Mathematical Preliminaries
random quantity #
sigma algebra #
If $\Omega$ is a given set, then a $\sigma-$algebra $\mathbb{F}$ on $\Omega$ is a family of subsets of $\Omega$ satisfying $
- $\phi\in \mathbb{F}$
- $F\in \mathbb{F}\Rightarrow F^{C}\in \mathbb{F}$
- $A_{1},A_{2},…\in \mathbb{F}\Rightarrow \cup_{i}A_{i}\in \mathbb{F}$
Measurable space: $(\Omega, \mathbb{F})$
Probably measure #
Defined on a measurable space $(\Omega,\mathbb{F})$
probability measure is a function P: $\mathbb{F}\rightarrow [0,1]$ such that
- $P(\phi)=0;P(\Omega)=1$
- For disjoint sets $A_{1},A_{2}…\in \mathbb{F}, P(\cup A_{i})=\sum\limits P(A_{i})$
Probability space: $(\Omega,\mathbb{F},P)$
The subsets of $\Omega$ which belong to $\mathbb{F}$ are called $\mathbb{F}-$measurable sets
smallest sig alg #
Given a family U of subsets of $\Omega$, the smallest $\sigma$ algebra containing it
$$ H_{U}=\cap {H;H\\ \sigma \text{-algebra of }\Omega,U\subset H} $$
The one generated by open sets is called the Borel $\sigma$ algebra on $\Omega$ and its elements are called Borel sets.
Lemma #
If X,Y are functions $\Omega\rightarrow R^{n}$ then Y is $H_{X}$ measurable iff there exists a Borel measurable function g such that $Y=g(X)$
Let there be a complete probability $(\Omega, \mathbb{F},P)$
A random variable X is $\mathbb{F}$ measurable function $X:\Omega\rightarrow R^{n}$
$$ \begin{align*} \mu_{X}(B)=P(X^{-1}(B))\\ E[X]=\int_{\Omega}X(w)dP(w)=\int_{R^{n}}xd\mu_{X}(x)\\ E[f(X)]=\int_{R^{n}}f(x)d \mu_{X}(x) \end{align*} $$
independence #
A collection of random variables {$X_{i}$} is independent if the collection of generated $\sigma$ algebras $H_{X}$ is independent