)P(B \\mid A)P(A)$\n",
    "\n",
    "     * $P(ABCD) = P(D|ABC)P(C|AB)P(B|A)P(A)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "事件独立\n",
    "\n",
    "* 一般情况$P(B)≠P(B \\mid A)$，但当$A$和$B$独立时，有\n",
    "$$P(AB) = P(B \\mid A )P(A) = P(A)P(B)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 说明：A B两事件独立，则说明A的发生对比B的发生没有影响，故而有P(B)=P(B∣A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* 抛甲、乙两枚硬币，事件$A$为甲硬币出现$H$，事件$B$为乙硬币出现$H$，$S=\\{HH,HT,TH,TT\\}$，$P(B \\mid A) = P(B) = \\frac{1}{2}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "全概率公式：走向贝叶斯\n",
    "\n",
    "* $P(B \\mid A) = \\frac{P(AB)}{P(A)}$中，$P(A)$有时候很难求，有无什么办法？\n",
    "\n",
    "* 对样本空间进行划分，$\\{1,2,3,4,5,6\\}$划分成$\\{1,2,3\\}$,$\\{4,5\\}$,$\\{6\\}$\n",
    "\n",
    "* 全概率公式：$P(A)=P(A \\mid B_1)P(B_1) + \\dots + P(A \\mid B_n)P(B_n)$\n",
    "\n",
    "* $n = 2$时，$$P(A) = P(A \\mid B)P(B) + P(A \\mid \\bar{B})P{(\\bar{B})}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 全概率公式有点类似后面的边缘概率，如"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\begin{align*}\n",
    "p(x) = \\sum_{z}p(x,z) &= p(x,z_1) + p(x,z_2) + \\dots + p(x,z_n) \\\\\n",
    "                      &= p(z_1)p(z_1 \\mid x) + p(z_2)p(z_2 \\mid x) + \\dots + p(z_n)p(z_n \\mid x)\n",
    "\\end{align*}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "贝叶斯公式\n",
    "\n",
    "* 最著名的贝叶斯公式 ($n = 2$)：\n",
    "\n",
    "$$P(B \\mid A) = \\frac{P(AB)}{P(A)} = \\frac{P(BA)}{P(A)} = \\frac{P(A \\mid B)P(B)}{P(A \\mid B)P(B) + P(A \\mid \\bar{B})P(\\bar{B})}$$\n",
    "\n",
    "* $n > 2$\n",
    "\n",
    "$$P(B_i \\mid A) = \\frac{P(A \\mid B_i)P(B_i)}{\\sum_{j=1}^nP(A \\mid B_j)P(B_j)},\\hspace{0.1cm}i = 1,2,\\dots,n$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 从贝叶斯公式中可以看出B的概率在知道A的情况下发生了变化，只有当A，B是两独立事件，则P(B)=P(B|A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "贝叶斯应用举例(经典)\n",
    "\n",
    "* 假设吸毒者每次检测呈阳性(+)的概率为99%，而不吸毒每次检测呈阴性(-)的概率为99%。假设某公司对全体雇员进行吸毒检测，已知0.5%的雇员吸毒。请问每位检测结果呈阳性的雇员吸毒的概率有多高？\n",
    "\n",
    "    * $P(D) = 0.005$，代表雇员吸毒的概率\n",
    "    * $P(N) = 0.095$，代表雇员不吸毒的概率\n",
    "    * $P(+ \\mid D) = 0.99$，代表吸毒者呈阳性检出率\n",
    "    * $P(+ \\mid N) = 0.01$，代表不吸毒者阳性检出率\n",
    "    * $P(+)$，代表不考虑其它因素的影响的阳性检出率\n",
    "    \n",
    "    $$\\begin{align*}\n",
    "        P(D \\mid +) &= \\frac{P(+ \\mid D)P(D)}{P(+)} \\\\\n",
    "        &= \\frac{P(+ \\mid D)P(D)}{P(+ \\mid D)P(D) + P(+ \\mid N)(P(N))} \\\\\n",
    "        &= 0.3322\n",
    "       \\end{align*}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "贝叶斯公式再思考\n",
    "\n",
    "* 通常，$P(A \\mid B) ≠ P(B \\mid A)$，但是如何确定两者的关系？(溯源)\n",
    "\n",
    "    * 可得$P(B \\mid A) = \\frac{P(A \\mid B)P(B)}{P(A)}$\n",
    "    * 先验概率，后验概率"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 说明：AB为独立等可能事件时，P(A∣B)=P(B∣A) \n",
    "# P(B)是先验概率，P(B|A)是后验概率，P(B)的概率在A发生的情况下，会发生变化，继而有了先验和后验之分\n",
    "# 整个贝叶斯公式是由条件概率和全概率公式组成\n",
    "# AB两事件存在关联，都可以使用贝叶斯公式，且一事件是另一事件的先验，另一事件则是后验"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "垃圾邮件过滤\n",
    "\n",
    "* 给定一封邮件，判定它是否属于垃圾邮件。用$D$来表示这封邮件，注意$D$由$N$个单词组成。用$h+$来表示垃圾邮件，$h-$表示正常邮件\n",
    "\n",
    "    * $P(h+ \\mid D) = \\frac{P(D \\mid h+)P(h+)}{P(D)}$\n",
    "    * $P(d_1,d_2,\\dots,d_n \\mid h+)$扩展为$\\dots \\times P(d_3 \\mid d_1,d_2,h+) \\times P(d_2 \\mid d_1,h+) \\times P(d_1 \\mid h+)$\n",
    "    * 朴素贝叶斯的含义$P(d_3 \\mid d_1,d_2,h+) \\times P(d_2 \\mid d_1,h+) \\times P(d_1 \\mid h+)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 说明：朴素贝叶斯假设词与词之间没有关联，相互独立(用于简化问题)，继而有公式化简得\n",
    "# P(dn|h+)...P(d3|h+)P(d2|h+)P(d1|h+)\n",
    "# 这边假定垃圾邮件的概率和不是垃圾邮件的概率都为0.5，这是先验概率，实际可以通过训练数据先得到一个概率，再用于预测新邮件\n",
    "# 而对于垃圾词库概率构建也可以通过训练数据来获得，或者直接使用现成的数据"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "朴素贝叶斯\n",
    "\n",
    "垃圾邮件过滤\n",
    "\n",
    "$$P(d_1 \\mid h+) \\times P(d_2 \\mid h+) \\times P(d_3 \\mid h+) \\times \\dots = 0.007 * 0.005 * \\dots$$\n",
    "\n",
    "$$P(d_1 \\mid h-) \\times P(d_2 \\mid h-) \\times P(d_3 \\mid h-) \\times \\dots = 0.00002 * 0.0045 * \\dots$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 说明：我们需要对垃圾邮件建立一个属于垃圾邮件词汇的语料库，并预先统计出其对应的垃圾邮件和不是垃圾邮件的概率\n",
    "# 并且假定是垃圾邮件的概率和不是垃圾邮件的概率相同，即h+=h-=1/2。\n",
    "# 实际过程使用log，取加法，不然越乘越小\n",
    "# 使用模型统计出各个词是垃圾邮件和不是垃圾邮件的概率\n",
    "# 然后再对新邮件计算其P(h+∣D)和P(h-∣D)，若P(h+∣D)>>P(h-∣D)，则说明其肯定是垃圾邮件"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 随机变量\n",
    "# 理解随机变量：https://www.zybang.com/question/63bf28b7ac8a1f2cf129192ee367d539.html"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "随机变量(随机事件的数量表现)\n",
    "\n",
    "* $S_2 = \\{HHH,HHT,HTH,THH,HTT,THT,TTH,TTT\\}$，以$X$记为三次投掷得到正面$H$的总数，即样本空间中每个样本点，$X$都有一个数与之对应。\n",
    "\n",
    "* $P(A) = P\\{HHT,HTH,THH\\} = P\\{X = 2\\} = 3/8$\n",
    "\n",
    "* 随机变量的取值可能离散或者连续，注意对比\n",
    "\n",
    "    * 离散随机变量\n",
    "    * 连续随机变量"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "一维离散随机变量(和连续的对比)\n",
    "\n",
    "* 分布率PMF：$P\\{X = x_k\\} = p_k,\\hspace{0.1cm}k=1,2,\\dots$\n",
    "\n",
    "    * $p_k \\ge 0$\n",
    "    * $\\sum p_k = 1$\n",
    "    \n",
    "* 分布函数：$F_X(x) = P\\{X \\le x\\}$，因此$P\\{x_1 < X \\le x_2\\} = P\\{X \\le x_2\\} - P\\{X \\le x_1\\} = F_X(x_2) - F_X(x_1)$，即已知$X$的分布函数，就知道$X$落在任一区间的概率"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 离散的随机变量，衡量的是满足要求的样本点的分布情况，也就是分布律\n",
    "# 一般是说在随机重复多次试验，获得的样本点分布情况\n",
    "# 引入随机变量，更是从概率到统计的进阶，可以量化计算\n",
    "# 分布函数：随机变量满足某种函数关系，通过变换随机变量，可以统计出任意随机变量的概率情况"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "一维连续随机变量(和离散的对比)\n",
    "\n",
    "* 概率密度PDF：$f(x)=\\frac{d}{dx}F_X(x)$\n",
    "\n",
    "    * $f_X(x) \\ge 0$\n",
    "    * $\\int_{-∞}^{+∞}f_X(x)dx = 1$\n",
    "    * $P(x \\le X \\le x+\\Delta x) ≈ f_X(x)\\Delta x$\n",
    "    \n",
    "* $P(x_1 \\le X \\le x_2) = F_X(x_2) - F_X(x_1) = \\int_{x_1}^{x_2}f_X(x)dx$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 连续型的随机变量可以理解为连续的离散点组成，在每个点的积分比为0\n",
    "# 比如有很多情况输出是连续的数值，这样情况一般是使用计算机统计数据，比如蒙特卡洛随机模拟\n",
    "# 对比连续值采用离散型的方法显然也不合适"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 随机变量数字特征"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "期望\n",
    "\n",
    "* 数学期望(离散)：$E(X)=\\sum_{k=1}^∞x_kp_k$\n",
    "\n",
    "* 数学期望(连续)：$E(X)=\\int_{-∞}^{+∞}xf_X(x)dx$\n",
    "\n",
    "* 随机变量函数的期望: $Y = g(X)$\n",
    "\n",
    "    * 离散：$E(Y) = E(g(X)) = \\sum_{k=1}^∞g(x_k)p_k$\n",
    "    * 连续：$E(Y) = E(g(X)) = \\int_{-∞}^{+∞}g(x)f_X(x)dx$\n",
    "    \n",
    "* 期望的性质：\n",
    "\n",
    "    * 常数的期望：$E(a) = a$\n",
    "    * 定理：$E(f(X)+g(X)) = E(f(X)) + E(g(x))$\n",
    "    * 期望的期望：$E[E[X]] = E[X]$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* 例：如果 $f_X(x) = 1$，$\\forall_x ∈ [0,1]$，其余是0，则有\n",
    "\n",
    "$$E(X) = \\int_{-∞}^{+∞}xf_X(x)dx = \\int_0^1xdx = \\frac{1}{2}$$\n",
    "\n",
    "    说明：这个是一个分段的铰链函数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 数学期望：衡量离散和连续样本点的加权平均，每个样本点的权重不再是等可能的情况\n",
    "# 随机变量函数的期望：套用了分布函数，随机变量满足某种变量关系\n",
    "# 定理：函数和的期望 = 期望的函数和"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "方差\n",
    "\n",
    "* 方差(离散)：$D(X) = E\\left\\{[X-E(X)^2]\\right\\} = \\sum_{k=1}^∞[x_k-E(X)]^2p_k$，用来度量$X$与$E(X)$的偏离程度\n",
    "\n",
    "* 方差(连续)：$D(X) = E\\left\\{[X-E(X)^2]\\right\\} = \\int_{-∞}^{+∞}[x-E(X)]^2f_X(x)dx$\n",
    "\n",
    "* 方差与期望的关系：$D(X) = E(X^2) - [E(X)]^2$\n",
    "\n",
    "$$推导：由\\begin{cases}\n",
    "               \\begin{align*}\n",
    "                    &E(a) = a \\\\\n",
    "                    &E(f(X)+g(X)) = E(f(X)) + E(g(x)) \\\\\n",
    "                    &E(E(X)) = E(X)\n",
    "                 \\end{align*}\n",
    "           \\end{cases}$$\n",
    "           \n",
    "$$\\Downarrow$$\n",
    "\n",
    "$$\\begin{align*}\n",
    "    D(X) &= E\\left\\{[X-E(X)^2]\\right\\} \\\\\n",
    "         &= E\\left\\{[X-E(X)][X-E(X)]\\right\\} \\\\\n",
    "         &= E\\left\\{X^2 - 2XE(X) + [E(X)]^2\\right\\} \\\\\n",
    "         &= E(X^2) - 2[E(X)]^2 + [E(X)]^2 \\\\\n",
    "         &= E(X^2) - [E(X)]^2\n",
    "\\end{align*}$$\n",
    "\n",
    "* * 例：如果 $f_X(x) = 1$，$\\forall_x ∈ [0,1]$，其余是0，则有\n",
    "\n",
    "$$E(X) = \\int_{-∞}^{+∞}xf_X(x)dx = \\int_0^1xdx = \\frac{1}{2}$$\n",
    "\n",
    "$$E(X^2) = \\int_{-∞}^{+∞}x^2f_X(x)dx = \\int_0^1x^2dx = \\frac{1}{3}$$\n",
    "\n",
    "$$D(X) = E(X^2) - [E(X)]^2 = \\frac{1}{3} - \\frac{1}{4} = \\frac{1}{12}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 说明：方差描述的样本点与其期望(也就是均值)的偏离程度，从直观的角度也能够理解通过这种方式能够描述样本点的分布\n",
    "# 方差描述的是样本点与均值的偏差的均值，也就是期望\n",
    "# 方差用偏差的平方，而不用绝对差，是因为绝对差不是连续的值，而且不方便计算\n",
    "# 为什么有了方差还要标准差？\n",
    "# 因为方差与标准差的量纲不一致，单位也不一样，自然不方便比较大小，所以使用开方的标准差了"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 人工智能中常见分布"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "常见分布\n",
    "\n",
    "离散\n",
    "\n",
    "* (0-1) 分布，$P\\{X = k\\} = p^k(1-p)^{1-k}$，$k = 0,1$\n",
    "\n",
    "    * $E(X) = p$\n",
    "    * $D(X) = p(1-p)$\n",
    "    \n",
    "* 二项分布：设将试验独立重复进行$n$次，每次试验中，事件$A$发生的概率均为$p$，则称这$n$次试验为$n$重贝努里试验。若以$X$表示$n$重贝努里试验事件$A$发生的次数，则称$X$服从参数为$n,p$的二项分布。\n",
    "\n",
    "$$P\\{X=k\\} = C_n^kp^k(1-p)^{n-k},\\hspace{0.1cm}k=0,1,\\dots,n$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "推导二项分布的均值和方差：\n",
    "\n",
    "因二项分布是多次0-1分布的推广，而从0-1分布的期望和方差分别为$E(X) = p$和$D(X) = p(1-p)$，因二项分布每次0-1试验都是相互独立的，固有\n",
    "\n",
    "$$E(X) = E(X_1) + E(X_2) + \\dots + E(X_n) = p + p + \\dots + p = np$$\n",
    "\n",
    "而方差有：\n",
    "\n",
    "$$D(X) = D(X_1) + D(X_2) + \\dots + D(X_n) = p(1-p) + p(1-p) + \\dots + p(1-p) = np(1-p)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 说明：二项分布是对0-1分布的多次重复试验"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "高斯分布(最美的分布)\n",
    "\n",
    "高斯分布(正太分布)\n",
    "\n",
    "* $f(x \\mid μ,\\sigma^2) = \\frac{1}{\\sigma\\sqrt{2\\pi}}e^{-\\frac{(x-μ)^2}{2\\sigma^2}}$，记做 $\\mathcal{N}(μ,\\sigma^2)$\n",
    "* $E(X) = μ$\n",
    "* $D(X) = \\sigma^2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 说明：μ决定了曲线的位置，σ^2决定了曲线的形状"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "for u,sigma in zip([0,0,0,-2],np.sqrt([0.2,1.0,5.0,0.5])):\n",
    "    x = np.linspace(-5,5,1000)\n",
    "    y = np.exp(-(x-u)**2/(2*sigma**2))/(sigma*np.sqrt(2*np.pi))\n",
    "    plt.plot(x,y,linewidth=2)"
   ]
  },
  {
   "attachments": {
    "Gaussian%20distribution.png": {
     "image/png": "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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![Gaussian%20distribution.png](attachment:Gaussian%20distribution.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "中心极限定理\n",
    "\n",
    "* 设随机变量$X_1,X_2,\\dots,X_n\\dots$独立同分布，且$E(X_k) = μ$，$D(X_k) = \\sigma^2$，则当$n$充分大时，$\\bar{X}$近似服从高斯分布：\n",
    "\n",
    "$$\\bar{X} = \\frac{1}{n}\\sum_{k=1}^nX_k \\sim \\mathcal{N}\\left(μ,\\frac{\\sigma^2}{n}\\right)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 说明：每个随机变量都服从高斯分布，并且相互独立，则其均值分布也服从高斯分布\n",
    "# 若每个随机变量对应的观测值服从同一个高斯分布，则这多个随机变量对应的均值也应服从高斯分布"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "多维随机变量\n",
    "\n",
    "* 以二维随机变量为例(离散和连续相互对比)\n",
    "\n",
    "* 离散\n",
    "    * 联合分布率：$P\\{X=x_i,Y=y_i\\} = p_{ij},\\hspace{0.1cm}i,j = 1,2,\\dots$\n",
    "    * 联合分布函数：$F(x,y) = \\sum_{x_i \\le x, y_j \\le y}p_{ij}$\n",
    "    * 边缘分布率：$P\\{X = x_i\\} = \\sum_{j=1}^∞p_{ij},\\hspace{0.1cm}i,j = 1,2,\\dots$\n",
    "    * 条件分布率：$P\\{X = x_i \\mid Y = y_j\\} = \\frac{P\\{X=x_i,Y=y_i\\}}{P\\{Y=y_j\\}}$ 回忆$\\left(P(B \\mid A) = \\frac{P(AB)}{P(A)}\\right)$\n",
    "    \n",
    "* 连续\n",
    "    * 联合概率密度：$f(x,y)$\n",
    "    * 联合分布函数：$F(x,y) = \\int_{-∞}^y\\int_{-∞}^xf(u,v)dudv$\n",
    "    * 边缘概率密度：$f_X(x) = \\int_{-∞}^{+∞}f(x,y)dy$\n",
    "    * 条件概率密度：$f_{X \\mid Y}(x \\mid y) = \\frac{f(x,y)}{f_Y(y)}$\n",
    "    \n",
    "* 协方差：$Cov(X,Y) = E\\{[X-E(X)][Y-E(Y)]\\} = E(XY) - E(X)E(Y)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 说明：积分近似几何意义：一重积分 - 面积；二重积分 - 体积；三重积分 - 质量、时空等\n",
    "# X=Y时，协方差Cov(X,Y) = 方差D(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "多元高斯分布\n",
    "\n",
    "多元概率密度函数：\n",
    "\n",
    "$$f_X(x_1,\\dots,x_k) = \\frac{1}{\\sqrt{(2\\pi)^k|\\mathbf{Σ}|}}e^{\\left(-\\frac{1}{2}(\\mathbf{x-μ})^TΣ^{-1}(\\mathbf{x-μ})\\right)}$$\n",
    "\n",
    "\n",
    "$$\\begin{align*}\n",
    "    \\mathbf{Σ} &=\\left[\n",
    "                    \\begin{matrix}\n",
    "                        Cov[X_1,X_1] & \\dots & Cov[X_1,X_n] \\\\\n",
    "                        \\vdots & \\ddots & \\vdots \\\\\n",
    "                        Cov[X_n,X_1] & \\dots & Cov[X_n,X_n]\n",
    "                    \\end{matrix}\n",
    "                 \\right] \\\\\n",
    "               &=\\left[\n",
    "                    \\begin{matrix}\n",
    "                        E[X_1^2] - E[X_1]E[X_1] & \\dots & E[X_1X_n] - E[X_1]E[X_n] \\\\\n",
    "                        \\vdots & \\ddots & \\vdots \\\\\n",
    "                        E[X_nX_1] - E[X_n]E[X_1] & \\dots & E[X_n^2] - E[X_n]E[X_n]\n",
    "                    \\end{matrix}\n",
    "                 \\right] \\\\\n",
    "               &=\\left[\n",
    "                    \\begin{matrix}\n",
    "                        E[X_1^2] & \\dots & E[X_nX_1] \\\\\n",
    "                        \\vdots & \\ddots & \\vdots \\\\\n",
    "                        E[X_nX_1] & \\dots & E[X_n^2]\n",
    "                    \\end{matrix}\n",
    "                 \\right] - \n",
    "                 \\left[\n",
    "                    \\begin{matrix}\n",
    "                        E[X_1]E[X_1] & \\dots & E[X_1]E[X_n] \\\\\n",
    "                        \\vdots & \\ddots & \\vdots \\\\\n",
    "                        E[X_n]E[X_1] & \\dots & E[X_n]E[X_n]\n",
    "                    \\end{matrix}\n",
    "                 \\right] \\\\\n",
    "               &= E[XX^T] - E[X]E[X]^T = \\dots = E[(X-E[X])(X-E[X])^T]\n",
    "\\end{align*}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "证明协方差矩阵半正定：\n",
    "\n",
    "已知：\n",
    "\n",
    "$$X = [X_1,X_2,\\dots,X_n]^T$$\n",
    "\n",
    "$$μ = [μ_1,μ_2,\\dots,μ_n]^T$$\n",
    "\n",
    "$$z = [z_1,z_2,\\dots,z_n]^T$$\n",
    "\n",
    "协方差矩阵$Σ$两边分别乘以任意向量$z^T$和$z$，得\n",
    "\n",
    "$$\\begin{align*}\n",
    "z^TΣz &=\n",
    "\\begin{bmatrix}\n",
    "    z_1 & \\dots & z_n\n",
    "\\end{bmatrix}E[(X-μ)(X-μ)^T]\n",
    "\\begin{bmatrix}\n",
    "    z_1 \\\\\n",
    "    \\vdots \\\\\n",
    "    z_n\n",
    "\\end{bmatrix} \\\\\n",
    "      &= \n",
    "\\begin{bmatrix}\n",
    "    z_1 & \\dots & z_n\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\n",
    "    E[(X_1-μ_1)(X_1-μ_1)] & \\dots & E[(X_1-μ_1)(X_n-μ_n)] \\\\\n",
    "    \\vdots & \\ddots & \\vdots \\\\\n",
    "    E[(X_n-μ_n)(X_1-μ_1)] & \\dots & E[(X_n-μ_n)(X_n-μ_n)]\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\n",
    "    z_1 \\\\\n",
    "    \\vdots \\\\\n",
    "    z_n\n",
    "\\end{bmatrix} \\\\\n",
    "      &= \n",
    "\\begin{bmatrix}\n",
    "    E[z_1(X_1-μ_1)(X_1-μ_1)] & \\dots & E[z_1(X_1-μ_1)(X_n-μ_n)] \\\\\n",
    "    \\vdots & \\ddots & \\vdots \\\\\n",
    "    E[z_n(X_n-μ_n)(X_1-μ_1)] & \\dots & E[z_n(X_n-μ_n)(X_n-μ_n)]\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\n",
    "    z_1 \\\\\n",
    "    \\vdots \\\\\n",
    "    z_n\n",
    "\\end{bmatrix} \\\\\n",
    "      &=\n",
    "\\begin{bmatrix}\n",
    "    E[z_1(X_1-μ_1)(X_1-μ_1)z_1] & \\dots & E[z_1(X_1-μ_1)(X_n-μ_n)z_1] \\\\\n",
    "    \\vdots & \\ddots & \\vdots \\\\\n",
    "    E[z_n(X_n-μ_n)(X_1-μ_1)z_n] & \\dots & E[z_n(X_n-μ_n)(X_n-μ_n)z_n]\n",
    "\\end{bmatrix} \\\\\n",
    "      &=\n",
    "\\left[\\sum_{i=1}^n\\sum_{j=1}^nE[z_i(X_i-μ_i)(X_j-μ_j)z_j]\\right] \\\\\n",
    "      &=\n",
    "\\left(\\sum_{i=1}^nE[z_i(X_i-μ_i)]\\right)\\left(\\sum_{j=1}^nE[z_j(X_j-μ_j)]\\right) \\\\\n",
    "      &=\n",
    "\\left(E[\\sum_{i=1}^nz_i(X_i-μ_i)]\\right)\\left(E[\\sum_{j=1}^nz_j(X_j-μ_j)]\\right)\n",
    "\\end{align*}$$\n",
    "\n",
    "因有$ \\sum_{i=1}^nz_i(X_i-μ_i) = \\sum_{j=1}^nz_j(X_j-μ_j)$，并令其为随机变量$Y$，进一步推导得：\n",
    "\n",
    "$z^TΣz = E[Y]^2 \\ge 0$，得正$Σ$半正定"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "证明协方差矩阵不仅是半正定矩阵，且是对称矩阵\n",
    "\n",
    "由对称矩阵的性质$\\mathbf{A}^T = \\mathbf{A}$，而协方差矩阵为$\\mathbf{Σ} = E[(X-E[X])(X-E[X])^T]$，对$E[(X-E[X])(X-E[X])^T]$进行转置，仍然是$E[(X-E[X])(X-E[X])^T]$，故而证得协方差矩阵为$\\mathbf{Σ}$是对称矩阵"
   ]
  }
 ],
 "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": "3.6.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}