Random Variable Introduction to Probability Theory Probability Space [Definition] : A Probability Space is a triplet $(\Omega, \mathcal{F},\mathbb{P})$ of three elements:
$\Omega$ is a set (called Sample Space) . $\mathcal{F}$ is a Sigma Algebra on $\Omega$. $\mathcal{F}$ is a collection of subsets of $\Omega$ and has the following properties: $\Omega, \varnothing \in \mathcal{F}$ $A\in \mathcal{F} \Rightarrow A^c \equiv \Omega \setminus A \in \mathcal{F}$ A sequence of sets ${A_i}{i=1}^{\infty}$ in which $A_i \in \mathcal{F}$, then $\bigcup{j=1}^{\infty} A_j \in \mathcal{F}$ $\mathbb{P}$ is a Probability Measure: $\mathbb{P}: \mathcal{F}\rightarrow [0,1]$ and has the following properties: (Normalization)$\mathbb{P}(\Omega) = 1$ (Countably additive)$A_1,A_2,\cdots \in \mathcal{F} \ , \ A_i\bigcap A_j=\varnothing \Rightarrow \mathbb{P}(\bigcup_{j=1}^{\infty} A_j) = \sum_{j=1}^{\infty} \mathbb{P}(A_j)$ [Definition] : A Measurable (real-valued) Function on a Measurable Space $(\Omega,\mathcal{F})$ is a function $f:\Omega \rightarrow \mathbb{R}$, which has the property:......
