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‎.gitignore‎

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*.out
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.DS_Store

‎README.md‎

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# MathStatsCode
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Codes for my mathematical statistics course
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Codes and Jupyter-notebooks for my course of Mathematical Statistics.

‎code_in_notes/exponential.do‎

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// file: exponential.do
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set more off
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clear
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set obs 100
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// 产生随机数
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gen x=-1*log(runiform())

‎code_in_notes/exponential.py‎

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#!/usr/bin/python3
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## file: exponential.py
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# 导入math包，使用math中的log()函数
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import math
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import random as rd
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# 获取5个均匀分布随机数
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x=[-1*math.log(rd.random()) for i in range(5)]
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print(x)
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# 导入numpy包，使用其中的log()函数
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import numpy as np
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import numpy.random as nprd
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# 获取5个均匀分布随机数
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x=-1*np.log(nprd.random(5))
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print(x)

‎code_in_notes/uniform.c‎

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// file: uniform.c
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#include<stdio.h>
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#include<stdlib.h> // 使用标准库<stdlib.h>，包含rand()
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int main(int argc, char const *argv[]) {
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// 打印出rand()所生成的最大整数
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printf("%d\n",RAND_MAX);
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// 设置seed
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srand(505);
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// 生成(0,1)上的10个随机数
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int i;
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for (i=0; i<5; ++i){
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double x=(double)rand()/RAND_MAX;
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printf("%f ",x);
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}
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printf("\n");
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// 如果每次都设置seed生成(0,1)上的5个随机数
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//那么每次生成的随机数都相同
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for (i=0; i<5; ++i){
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srand(505);
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double x=(double)rand()/RAND_MAX;
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printf("%f ",x);
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}
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printf("\n");
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return 0;
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}

‎code_in_notes/uniform.do‎

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// file: uniform.do
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// 关闭--more--
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set more off
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// 清楚工作区内所有数据
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clear
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// 设置样本量
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set obs 100
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// 设置seed
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set seed 505
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// 产生随机数
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gen x=runiform()

‎code_in_notes/uniform.py‎

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#!/usr/bin/python3
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## file: uniform.py
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# 导入random包
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import random as rd
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# 设置seed
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rd.seed(505)
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# 获取5个均匀分布随机数
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x=[rd.random() for i in range(5)]
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print(x)
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# 导入numpy中的random包
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import numpy.random as nprd
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# 设置seed
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nprd.seed(505)
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# 获取5个均匀分布随机数
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x=nprd.random(5)
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print(x)

‎notebook_julia/Julia.ipynb‎

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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Julia的安装\n",
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"Julia是一个开源的数值计算软件，由于具有JIT等特性，所以具有非常高的性能。我们可以从Julia官网：https://julialang.org 下载和安装。\n",
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"Julia的界面非常简单，不适合编写程序，因而我们一般通过配置Atom, sublime等编辑器写Julia程序。此外，通过安装Jupyter，再在Julia中安装IJulia也可以使用Jupyter调用Julia。安装IJulia的方法很简单，只需要在Julia的命令界面数据「Pkg.add(\"IJulia\")」即可，其中「Pkg.add()」即安装Julia扩展包的命令。\n",
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"为了展示方便，我们仍然推荐使用Jupyter作为学习工具。比如此文档就是使用Jupyter写作。\n",
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"# 使用Julia生成随机数\n",
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"在Julia中，可以使用rand()函数生成随机数。最基本的是生成一个在$(0,1)$区间内的均匀分布的随机数，使用此均匀分布随机数，给定任意的分布函数$F$，可以生成服从$F$的随机数。"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"一个(0,1)区间内的随机数：0.37548057506738663\n",
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"20个(0,1)区间内的随机数：\n",
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"[0.719472, 0.447485, 0.101696, 0.166868, 0.042199, 0.29615, 0.957232, 0.688239, 0.583882, 0.30275, 0.718666, 0.945527, 0.150395, 0.650269, 0.811649, 0.770101, 0.681383, 0.066861, 0.0560345, 0.760834]"
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]
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}
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],
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"source": [
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"## 生成一个x~Uniform(0,1)\n",
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"x=rand()\n",
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"print(\"一个(0,1)区间内的随机数：\",x,\"\\n\")\n",
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"\n",
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"## 生成20个z~Uniform(0,1)\n",
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"z=rand(20)\n",
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"print(\"20个(0,1)区间内的随机数：\",\"\\n\")\n",
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"print(z)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"如果我们需要生成一个服从分布函数$F: R\\rightarrow [0,1]$的随机数，那么只要首先生成一个$(0,1)$的随机数$u$，并令$x=F^{-1}(u)$，那么新生成的$x$即服从$F$的分布。比如，指数分布的分布函数为$1-e^{-\\frac{1}{b}\\cdot x}$，其中b为一个参数，因而我们可以使用$x=-b\\cdot \\ln(u)$来生成服从指数分布的随机数。"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {
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"scrolled": true
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"20个服从指数分布F(x)=1-exp{-(1/b)*x}的随机数：\n",
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"[2.58645, 11.1848, 0.918512, 0.543136, 3.53971, 0.259657, 1.99849, 0.350091, 3.09114, 0.165935, 0.0657694, 0.949447, 9.25605, 0.393429, 10.0648, 1.87895, 2.65323, 6.0845, 6.48842, 2.73542]"
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]
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}
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],
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"source": [
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"## 设定参数\n",
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"b=3\n",
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"\n",
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"## 生成20个x~F(x)=1-exp{-(1/b)*x}\n",
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"x=-1*log.(rand(20))*b\n",
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"print(\"20个服从指数分布F(x)=1-exp{-(1/b)*x}的随机数：\\n\")\n",
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"print(x)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"我们可以使用经验分布函数（Empirical distribution function）与理论的分布函数比较，来判断我们生成的随机数是否满足某一分布。经验分布函数的定义为：$\\hat{F}(x)=\\frac{1}{N}\\cdot \\sum_{i=1}^N 1\\{X_i \\leq x\\}$，也就是给定一个$x$，其经验分布函数的值为样本中小于等于$x$的比例，比如："
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]
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},
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{
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"cell_type": "code",
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"execution_count": 20,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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" x | 分布函数 | 经验分布函数| 差的绝对值\n",
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"0.1 |0.0327838995179941 | 0.035 | 0.002216100482005906 \n",
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"0.3 |0.09516258196404037 | 0.1 | 0.004837418035959634 \n",
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"0.5 |0.15351827510938598 | 0.185 | 0.031481724890614016 \n",
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"0.7 |0.20811043366321835 | 0.255 | 0.04688956633678165 \n",
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"0.9 |0.2591817793182821 | 0.295 | 0.03581822068171786 \n",
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"1.1 |0.3069593799135585 | 0.325 | 0.018040620086441528 \n",
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"1.3 |0.3516556589984903 | 0.4 | 0.04834434100150975 \n",
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"1.5 |0.3934693402873666 | 0.455 | 0.06153065971263344 \n",
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"1.7 |0.43258633120299617 | 0.495 | 0.06241366879700383 \n",
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"1.9 |0.46918054943798604 | 0.54 | 0.070819450562014 \n",
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"2.1 |0.5034146962085905 | 0.575 | 0.07158530379140948 \n",
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"2.3 |0.5354409796390884 | 0.625 | 0.08955902036091157 \n",
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"2.5 |0.5654017914929217 | 0.655 | 0.08959820850707834 \n",
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"2.7 |0.5934303402594009 | 0.68 | 0.08656965974059916 \n",
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"2.9 |0.6196512434107414 | 0.715 | 0.09534875658925857 \n",
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"3.1 |0.6441810814626581 | 0.735 | 0.09081891853734192 \n",
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"3.3 |0.6671289163019204 | 0.745 | 0.0778710836980796 \n",
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"3.5 |0.6885967760854023 | 0.76 | 0.07140322391459775 \n",
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"3.7 |0.708680108866529 | 0.77 | 0.06131989113347103 \n",
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"3.9 |0.7274682069659874 | 0.785 | 0.05753179303401268 \n",
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"4.1 |0.7450446039734899 | 0.795 | 0.04995539602651011 \n",
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"4.3 |0.7614874461456975 | 0.8 | 0.03851255385430252 \n",
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"4.5 |0.7768698398515702 | 0.81 | 0.033130160148429844 \n",
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"4.7 |0.7912601766099203 | 0.825 | 0.03373982339007964 \n",
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"4.9 |0.8047224371643142 | 0.835 | 0.030277562835685723 \n",
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"5.1 |0.8173164759472653 | 0.845 | 0.027683524052734665 \n",
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"5.3 |0.8290982871984751 | 0.85 | 0.020901712801524863 \n",
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"5.5 |0.8401202539203061 | 0.86 | 0.019879746079693894 \n",
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"5.7 |0.8504313807773649 | 0.87 | 0.019568619222635086 \n",
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"5.9 |0.8600775119756906 | 0.87 | 0.009922488024309395 \n",
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"6.1 |0.8691015350902557 | 0.88 | 0.010898464909744332 \n",
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"6.3 |0.877543571747018 | 0.895 | 0.017456428252981993 \n",
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"6.5 |0.8854411560073123 | 0.895 | 0.009558843992687693 \n",
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"6.7 |0.8928294012476933 | 0.9 | 0.007170598752306745 \n",
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"6.9 |0.8997411562771962 | 0.9 | 0.0002588437228038254 \n",
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"7.1 |0.9062071513861237 | 0.91 | 0.0037928486138762985 \n",
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"7.3 |0.9122561349757057 | 0.91 | 0.002256134975705648 \n",
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"7.5 |0.9179150013761012 | 0.92 | 0.0020849986238988816 \n",
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"7.7 |0.9232089104210319 | 0.92 | 0.0032089104210318853 \n",
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"7.9 |0.9281613993106868 | 0.92 | 0.008161399310686712 \n",
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"8.1 |0.9327944872602503 | 0.925 | 0.007794487260250227 \n",
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"8.3 |0.9371287733993328 | 0.925 | 0.012128773399332715 \n",
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"8.5 |0.9411835283575701 | 0.94 | 0.001183528357570185 \n",
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"8.7 |0.9449767799435927 | 0.94 | 0.004976779943592802 \n",
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"8.9 |0.9485253932982992 | 0.945 | 0.003525393298299262 \n",
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"9.1 |0.9518451458788036 | 0.955 | 0.0031548541211963155 \n",
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"9.3 |0.9549507976064422 | 0.955 | 4.92023935577679e-5 \n",
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"9.5 |0.9578561564907236 | 0.955 | 0.0028561564907236825 \n",
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"9.7 |0.9605741400209925 | 0.96 | 0.0005741400209925418 \n",
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"9.9 |0.96311683259876 | 0.96 | 0.0031168325987600554 \n",
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"Mean absolute bias:0.03123553671578694"
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]
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}
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],
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"source": [
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"## 设定参数\n",
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"b=3\n",
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"\n",
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"## 生成200个x~F(x)=1-exp{-(1/b)*x}\n",
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"x=-1*log.(rand(200))*b\n",
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"\n",
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"## 给定一些点，在这些点上计算分布函数和经验分布函数\n",
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"x_eval=[i for i in 0.1:0.2:9.9] #0.1,0.3,...,9.9\n",
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"\n",
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"## 计算理论的分布函数\n",
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"F=1-exp.(-1/b.*x_eval)\n",
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"\n",
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"## 给定z计算经验分布函数\n",
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"function empirical_F(x,z::Float64)::Float64\n",
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" return mean(x.<=z)\n",
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"end\n",
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"\n",
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"## 计算经验分布函数\n",
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"Fhat=[empirical_F(x,z) for z in x_eval]\n",
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"\n",
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"## 计算经验分布函数与真实的分布函数之间的绝对差异\n",
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"bias=mean(abs.(Fhat-F))\n",
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"## 打印两个分布函数及其绝对差异，以及平均的绝对差异\n",
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"print(\" x | 分布函数 | 经验分布函数| 差的绝对值\\n\")\n",
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"for i in 1:1:length(x_eval)\n",
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" print(\"$(x_eval[i]) |$(F[i]) | $(Fhat[i]) | $(abs(F[i]-Fhat[i])) \\n\")\n",
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"end\n",
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"print(\"Mean absolute bias:\",bias)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"display_name": "Julia 0.6.0",
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"language": "julia",
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"name": "julia-0.6"
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"language_info": {
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"file_extension": ".jl",
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"mimetype": "application/julia",
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"name": "julia",
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"version": "0.6.0"
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},
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"latex_envs": {
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"LaTeX_envs_menu_present": true,
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"autocomplete": true,
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"bibliofile": "biblio.bib",
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"cite_by": "apalike",
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