Random variables §

Def A real random variable is a measurable function $X:(\Omega ,\mathcal{F})\mapsto (\mathbb{R},\mathcal{B})$.

Where $\mathcal{B}$ is the usual Borel $\sigma$-algebra.

Examples §

Given a measurable set $A\in \mathcal{F}$, its characteristic function is a random variable:

$$1_A(\omega ):=\{\begin{array}{l}1\text{ if }\omega \in A\\ 0\text{ if }\omega \notin A\end{array}$$

Infact, if we study the preimages, every Borel set $B\in \mathcal{B}$:

$$1_A(B{)}^{-1}=\{\begin{array}{l}\varnothing \text{ if }1,0\notin B\\ \Omega \text{ if }1,0\in B\\ A\text{ if }1\in B,0\notin B\\ A^C\text{ if }1\notin B,0\in B\end{array}$$

which are all measurable.

The law of a random variable §

To each random variable $X$ on a given probability space we can associate a probability mesure ${\mathcal{L}}_X$, that we refer to as the law, or the distribution of $X$.

Def The law of a random variable $X$ on a probability space $(\Omega ,\mathcal{F},\mathbb{P})$ is the probability mesure ${\mathcal{L}}_X:\mathcal{B}\mapsto [0,1]$ on $(\mathbb{R},\mathcal{B})$ defined by:

$${\mathcal{L}}_X(A):=\mathbb{P}(X^{-1}(A))=\mathbb{P}(X\in A)$$

where the event in the last term is a standard notation for $\{\omega \in \Omega :X(\omega )\in A\}$.

Since the sets $(-\infty ,a]$ with $a\in \mathbb{R}$ form a Pi-system that generates $\mathcal{B}$, it follow from the uniqueness lemma the law is uniquely specified by its value on such sets:

$${\mathcal{L}}_X((-\infty ,x]):=\mathbb{P}(X^{-1}((-\infty ,x]))=\mathbb{P}(X\le x)$$

We can define a function $F_X:\mathbb{R}\mapsto [0,1]$ called distribution function of $X$ defined by $F_X(x):=\mathbb{P}(X\le x)$.