[Probability Theory_3] Week3

서울대학교 통계학과 대학원 2021년도 1학년 1학기 확률론1 수업의 정리내용입니다.

reference: Lecture note (Prof.sangSangyeol Lee),
Probability: Theory and Examples, Rick Durrett, Version 5 January 11, 2019

$\textbf{Definition[simple]}$

A r.v. $X$ is simple if $X$ takes a finite number of values

$\textbf{Theorem}$

If $X \geqslant 0, \exists $sequence of simple r.v.s $\lbrace X_n \rbrace$ s.t. $0 \leqslant X_n \leqslant X_{n+1} \nearrow X$

$\textbf{Definition}$

Let $X$ be a r.v. then $\sigma(X) = \lbrace (X \in B): B \in \mathscr{B}(\mathbb{R}) \rbrace ( \subset \mathcal{F})$

$\textbf{Definition}$

A r.v. $X$ is $\mathcal{G}$-measurable, if $\forall B \in \mathscr{B}(\mathbb{R}), (X \in B) \in\mathcal{G} $

$\textbf{Theorem}$

Assume $X$ is a r.v. and r.v. $Y \in \sigma(X)$, then $\exists$Borel of $f: \mathbb{R} \rightarrow \mathbb{R}$ s.t. $Y = f(X)$

$\textbf{Definition[Distribution]}$

A function $F: \mathbb{R} \rightarrow \mathbb{R} $ is a distribution if

1. $F$ is non-decreasing
2. $F$ is right-continuous
3. $F$ has left-limits
4. $ lim_{x \rightarrow \infty }F = 1$ and $ lim_{x \rightarrow -\infty }F = 0$
   
   

$\textbf{Definition}$

Let $F$ be a distribution. Define for $u \in [0,1]$,

$F^{-1}(u) = inf \lbrace x \in \mathbb{R} : F(x) \geqslant u \rbrace$

$\textbf{Definitionp[field]}$

A collection $\mathcal{F}$ of subsets of $\Omega$ is a field if

1. $\Omega \in \mathcal{F}$
2. $A \in \mathcal{F} \rightarrow A^c \in \mathcal{F}$
3. $A_1, \cdots,A_n, n \geqslant 1 \rightarrow \cup^n_{k=1} A_k \in \mathcal{F}$
   
   

$\textbf{Definitionp[Probability measure]}$

$P: \mathcal{F} \rightarrow [0,1]$ is a probability measure if

1. $\forall A \in \mathcal{F}, 0 \leqslant P(A) \leqslant P(\Omega) \leqslant 1$
2. $\forall$disjoint $A_i, i \geqslant 1,$ with $A = \cup^{\infty}{i=1} A_i$, if $A \in \mathcal{F}$, then $P(A) = \Sigma^{\infty}{i=1} P(A_i)$
   
   

$\textbf{Fact}$

$Q=P$ on $\mathcal{F}$, then $\exists$unique prob mea $P^*$ on $\sigma (\mathcal{F})$

$\textbf{Difinition[Borel measure]}$

Any (Prob) measure $\mu$ on $(\mathbb{R}, \mathscr{B}(\mathbb{R}))$ is a (Prob) Borel measure.

$\textbf{Theorem}$

For any dist $F$, $\exists !$prob mea $\mu$ s.t. $F(x) = \mu(-\infty , x], \forall x$