6_Sample_and_Sampling_Distribution.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 总体、样本"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "样本与总体的概念，在前面已经多次接触到，只是没有非常明确的给出定义。例如我们一般假设某个学校全体学生的身高，用随机变量X表示，近似服从正态分布，这里的正态分布就是\"总体\"的分布，字面意思就是所有待研究对象的集合。在实际的数据分析中，我们通过观察或其他测量方式得到的数据一般都只是待研究对象的一个子集，这个子集就是一个样本（可以包含多个个体）。例如通过某种方式，从全体学生中找出100名学生，这100名学生就是一个样本。\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 概念\n",
    "1. 总体： 研究对象的全部个体\n",
    "2. 个体： 总体中的一个\n",
    "3. 总体容量： 总体中的个体的多少\n",
    "4. 有限总体：容量有限的总体\n",
    "5. 无限总体：容量无限(很大)的总体"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 样本、总体、随机变量\n",
    "1. 样本：总体中的一些个体组成样本\n",
    "2. 随机变量：研究总体某个指标X(如，身高)，对于不同的个体有不同的取值，这些取值构成一个分布。因此可以称$X$是一个随机变量\n",
    "3. 有时候直接将$X$称为总体. 假设X的分布函数为$F(x)$,也称总体$X$具有分布$F(x)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 简单随机样本\n",
    "  (简单地说就是，独立同分布的样本)\n",
    "  \n",
    "![](images/6_1.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 简单随机抽样\n",
    "1. 简单随机抽样：获取简单随机样本\n",
    "2. 如何进行抽样  \n",
    "    + 放回抽样：样本容量有限(小)\n",
    "    + 不放回抽样：样本容量无限(很大)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 统计量\n",
    "## 定义\n",
    ">统计量：样本的不含有任何未知参数的**函数**(如，平均值函数等)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 常用统计量\n",
    "1. 样本均值：$$\\bar{X} = \\frac{1}{n} \\sum_{i=1}^{n}X_i$$\n",
    "2. 样本方差(注意分母是n-1): $$S^2 = \\frac{1}{n-1} \\sum_{i=1}^{n}(X_i - \\bar{X})^2$$  \n",
    "样本标准差：$$S = \\sqrt{S^2}$$  \n",
    "3. 样本矩：  \n",
    "k阶矩：$$A_k = \\frac{1}{n} \\sum_{i=1}^{n}X_i^k$$  \n",
    "k阶中心距：$$B_k = \\frac{1}{n} \\sum_{i=1}^{n}(X_i - \\bar{X}^k)$$  \n",
    "$$k = 1,2,\\cdots$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 相关问题\n",
    "#### $\\bar{X} = \\mu$？\n",
    "![](images/6_2.png)\n",
    "#### 样本方差分母为什么的n-1\n",
    "##### 无偏估计\n",
    "1. 简单定义：估计量的期望等于估计参数的真值\n",
    "2. 理解\n",
    "设想一下，想知道全体女性的身高均值$\\mu$ ,但是我们没有办法对每个女性进行测量，只能抽一部分人来估计全体女性的身高，那么根据抽样的数据如何进行推断？什么样的推断方法才称得上‘准确’？\n",
    "<br>\n",
    "比如：我们得到的样本女性身高为：\n",
    "<center>${x_1,x_2···,x_n}$</center>\n",
    "那么，\n",
    "<br>\n",
    "<center>$\\bar{x} = \\frac{x_1+x_2+···+x_n}{n}$</center>\n",
    "<br>\n",
    "对于$\\mu$来说是一个不错的估计，因为它是无偏估计。(即，$\\bar{X}$可以近似的代替$\\mu$)\n",
    "<br>\n",
    "3. 方差$\\sigma^2$的无偏估计    \n",
    "假设：$\\mu$已知，而$\\sigma^2$未知\n",
    "<br>\n",
    "由方差的定义有(对于单个样本)： <br> \n",
    "$E[(X_i - \\mu)^2] = \\sigma^2, i= 1,2,···,n$\n",
    "<br>\n",
    "$=>$\n",
    "<br>\n",
    "(对于所有样本)\n",
    "<br>\n",
    "$E[\\frac{1}{n} \\displaystyle \\sum_{i=1}^{n}(X_i - \\mu)^2] = \\frac{1}{n}\\displaystyle \\sum_{i=1}^{n}E(X_i-\\mu)^2 = \\frac{1}{n} \\times n\\sigma^2 = \\sigma^2$\n",
    "<br>\n",
    "$=>$\n",
    "<br>\n",
    "$\\frac{1}{n} \\displaystyle \\sum_{i=1}^{n}(X_i - \\mu)^2$ 是方差$\\sigma^2$的一个无偏估计    \n",
    "\n",
    "这个结果符合直觉，并且在数学上也是显而易见的。  \n",
    "现在，我们考虑随机变量$X$的数学期望$\\mu$是未知的情形。这时，我们会倾向于无脑直接用样本均值$\\bar{X}$替换掉上面式子中的$\\mu$。这样做有什么后果呢？后果就是，  \n",
    "**如果直接使用$\\frac{1}{n} \\displaystyle \\sum_{i=1}^{n}(X_i - \n",
    "\\bar{X})^2$作为估计，那么你会倾向于低估方差！**  \n",
    "这是因为：  \n",
    "$\\frac{1}{n} \\displaystyle \\sum_{i=1}^{n}(X_i - \\bar{X})^2 $  \n",
    "$=\\frac{1}{n} \\displaystyle \\sum_{i=1}^{n}[(X_i - \\mu) + (\\mu - \\bar{X})]^2$   $=\\frac{1}{n} \\displaystyle \\sum_{i=1}^{n}(X_i - \\mu)^2 + \\frac{2}{n} \\displaystyle \\sum_{i=1}^{n}(X_i -\\mu)(\\mu-\\bar{X})+\\frac{1}{n} \\displaystyle \\sum_{i=1}^{n}(\\mu-\\bar{X})^2$  \n",
    "$=\\frac{1}{n} \\displaystyle \\sum_{i=1}^{n}(X_i -\\mu)^2 + 2(\\bar{X}-\\mu)(\\mu-\\bar{X})+(\\mu-\\bar{X})^2\\\\= \\frac{1}{n} \\displaystyle \\sum_{i=1}^{n}(X_i - \\mu)^2 - (\\mu - \\bar{X})^2$\n",
    "\n",
    "换言之，除非正好${\\bar{X} = \\mu}$，否则我们一定有\n",
    "$\\frac{1}{n} \\displaystyle \\sum_{i=1}^{n}(X_i - \\bar{X})^2  < \\frac{1}{n} \\displaystyle \\sum_{i=1}^{n}(X_i - \\mu^2 $\n",
    "\n",
    "这个不等式说明了为什么直接使用$\\frac{1}{n} \\displaystyle \\sum_{i=1}^{n}(X_i - \\bar{X})^2 $会导致结果出现偏差  \n",
    "那么，在不知道随机变量真实数学期望的前提下，如何“正确”的估计方差呢？  \n",
    "##### 样本方差\n",
    "定义：  \n",
    "设$X_1,…,X_n$是随机变量X的$n$个样本，则样本方差定义为\n",
    "$s^2 = \\frac{1}{n-1}\\displaystyle \\sum_{i=1}^{n}(X_i - n\\bar{X})^2$  \n",
    "其中，$\\bar{X}为样本均值$  \n",
    "根据定义可以得出：\n",
    "$s^2 = \\frac{1}{n-1}\\displaystyle \\sum_{i=1}^{n}(X_i - n\\bar{X})^2 \\\\= \\frac{1}{n-1}\\displaystyle \\sum_{i=1}^{n}(X_i^2 -  2n\\bar{X}^2 + \\bar{X}^2) \\\\= \\frac{1}{n-1}(\\displaystyle \\sum_{i=1}^{n}X_i^2 - n\\bar{X}^2)$   \n",
    "无偏性：  \n",
    "$$E(s^2) = \\dfrac{1}{n-1}(n\\sigma^2 + n\\mu^2 - n(\\dfrac{\\sigma^2}{n} + \\mu^2)) = \\sigma^2$$  \n",
    "其中：\n",
    "$\\displaystyle \\sum_{i=1}^{n}E(X_i^2) = D(X) + [E(X)]^2 = (\\sigma^2 + \\mu^2)\\times n$  \n",
    "$E(\\bar{X}^2) = D(\\bar{X}) + [E(\\bar{X})]^2 = D(\\frac{X_1+X_2+…+X_n}{n}) + \\mu^2 = \\frac{1}{n^2}D(X_1+X_2+…+X_n) + \\mu^2 = \\frac{1}{n^2}n\\sigma^2 + \\mu^2 = \\frac{\\sigma^2}{n} + \\mu^2\\$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 抽样分布\n",
    "## 定义\n",
    ">当总体X服从一般分布（如指数分布、均匀分布等），要得出统计量的分布是很困难的；当总体X服从正态分布时，统计量$\\bar{X}$、$S^2$是可以计算的，且服从一定的分布。这些分布就是下面要介绍的三大抽样分布——$χ^2$分布，$t$分布，$F$分布。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 分位数/分位点\n",
    "分位数是一个非常重要的概念，一开始也有点难理解。首先要明确一点，分位数分的是**面积**。更准确的说，分位数分的是**某个特定分布的概率密度函数曲线下的面积**。每给定一个分位数，这个概率密度函数曲线就被该点一分为二。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 四分位数（Quartile）\n",
    ">四分位数（Quartile）也称四分位点，是指在统计学中把所有数值由小到大排列并分成四等份，处于三个分割点位置的数值。  \n",
    "例如1， 3， 5， 7， 9， 11，其3个四分位点分别是3，6，9。分别叫做第一四分位数（$Q_1$或者$x_{0.25}$），第二四分位数（$Q_2$或者$x_{0.5}$），第三四分位数（$Q_3$或者$x_{0.75}$）。(认识最多的就是$Q_2$,也就是中位数)\n",
    "\n",
    "对于概率密度函数来说，四分位点就是将概率密度曲线下的面积均分为4部分的点。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 利用pd.DataFrame().describe()实现统计数据显示(包括分位数)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-28T12:54:39.857521Z",
     "start_time": "2019-10-28T12:54:39.820347Z"
    }
   },
   "outputs": [
    {
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       "               0\n",
       "count  10.000000\n",
       "mean   46.700000\n",
       "std    23.127905\n",
       "min     3.000000\n",
       "25%    37.500000\n",
       "50%    46.500000\n",
       "75%    59.750000\n",
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   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "\n",
    "# 获取1到100的10个随机整数\n",
    "sample = np.random.randint(1, 100, 10)\n",
    "# 显示统计数据：平均值，标准差，最小值，分位数\n",
    "pd.DataFrame(sample).describe()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 上$\\alpha$分位数(Upper Percentile)\n",
    ">上$α$分位数是概率密度函数定义域内的一个数值，这个数值将概率密度函数曲线下的面积沿x轴分成了两个部分，其中该点右侧部分概率密度函数曲线与x轴围成的面积等于$α$。  \n",
    "\n",
    "![](images/6_4.png)\n",
    "由于概率密度函数曲线下的面积就是概率，因此上α分位数中的α既是该点右侧区域的面积，也是在这个分布中取到所有大于该点的值的概率。  \n",
    "即：$$p(x≥x_{\\alpha}) = \\alpha$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "此时有两个值，一个是$α$，另一个是$x_α$。这两个值中确定其中一个，另一个值也就确定了。因此我们可以通过一个给定的α值，求在某个特定分布中的上$α$分位数，即$x_α$，的值；也可以在某个特定分布中，任意给定一个定义域内的点x，求取到比该点的值更大的值的概率，即该点的$α$值。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## $\\chi^2$分布\n",
    "### 定义\n",
    "![](images/6_3.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 概率密度(了解)\n",
    "![](images/6_5.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](images/6_6.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "由上图可以看出，随着自由度n的增加，卡方分布越近似正态分布"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 性质\n",
    "![](images/6_8.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "证明1：  \n",
    "$E(\\chi^2)\\\\=E(\\sum_{i=1}^{n}X_i^2)\\\\=\\sum_{i=1}^{n}E(X_i^2)\\\\=\\sum_{i=1}^{n}[D(X) + [E(x)]^2 ]\\\\=\\sum_{i=1}^{n}D(X)\\\\=n$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$D(\\chi^2)\\\\=D(\\sum_{i=1}^{n}X_i^2)\\\\=\\sum_{i=1}^{n}D(X_i^2)\\\\=\\sum_{i=1}^{n}[E(X_i^4) - [E(X_i^2)]^2]\\\\=\\sum_{i=1}^{n}(3-1)\\\\=2n$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "证明2：  \n",
    "$$Y_1 = \\sum_{i=1}^{n_1}X_i$$  \n",
    "$$Y_2 = \\sum_{i=1}^{n_2}X_i$$  \n",
    "由于：$$X_i \\sim N(0, 1)$$  \n",
    "所以：  \n",
    "$$Y_1 + Y_2 = \\sum_{i=1}^{n_1 + n_2}X_i$$  \n",
    "即：  \n",
    "$$Y_1 + Y_2 \\sim \\chi^2(n_1 + n_2)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 上$\\alpha$分位数"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](images/69.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$P{ [\\chi^2>\\chi^2_{\\alpha}(n)] } = \\int_{\\chi^2_{\\alpha}(n)}^{\\infty}f(y)dy =\\alpha$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 上$\\alpha$分位数表以及如何查看"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](images/70.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. 确定自由度(n)\n",
    "2. 确定分位点$\\alpha$\n",
    "3. 读出数据  \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](images/6_9.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## t分布"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 定义"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](images/6_10.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-30T09:00:41.952833Z",
     "start_time": "2019-10-30T09:00:41.905070Z"
    }
   },
   "source": [
    "![](images/6_11.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 上$\\alpha$分位数(上侧分位数)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](images/6_12.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 性质\n",
    "1. t分布的概率密度函数为偶函数\n",
    "2. 对于任意实数x，当$n->\\infty$,t分布近似于标准正态分布$N(0, 1)$\n",
    "$$\\lim_{n->\\infty}f_T(x) = \\frac{1}{\\sqrt{2\\pi}}e^{-\\frac{1}{2}x^2}$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](images/6_17.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](images/6_13.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](images/6_14.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 例题\n",
    "![](images/6_15.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "![](images/6_16.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## F分布"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 定义\n",
    "\n",
    "![](images/6_18.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 上$\\alpha$分位数"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](images/6_19.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 如何查表\n",
    "![](images/6_20.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 小结"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](images/6_21.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 单个正态总体的抽样分布"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](images/611.png)\n",
    "![](images/612.png)\n",
    "![](images/613.png)\n",
    "![](images/614.png)\n",
    "![](images/615.png)\n",
    "![](images/616.png)\n",
    "![](images/617.png)\n",
    "![](images/618.png)\n",
    "![](images/619.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 两个正态总体的抽样分布"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](images/620.png)\n",
    "![](images/621.png)\n",
    "![](images/622.png)\n",
    "![](images/623.png)\n",
    "![](images/624.png)\n",
    "![](images/625.png)\n",
    "![](images/626.png)\n",
    "![](images/627.png)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# python实现"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-30T10:43:29.993930Z",
     "start_time": "2019-10-30T10:43:20.506079Z"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy import stats"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 统计数据"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-30T02:17:29.756949Z",
     "start_time": "2019-10-30T02:17:29.740169Z"
    }
   },
   "outputs": [],
   "source": [
    "def get_stats(dist_sample):\n",
    "    sample = pd.DataFrame(dist_sample, columns=['sample'])\n",
    "    sample['mean'] = np.mean(sample['sample'])\n",
    "    sample['X_i - X_mean'] = sample['sample'] - sample['mean']\n",
    "    sample['(X_i - X_mean)^2'] = np.power(sample['X_i - X_mean'], 2)\n",
    "    sample['S^2'] = np.std(dist_sample)\n",
    "    sample['B_2'] = ((np.alen(dist_sample)-1) / np.alen(dist_sample)) * sample['S^2']\n",
    "    return sample"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-30T02:17:56.651518Z",
     "start_time": "2019-10-30T02:17:56.561907Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "     sample      mean  X_i - X_mean  (X_i - X_mean)^2       S^2       B_2\n",
      "0  0.030305  0.589302     -0.558997          0.312478  0.321702  0.289532\n",
      "1  0.932017  0.589302      0.342714          0.117453  0.321702  0.289532\n",
      "2  0.691355  0.589302      0.102053          0.010415  0.321702  0.289532\n",
      "3  0.956281  0.589302      0.366979          0.134673  0.321702  0.289532\n",
      "4  0.571374  0.589302     -0.017928          0.000321  0.321702  0.289532\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "10"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "uniform_sample = np.random.rand(10)\n",
    "print(get_stats(uniform_sample).head())\n",
    "_ = pd.Series(uniform_sample)\n",
    "np.alen(uniform_sample)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-30T02:18:16.272289Z",
     "start_time": "2019-10-30T02:18:16.245255Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "     sample      mean  X_i - X_mean  (X_i - X_mean)^2       S^2       B_2\n",
      "0 -1.386327 -0.499904     -0.886423          0.785746  0.621319  0.559187\n",
      "1  0.867403 -0.499904      1.367307          1.869528  0.621319  0.559187\n",
      "2 -0.207902 -0.499904      0.292002          0.085265  0.621319  0.559187\n",
      "3 -0.450259 -0.499904      0.049644          0.002465  0.621319  0.559187\n",
      "4 -0.065373 -0.499904      0.434531          0.188817  0.621319  0.559187\n"
     ]
    }
   ],
   "source": [
    "normal_sample = np.random.randn(10)\n",
    "print(get_stats(normal_sample).head())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## $\\chi^2$分布"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](images/6_22.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 绘图"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-30T02:48:03.990610Z",
     "start_time": "2019-10-30T02:48:03.605479Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def chi2_distribution(df_list=[1,2,3,4]):\n",
    "    \"\"\"\n",
    "    实现不同自由度的卡方分布\n",
    "    : df :degree freedom 自由度，也就是卡方分布的n\n",
    "    \"\"\"\n",
    "    # 参见chi^2函数\n",
    "    chi2_dis_1 = stats.chi2(df_list[0])\n",
    "    chi2_dis_2 = stats.chi2(df_list[1])\n",
    "    chi2_dis_3 = stats.chi2(df_list[2])\n",
    "    chi2_dis_4 = stats.chi2(df_list[3])\n",
    "    \n",
    "    # 设置x轴\n",
    "    x_1 = np.linspace(chi2_dis_1.ppf(0.65),\n",
    "                      chi2_dis_1.ppf(0.999), 100)\n",
    "    x_2 = np.linspace(chi2_dis_2.ppf(0.0001),\n",
    "                      chi2_dis_2.ppf(0.9999), 100)\n",
    "    x_3 = np.linspace(chi2_dis_3.ppf(0.001),\n",
    "                      chi2_dis_3.ppf(0.999), 100)\n",
    "    x_4 = np.linspace(chi2_dis_4.ppf(0.001),\n",
    "                      chi2_dis_4.ppf(0.999), 100)\n",
    "    \n",
    "    # 计算概率密度函数\n",
    "    pdf_1 = chi2_dis_1.pdf(x_1)\n",
    "    pdf_2 = chi2_dis_2.pdf(x_2)\n",
    "    pdf_3 = chi2_dis_3.pdf(x_3)\n",
    "    pdf_4 = chi2_dis_4.pdf(x_4)\n",
    "    \n",
    "    # 画图\n",
    "    fig, ax = plt.subplots(1,1)\n",
    "    ax.plot(x_1, pdf_1, 'g-', lw=2, label='df={}'.format(df_list[0]))\n",
    "    ax.plot(x_2, pdf_2, 'r-', lw=2, label='df={}'.format(df_list[1]))    \n",
    "    ax.plot(x_3, pdf_3, 'b-', lw=2, label='df={}'.format(df_list[2]))    \n",
    "    ax.plot(x_4, pdf_4, 'y-', lw=2, label='df={}'.format(df_list[3]))    \n",
    "    \n",
    "    plt.ylabel('Probablity')\n",
    "    plt.title(r'PDF of $\\chi^2 Dist.$')\n",
    "    ax.legend(loc='best', frameon=False)\n",
    "    plt.show()\n",
    "\n",
    "chi2_distribution([1,4,10,20])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 分位数计算\n",
    "例如计算:  \n",
    "\n",
    "$$P{ [\\chi^2(10)>20.483] } = ?$$\n",
    "即，$$\\chi^2_\\alpha(10)=20.483$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-30T10:56:33.436146Z",
     "start_time": "2019-10-30T10:56:33.409457Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.026155741012841877\n",
      "0.025001449464351087\n"
     ]
    }
   ],
   "source": [
    "# 计算概率密度\n",
    "def calculate_cdf(df, upper_percent):\n",
    "    \"\"\"\n",
    "    准确度比较低\n",
    "    : df :自由度\n",
    "    : upper_percentile: 上α分位数值\n",
    "    \"\"\"\n",
    "    return stats.chi2.cdf(df, upper_percent)\n",
    "\n",
    "# 计算自由度为10， 分位点α为20.483的概率\n",
    "print(calculate_cdf(10, 20.483))\n",
    "\n",
    "def calculate_cdf_2(df, upper_percent):\n",
    "    \"\"\"\n",
    "    准确度高\n",
    "    : sf:survival function\n",
    "    \"\"\"\n",
    "    return stats.chi2.sf(upper_percent, df)\n",
    "print(calculate_cdf_2(10, 20.483))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "例如计算：\n",
    "$$P[\\chi^2(10)>\\chi^2_{\\alpha}(10)] = 0.05$$\n",
    "$$求\\chi^2_{\\alpha(10)} = ?$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-30T11:00:49.299035Z",
     "start_time": "2019-10-30T11:00:49.286774Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "18.30703805327515\n"
     ]
    }
   ],
   "source": [
    "def calculate_upper_percentile(df, probablity):\n",
    "    \"\"\"\n",
    "    ：df：自由度\n",
    "    ：probability：概率\n",
    "    \"\"\"\n",
    "    return stats.chi2.isf(probablity, df)\n",
    "\n",
    "print(calculate_upper_percentile(10, 0.05))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## t分布"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](images/6_23.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 绘图"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-30T11:31:17.660112Z",
     "start_time": "2019-10-30T11:31:17.174514Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def t_distribution(df_list=[1,2,3,4]):\n",
    "    \"\"\"\n",
    "    实现不同自由度的卡方分布\n",
    "    : df :degree freedom 自由度，也就是卡方分布的n\n",
    "    \"\"\"\n",
    "    # 创建t分布函数\n",
    "    t_dis_1 = stats.t(df_list[0])\n",
    "    t_dis_2 = stats.t(df_list[1])\n",
    "    t_dis_3 = stats.t(df_list[2])\n",
    "#     t_dis_4 = stats.t(df_list[3])\n",
    "    \n",
    "    # 创建标准正态分布函数\n",
    "    norm_dist = stats.norm()\n",
    "    \n",
    "    # 设置x轴\n",
    "#     x_1 = np.linspace(t_dis_1.ppf(0.004),\n",
    "#                       t_dis_1.ppf(0.999), 1000)\n",
    "#     x_2 = np.linspace(t_dis_2.ppf(0.001),\n",
    "#                       t_dis_2.ppf(0.999), 1000)\n",
    "#     x_3 = np.linspace(t_dis_3.ppf(0.001),\n",
    "#                       t_dis_3.ppf(0.999), 1000)\n",
    "#     x_4 = np.linspace(t_dis_4.ppf(0.001),\n",
    "#                       t_dis_4.ppf(0.999), 1000)\n",
    "    x = np.linspace(-4, 4, 100)\n",
    "    \n",
    "    # 计算概率密度函数\n",
    "    pdf_1 = t_dis_1.pdf(x)\n",
    "    pdf_2 = t_dis_2.pdf(x)\n",
    "    pdf_3 = t_dis_3.pdf(x)\n",
    "#     pdf_4 = t_dis_4.pdf(x)\n",
    "    pdf_5 = norm_dist.pdf(x)\n",
    "    \n",
    "    # 画图\n",
    "    fig, ax = plt.subplots(1,1)\n",
    "    ax.plot(x, pdf_1, 'g-', lw=2, label=r'$X \\sim t({})$'.format(df_list[0]))\n",
    "    ax.plot(x, pdf_2, 'r-', lw=2, label=r'$X \\sim t({})$'.format(df_list[1]))    \n",
    "    ax.plot(x, pdf_3, 'b-', lw=2, label=r'$X \\sim t({})$'.format(df_list[2]))    \n",
    "#     ax.plot(x, pdf_4, 'y-', lw=2, label=r'X \\sim \\chi^2({})'.format(df_list[3]))    \n",
    "    ax.plot(x, pdf_5, 'k-', lw=2, label=r'$X\\sim N(0,1)$')    \n",
    "    \n",
    "    plt.ylabel('Probablity')\n",
    "    plt.title(r'PDF of t Dist.')\n",
    "    ax.legend(loc='best', frameon=False)\n",
    "    plt.show()\n",
    "\n",
    "t_distribution([1,5,10])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 分位数计算\n",
    "例如计算:  \n",
    "\n",
    "$$P{ [t(10)>1.8125] } = ?$$\n",
    "即，$$t_\\alpha(10)=1.8125$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-30T11:22:55.653177Z",
     "start_time": "2019-10-30T11:22:55.638059Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.04999682852392339\n"
     ]
    }
   ],
   "source": [
    "def calculate_cdf_2(df, upper_percent):\n",
    "    \"\"\"\n",
    "    计算t分布的上侧分位数\n",
    "    准确度高\n",
    "    : sf:survival function\n",
    "    \"\"\"\n",
    "    return stats.t.sf(upper_percent, df)\n",
    "print(calculate_cdf_2(10, 1.8125))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "例如计算：\n",
    "$$P[t(10)>t_{\\alpha}(10)] = 0.05$$\n",
    "$$求t_{\\alpha(10)} = ?$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-30T11:23:57.351031Z",
     "start_time": "2019-10-30T11:23:57.336789Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.8124611228107341\n"
     ]
    }
   ],
   "source": [
    "def calculate_upper_percentile(df, probabilty):\n",
    "    \"\"\"\n",
    "    计算t分布上侧分位数对应的值\n",
    "    : df:自由度\n",
    "    : probability:概率\n",
    "    \"\"\"\n",
    "    return stats.t.isf(probabilty, df)\n",
    "print(calculate_upper_percentile(10, 0.05))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### F分布"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](images/6_24.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-30T11:34:32.995257Z",
     "start_time": "2019-10-30T11:34:32.587492Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def t_distribution(df_list=[1,2,3,4], df_list_2=[1,2,3,4]):\n",
    "    \"\"\"\n",
    "    实现不同自由度的F分布\n",
    "    : df_list :degree freedom 自由度，也就是F分布的n1\n",
    "    : df_list_2 :degree freedom 自由度，也就是F分布的n_2\n",
    "    \"\"\"\n",
    "    # 创建t分布函数\n",
    "    f_dis_1 = stats.f(df_list[0], df_list_2[0])\n",
    "    f_dis_2 = stats.f(df_list[1], df_list_2[1])\n",
    "    f_dis_3 = stats.f(df_list[2], df_list_2[2])\n",
    "#     t_dis_4 = stats.t(df_list[3])\n",
    "    \n",
    "    # 设置x轴\n",
    "#     x_1 = np.linspace(t_dis_1.ppf(0.004),\n",
    "#                       t_dis_1.ppf(0.999), 1000)\n",
    "#     x_2 = np.linspace(t_dis_2.ppf(0.001),\n",
    "#                       t_dis_2.ppf(0.999), 1000)\n",
    "#     x_3 = np.linspace(t_dis_3.ppf(0.001),\n",
    "#                       t_dis_3.ppf(0.999), 1000)\n",
    "#     x_4 = np.linspace(t_dis_4.ppf(0.001),\n",
    "#                       t_dis_4.ppf(0.999), 1000)\n",
    "    x = np.linspace(0.01, 10, 100)\n",
    "    \n",
    "    # 计算概率密度函数\n",
    "    pdf_1 = f_dis_1.pdf(x)\n",
    "    pdf_2 = f_dis_2.pdf(x)\n",
    "    pdf_3 = f_dis_3.pdf(x)\n",
    "#     pdf_4 = t_dis_4.pdf(x)\n",
    "    \n",
    "    # 画图\n",
    "    fig, ax = plt.subplots(1,1)\n",
    "    ax.plot(x, pdf_1, 'g-', lw=2, label=r'$X \\sim F({}, {})$'.format(df_list[0], df_list_2[0]))\n",
    "    ax.plot(x, pdf_2, 'r-', lw=2, label=r'$X \\sim F({}, {})$'.format(df_list[1], df_list_2[1]))    \n",
    "    ax.plot(x, pdf_3, 'b-', lw=2, label=r'$X \\sim F({}, {})$'.format(df_list[2], df_list_2[2]))    \n",
    "#     ax.plot(x, pdf_4, 'y-', lw=2, label=r'X \\sim \\chi^2({})'.format(df_list[3]))    \n",
    "    \n",
    "    plt.ylabel('Probablity')\n",
    "    plt.title(r'PDF of F Dist.')\n",
    "    ax.legend(loc='best', frameon=False)\n",
    "    plt.show()\n",
    "\n",
    "t_distribution([1,30,100], [30,5,100])"
   ]
  }
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