Basic probability theory

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 $B\subseteq A$.