2026-09-07-Note

课程导航

课程首页：Probability Theory
后续内容：概率公理与计数方法
课程资料：Lecture slides Ch1 Part 1；Casella & Berger, Statistical Inference, Ch1

Foundations of Set Theory and Probability / 集合论与概率论基础

1. From Random Experiments to Probability Models / 从随机试验到概率模型

随机试验（random experiment）的可能结果是明确的，但试验前无法确定哪一个结果会发生。概率模型的基本结构为

$$\text{随机试验}⟶\text{样本空间 }S⟶\text{事件 }A⟶\text{概率 }P(A).$$

2. Sample Spaces and Events / 样本空间与事件

Definition

For a random experiment, the sample space $S$ is the set of all possible basic outcomes.

A set is countable if its elements can be placed in one-to-one correspondence with a subset of the integers.

An event is any subset of the sample space $S$. If the realized outcome belongs to $A\subseteq S$, we say that event $A$ occurs.

The set containing no outcomes is denoted by $\varnothing$ and represents an impossible event.

样本空间可以是有限集、可数无限集或不可数集。事件发生，是指试验实现的基本结果属于对应子集。

集合包含与相等满足

$$\begin{array}{rl}A\subseteq B& ⟺(x\in A\Rightarrow x\in B),\\ A=B& ⟺A\subseteq B\text{ 且 }B\subseteq A.\end{array}$$

证明两个集合相等时，通常使用双向包含。

3. Set Operations / 集合运算

3.1 Basic Operations / 基本运算

$$\begin{array}{rl}A\cup B& =\{x:x\in A\text{ 或 }x\in B\},\\ A\cap B& =\{x:x\in A\text{ 且 }x\in B\},\\ A^c& =\{x\in S:x\notin A\}.\end{array}$$

并、交运算满足交换律、结合律和分配律。

3.2 De Morgan’s Laws / De Morgan 律

$$\begin{array}{rl}(A\cup B{)}^c& =A^c\cap B^c,\\ (A\cap B{)}^c& =A^c\cup B^c.\end{array}$$

对可数个集合，

$$\begin{array}{rl}{\left(\bigcup _i{A}_i\right)}^c& =\bigcap _i{A}_i^c,\\ {\left(\bigcap _i{A}_i\right)}^c& =\bigcup _i{A}_i^c.\end{array}$$

其中，${\bigcup }_{i=1}^{\infty }{A}_i$ 表示至少属于一个 $A_i$；${\bigcap }_{i=1}^{\infty }{A}_i$ 表示属于每一个 $A_i$。

4. Partitions of the Sample Space / 样本空间的分割

Definition

Events $A_1,\dots ,A_n$ are mutually exclusive (pairwise disjoint) if

$$A_i\cap A_j=\varnothing (i\ne j).$$

Events $A_1,\dots ,A_n$ are collectively exhaustive if

$$\bigcup _{i=1}^n{A}_i=S.$$

A collection $A_1,A_2,\dots$ is a partition of $S$ if the sets are pairwise disjoint and collectively exhaustive.

分割是全概率公式与 Bayes 公式的基础。

5. Sigma-Algebras / σ-代数

Definition

A collection $\mathcal{B}$ of subsets of $S$ is a $\sigma$-algebra if

1. $\varnothing \in \mathcal{B}$;
2. $A\in \mathcal{B}\Rightarrow A^c\in \mathcal{B}$;
3. $A_1,A_2,\dots \in \mathcal{B}\Rightarrow {\bigcup }_{i=1}^{\infty }{A}_i\in \mathcal{B}$.

由 De Morgan 律，$\sigma$-algebra 也对可数交封闭。幂集 $2^S$ 与平凡 $\sigma$-algebra $\{\varnothing ,S\}$ 都满足定义。