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 \(B\subseteq A\).