Random variable

Contents

Let

(\Omega, \mathcal{F})

be a measurable space and let (

\mathbb{R}, \mathcal{B}(\mathbb{R})

) be the real line with system

\mathbb{B}(\mathbb{R})

of Borel sets. \\A real function

\xi = \xi (\omega)

defined on

(\Omega, \mathcal{F})

is an

\mathcal{F}

-measurable function, or a ---random variable---, if

\begin{equation*}\{ \omega: \xi(\omega) \in B \} \in \mathcal{F}\end{equation*}

for every

B \in \mathcal{B}(\mathbb{R})

; or, equivalently, if the inverse image

\begin{equation*}\xi^{-1} (B) \equiv \{ \omega: \xi(\omega) \in B \}\end{equation*}

is measurable set in

\Omega

.We may say that a random variable is a numerical property of an experiment, with a value depending on "chance". The requirement of measurability is fundamental, for the following reason. If a probability measure

P

is defined on

(\Omega, \mathcal{F})

, it then makes sense to speak of the probability of the event

\{ \xi(omega) \in B\}

that the value of the random variable belongs to a Borel set B. \\The simplest example of a random variable is the indicator

1_A(\omega)

of an arbitrary (measurable) set

A \in \mathcal{F}

Definition

Definition
A probability measure

P_{\xi}

on (

\mathbb{R}, \mathcal{B}(\mathbb{R})

) with

\begin{equation*}P_{\xi} (B) = P \{ \omega : \xi(\omega) \in B \}\end{equation*}

is called the ---probability distribution--- of

\xi

on (

\mathbb{R}, \mathcal{B}(\mathbb{R})

).The function

\begin{equation*}F_{\xi} (x) = P (\omega: \xi(\omega) \leq x , \quad x \in \mathbb{R}\end{equation*}

is called the ---distribution function--- of

\xi

. \\A random variable

\xi

is called ---continuous--- if its distribution function

F_{\xi}(x)

is continuous for

x \in \mathbb{R}

. A random variable

\xi

is called ---absolutely continouous--- if there is a nonnegative function

f = f_{\xi}(x)

, called its density, such that

\begin{equation*}F_{\xi} (x) = \int^x_{-\infty} f_{\xi}(y) dy, \quad x \in \mathbb{R}\end{equation*}

Measurability

To establish that a function

\xi = \xi (\omega)

is a random variable, we have to verify property

\{ \omega: \xi(\omega) \in B \} \in \mathcal{F}

in the definition for all sets

B \in \mathcal{F}

. The following lemma shows that the class of such "test" sets can be considerably narrowed. \\

Lemma .

\begin{equation*}\{ \omega: \xi (\omega) \in E\} \in \mathcal{F}\end{equation*}

for all

E \in \mathcal{E}

\quad

Proof.

\Longrightarrow

(necessity) is evident.

\Longleftarrow

To prove the sufficiency we use the principle of appropriate sets. Let

\mathcal{D}

be the system of those Borel sets

D

\mathcal{B}(\mathbb{R})

for which

\xi^{-1} (D) \in \mathcal{F}

. The operation

\xi^{-1}

is easily shown to preserve the set-theoretic operations of union, intersection and complement:

\begin{equation*}\xi^{-1} \left( \bigcup_{\alpha} B_{\alpha} \right) = \bigcup_{\alpha} \xi^{-1} (B_{\alpha}),\end{equation*}

\begin{equation*}\xi^{-1} \left( \bigcap_{\alpha} B_{\alpha} \right) = \bigcap_{\alpha} \xi^{-1} (B_{\alpha}),\end{equation*}

\begin{equation*}\overline{\xi^{-1} (B_{\alpha})} = \xi^{-1} (\overline{B_{\alpha}}).\end{equation*}

It follows that

\mathcal{D}

is a

\sigma

-algebra. Therefore

\begin{equation*}\mathcal{E} \subseteq \mathcal{D} \subseteq \mathcal{B}(\mathbb{R})\end{equation*}

and

\begin{equation*}\sigma(\mathcal{E}) \subseteq \sigma(\mathcal{D}) = \mathcal{D} \subseteq \mathcal{B}(\mathbb{R}).\end{equation*}

But

\sigma(E) = \mathcal{B}(\mathbb{R})

and consequently

\mathcal{D} = \mathcal{B}(\mathbb{R})

\Box

Corollary
A necessary and sufficient condition for

\xi = \xi(\omega)

to be a random variable is that

\begin{equation*}\{ \omega: \xi(\omega) < x \} \in \mathcal{F}, \quad \forall x \in \mathbb{R}\end{equation*}

\begin{equation*}\{ \omega: \xi(\omega) \leq x \} \in \mathcal{F}, \quad \forall x \in \mathbb{R}\end{equation*}

The proof is immediate, since each of the systems

\begin{equation*}\mathcal{E}_1 = \{ x: x < c, c \in \mathbb{R} \},\end{equation*}

\begin{equation*}\mathcal{E}_2 = \{ x: x \leq c, c \in \mathbb{R} \},\end{equation*}

generates the

\sigma

-algebra

\mathcal{B}(\mathbb{R})

\sigma(\mathcal{E}_1) = \sigma(\mathcal{E}_2)

\Box

References

[MörtersPeres]
Peter Mörters and Yuval Peres. Brownian Motion. Cambridge University Press, 2010.