"Basic probability theory"의 두 판 사이의 차이
둘러보기로 가기
검색하러 가기
imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) |
||
| 1번째 줄: | 1번째 줄: | ||
==introduction== | ==introduction== | ||
| − | * Let | + | * Let <math>(\Omega, \mathcal{F}, P)</math> be probability space |
| − | * A real-valued function | + | * A real-valued function <math>X : \Omega\to \mathbb{R}</math> is called a random variable |
| − | * let | + | * let <math>A\subseteq \mathbb{R}</math> be the range of <math>X</math>, <math>A=\{s|X(s)=x,s\in S\}</math>. We call <math>A</math> the space of <math>X</math> |
| − | * | + | * <math>\{X=x\}</math> denote the subset <math>\{s|X(s)=x\}</math> of <math>\mathbb{R}</math> |
| − | * the induced probability measure | + | * the induced probability measure <math>P_X : \mathbb{R}\to [0,1]</math> |
| − | * probability density function | + | * probability density function <math>f : \mathbb{R}\to [0,\infty)</math> of <math>X</math> satisfies |
| − | + | :<math> | |
P_{X}(X\in A)=\int_A f(x)\, dx=1 | P_{X}(X\in A)=\int_A f(x)\, dx=1 | ||
| − | + | </math> | |
and | and | ||
| − | + | :<math> | |
P_{X}(X\in B)=\int_B f(x)\, dx | P_{X}(X\in B)=\int_B f(x)\, dx | ||
| − | + | </math> | |
| − | for | + | for <math>B\subseteq A</math>. |
[[분류:Probability]] | [[분류:Probability]] | ||
[[분류:migrate]] | [[분류:migrate]] | ||
2020년 11월 16일 (월) 04:26 기준 최신판
introduction
- Let \((\Omega, \mathcal{F}, P)\) be probability space
- A real-valued function \(X : \Omega\to \mathbb{R}\) is called a random variable
- let \(A\subseteq \mathbb{R}\) be the range of \(X\), \(A=\{s|X(s)=x,s\in S\}\). We call \(A\) the space of \(X\)
- \(\{X=x\}\) denote the subset \(\{s|X(s)=x\}\) of \(\mathbb{R}\)
- the induced probability measure \(P_X : \mathbb{R}\to [0,1]\)
- probability density function \(f : \mathbb{R}\to [0,\infty)\) of \(X\) satisfies
\[ P_{X}(X\in A)=\int_A f(x)\, dx=1 \] and \[ P_{X}(X\in B)=\int_B f(x)\, dx \] for \(B\subseteq A\).