概率论与数理统计

本书包括随机事件与概率、一维随机变量及其分布、多维随机变量及其分布、随机变量的数字特征、大数定律及中心极限定理,统计量及其分布、参数估计、假设检验等内容。每一章有章节思维导图，每节后面附有相应的习题及部分考研真题。本书可作为高等学校理工科、经济类、管理类和其他非数学专业教学用书，也适合数学专业专科的概率论与数理统计课程的教学需要。

第一章 随机事件与概率···········································································································1 第一节 随机事件及其运算·································································································1 一、随机试验与样本空间 ··································································································1 二、随机事件 ··················································································································2 三、随机事件的关系与运算 ······························································································3 习题 1.1 ···························································································································6 第二节 随机事件的概率 ····································································································7 一、古典概型 ··················································································································8 二、几何概型 ················································································································10 三、频率与概率 ·············································································································12 四、概率的公理化定义及其运算性质 ···············································································13 习题 1.2 ·························································································································16 第三节 条件概率与全概率公式·······················································································17 一、条件概率 ················································································································17 二、乘法定理 ················································································································19 三、全概率公式与贝叶斯公式 ·························································································20 习题 1.3 ·························································································································23 第四节 随机事件的独立性·······························································································25 习题 1.4 ·························································································································28 第五节 伯努利概型··········································································································29 习题 1.5 ·························································································································32 第一章思维导图··················································································································33 第二章 随机变量及其概率分布····························································································34 第一节 随机变量的定义 ··································································································34 习题 2.1 ·························································································································36 第二节 离散型随机变量及其概率分布 ···········································································36 一、两点分布(0-1 分布) ·································································································37 二、二项分布 ················································································································38 三、泊松分布 ················································································································40 四、超几何分布 ·············································································································42 概率论与数理统计 6 五、几何分布 ················································································································44 习题 2.2 ·························································································································45 第三节 随机变量的分布函数···························································································46 习题 2.3 ·························································································································50 第四节 连续型随机变量及其概率密度函数····································································51 一、均匀分布 ················································································································53 二、指数分布 ················································································································54 三、正态分布 ················································································································55 习题 2.4 ·························································································································59 第五节 随机变量函数的概率分布···················································································62 一、离散型随机变量函数的概率分布 ···············································································62 二、连续型随机变量函数的概率密度函数 ·········································································64 习题 2.5 ·························································································································68 第二章思维导图··················································································································69 第三章 多维随机变量及其联合概率分布··············································································70 第一节 二维随机变量 ······································································································70 习题 3.1 ·························································································································75 第二节 边缘分布··············································································································76 习题 3.2 ·························································································································82 第三节 条件分布··············································································································83 习题 3.3 ·························································································································90 第四节 随机变量的独立性·······························································································91 习题 3.4 ·························································································································94 第五节 二维随机变量函数的概率分布 ···········································································95 一、二维离散型随机变量函数的概率分布 ·········································································95 二、二维连续型随机变量函数的概率分布 ·········································································96 习题 3.5 ·······················································································································103 第三章思维导图················································································································106 第四章 随机变量的数字特征 ·······························································································107 第一节 数学期望············································································································107 一、离散型随机变量的数学期望 ····················································································107 二、连续型随机变量的数学期望 ····················································································109 三、随机变量函数的数学期望 ·······················································································110 四、数学期望的性质 ····································································································113 习题 4.1 ·······················································································································115 目 录 7 第二节 方差····················································································································116 一、随机变量的方差 ····································································································117 二、方差的性质 ···········································································································120 习题 4.2 ·······················································································································122 第三节 协方差与相关系数·····························································································123 一、协方差 ··················································································································124 二、相关系数 ··············································································································125 习题 4.3 ·······················································································································128 第四节 随机变量的其他数字特征·················································································130 习题 4.4 ·······················································································································134 第四章思维导图················································································································135 第五章 大数定律与中心极限定理························································································135 第一节 大数定律············································································································135 一、切比雪夫不等式 ····································································································135 二、大数定律 ··············································································································136 习题 5.1 ·······················································································································139 第二节 中心极限定理 ····································································································140 习题 5.2 ·······················································································································143 第五章思维导图················································································································145 第六章 数理统计的基本思想 ·······························································································146 第一节 总体与样本········································································································146 一、总体与样本的基本概念 ··························································································146 二、统计量的抽样分布 ·································································································148 习题 6.1 ·······················································································································152 第二节 数理统计的三大分布·························································································153 一、 2 χ 分布 ················································································································153 二、t 分布 ···················································································································155 三、F 分布 ··················································································································156 习题 6.2 ·······················································································································159 第三节 正态总体的抽样分布定理·················································································159 习题 6.3 ·······················································································································164 第六章思维导图················································································································165 第七章 参数估计···················································································································166 第一节 参数的点估计 ····································································································166 一、矩估计法 ··············································································································167 概率论与数理统计 8 二、最大似然估计法 ····································································································170 三、点估计的评价标准 ·································································································175 习题 7.1 ·······················································································································179 第二节 区间估计············································································································181 一、区间估计的概念 ····································································································182 二、正态总体参数的区间估计 ·······················································································184 三、单侧置信区间 ········································································································191 习题 7.2 ·······················································································································193 第七章思维导图················································································································194 第八章 假设检验···················································································································195 第一节 假设检验的基本概念·························································································195 习题 8.1 ·······················································································································199 第二节 单个正态总体参数的假设检验 ·········································································199 一、关于正态总体期望 μ 的假设检验 ·············································································200 二、关于正态总体方差 σ 2 的假设检验 ············································································202 习题 8.2 ·······················································································································204 第三节 两个正态总体参数的假设检验 ·········································································205 一、关于两个正态总体期望值相等的假设检验 ································································205 二、关于两个正态总体方差相等的假设检验 ···································································207 习题 8.3 ·······················································································································209 第四节 单边假设检验 ····································································································210 一、单个正态总体期望的单边检验 ·················································································211 二、单个正态总体方差的单边检验 ·················································································213 三、区间估计与假设检验的关系 ····················································································215 习题 8.4 ·······················································································································215 第五节 非参数假设检验 ································································································216 一、皮尔逊 2 χ 拟合检验法 ····························································································216 二、柯尔莫哥洛夫(Kolmogolov)检验法 ··········································································218 三、夏皮洛-威尔克(Shapior-Wilk)检验法(W 检验) ·························································219 四、独立性检验 ···········································································································220 习题 8.5 ·······················································································································222 第八章思维导图················································································································223 附录 A 几种常用的概率分布······························································································224 附录 B 泊松分布数值表 ·······································································································226 目 录 9 附录 C 标准正态分布函数数值表························································································229 附录 D t 分布临界值表·········································································································230 附录 E 2 χ 分布临界值表······································································································231 附录 F F 分布临界值表········································································································233 附录 G 相关系数显著性检验表 ···························································································238 参考文献··································································································································239