Random Variable

Given a measurable space $(\Omega ,\mathcal{F})$, a random variable is a function $f:\Omega \to \mathbb{R}$ if $\{\omega :X(\omega )\le c\}$ is $\mathcal{F}$-Measurable for any $c\in \mathbb{R}$.

We call a r.v. an extended r.v. if the range of $f$ considers $\infty$.

Note that the Borel Sigma Field can be generated by $\{(-\infty ,c]:c\in \mathbb{R}\}$. Therefore, for any Borel Set $B$, $X^{-1}(B):=\{\omega :X(\omega )\in B\}$ is $\mathcal{F}$-measurable.

Law

Let the measure space be a Probability Space, then a r.v. introduces a probability law:

$$P_X:\mathcal{B}\to [0,1],B\mapsto P_X(B)=P(X\in B).$$

This is also called the distribution of $X$. Note that a law is different from a Cumulative Distribution Function, but they uniquely determine each other.

Importantly, $P_X$ is a probability Measure. So $X$ provides a mapping between two probability spaces:

$$(\Omega ,\mathcal{F},P)\stackrel{X}{⟷}(\mathbb{R},\mathcal{B},P_X).$$

More concretely, the indicator function $1_A$ is a r.v. for all $A\in \mathcal{F}$. And a simple r.v. is a linear combination of indicator functions.

Properties

• A Random Variable is a $(\mathcal{F},\mathcal{B})$-measurable function where $\mathcal{F}$ is the Sigma Field on the Probability Space and $\mathcal{B}$ is the Borel Sigma Field on the real line.
• A continuous function $f:\mathbb{R}\to \mathbb{R}$ is Borel-measurable.
• The composition of measurable functions is measurable.
  • Specifically, given a continuous function $f$ and Random Variable $X$, $f(X)$ is a Random Variable.
  • Further, given a continuous multivariate function $f:{\mathbb{R}}^n\to \mathbb{R}$ and Random Variables $X_1,\dots ,X_n$, $f(X_1,\dots ,X_n)$ is a Random Variable.
    • In particular, the sum and product of Random Variables are Random Variables.

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• Given a sequence of r.v.s $X_n$, the following limits are r.v.s: ${\sup }_n{X}_n$, ${\inf }_n{X}_n$, ${\mathrm{limsup}}_n{X}_n$, ${\mathrm{liminf}}_n{X}_n$.