概率论与随机过程笔记（1）：样本空间与概率

概率论与随机过程笔记（1）：样本空间与概率

2019-10-27

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这部分的笔记依据Dimitri P. Bertsekas和John N. Tsitsiklis的《概率导论》第1章内容（不包括1.6节组合数学的内容）。鉴于线性代数的笔记中大量latex公式输入中，切换中英文输入法浪费了很多时间，所以概率笔记会用英文完成。

1.1 集合（sets）

【集合的定义】A set is a collection of objects, which are the elements of the set. If $S$ is a set and $x$ is an element of $S$, we write $x\in S$. If $x$ is not an element of $S$, we write $x\notin S$. A set can have no elements, in which case it is called the empty set, denoted by $\varnothing$

【集合的表示方法】Sets can be specified in a variety of ways:
S = { x 1 , x 2 , ⋯   , x n } S=\{x_1,x_2,\cdots,x_n\} S={ x1​,x2​,⋯,xn​} S = { x 1 , x 2 , ⋯   } S=\{x_1,x_2,\cdots\} S={ x1​,x2​,⋯} { x ∣ x    s a t i s f i e s    P } \{x \vert x \; satisfies \; P \} { x∣xsatisfiesP}
The symbol “$|$” is to be read as “such that.”

【集合之间的关系】If every element of a set $S$ is also an element of a set $T$, we say that $S$ is a subset of $T$, and we write $S\subset T$ or $T\supset S$. If $S\subset T$ and $T\subset S$, the two sets are equal, and we write $S=T$.

【空间 $\Omega$】It is also expedient to introduce a universal set, denoted by $\Omega$, which contains all objects that could conceivably be of interest in a particular context. Having specified the context in terms of a universal set $\Omega$, we only consider sets $S$ that are subsets of $\Omega$.

【补集】The complement of a set $S$, with respect to the universe $\Omega$, is the set { x ∈ Ω ∣ x ∉ S } \{x\in \Omega \vert x \notin S\} { x∈Ω∣x∈/​S} of all the elements of $\Omega$ that do not belong to $S$, and is denoted by $S^c$. Note that ${\Omega }^c=\varnothing$.

【集合的交和并】The union of two sets $S$ and $T$ is the set of all elements that belong to $S$ or $T$ (or both), and is denoted by $S\cup T$. The intersection of two sets $S$ and $T$ is the set of all elements that belong to both $S$ and $T$, and is denoted by $S\cap T$. Thus, S ∪ T = { x    ∣    x ∈ S    o r    x ∈ T } S \cup T=\{x \;\vert \;x \in S \;or \;x \in T\} S∪T={ x∣x∈Sorx∈T} S ∩ T = { x    ∣    x ∈ S    a n d    x ∈ T } S \cap T=\{x \;\vert \;x \in S \;and \;x \in T\} S∩T={ x∣x∈Sandx∈T} ⋃ n = 1 ∞ = S 1 ∪ S 2 ⋯ = { x    ∣    x ∈ S n    f o r    s o m e    n } \bigcup_{n=1}^\infty = S_1 \cup S_2 \cdots = \{x\;\vert\;x \in S_n \;for \;some \;n\} n=1⋃∞​=S1​∪S2​⋯={ x∣x∈Sn​forsomen} ⋂ n = 1 ∞ = S 1 ∩ S 2 ⋯ = { x    ∣    x ∈ S n    f o r    a l l e    n } \bigcap_{n=1}^\infty = S_1 \cap S_2 \cdots = \{x\;\vert\;x \in S_n \;for \;alle \;n\} n=1⋂∞​=S1​∩S2​⋯={ x∣x∈Sn​forallen}
【不想交 & 分割】Two sets are said to be disjoint if their intersection is empty. More generally, several sets are said to be disjoint if no two of them have a common element. A collection of sets is said to be a partition of a set $S$ if the sets in the collection are disjoint and their union is $S$.

If $x$ and $y$ are two objects. we use $(x.y)$ to denote the ordered pair of $x$ and $y$. The set of scalars (real numbers) is denoted by $\mathbb{R}$: the set of pairs (or triplets) of scalars, i.e … the two-dimensional plane (or three-dimensional space, respectively) is denoted by ${\mathbb{R}}^2$ (or ${\mathbb{R}}^3$. respectively).

Sets and the associated operations are easy to visualize in terms of Venn diagrams. as illustrated in Fig. 1.1.

【矩阵代数】Set operations have several properties, which are elementary consequences of the definitions. Some exa1nples are:

• $S\cup T=T\cup S$
• $S\cup (T\cup U)=(S\cup T)\cup U$
• $S\cap (T\cup U)=(S\cap T)\cup (S\cap U)$
• $S\cup (T\cap U)=(S$