Random Variables

Intuitively, a random variable represents a numerical value determined by the outcome of a random experiment. For example, if we roll a die, the outcome is “the face that shows up”, and a random variable $X$ could be “the number of dots on the face”.

In our formal framework, we define a random variable as a function mapping the sample space to real numbers. But not just any function — it must preserve the structure of our probability space.

Definition

Definition: Measurable Mapping

Let $(\Omega, \cF)$ and $(S, \cA)$ be two measurable spaces. A mapping $X: \Omega \to S$ is called measurable if for any $A \in \cA$,

$$X^{-1}(A) = \{\omega \in \Omega : X(\omega) \in A\} \in \cF$$

The preimage of any measurable set is measurable.

If $(S, \cA) = (\R, \cB)$, then $X$ is called a random variable.

Definition: Random Variable

Given a probability space $(\Omega, \cF, \Pr)$, a function $X: \Omega \to \R$ is called a random variable if for every Borel set $B \in \cB$, the inverse image of $B$ is an event in $\cF$. That is:

$$X^{-1}(B) = \{\omega \in \Omega : X(\omega) \in B\} \in \cF \quad \text{for all } B \in \cB.$$

Such a function is said to be $\cF$-measurable.

When we discuss events defined by the value of $X$, we often use the shorthand notation $\{X \in B\}$ for $\{\omega : X(\omega) \in B\}$.

Why do we need this condition?

We want to be able to answer probabilistic questions about $X$, such as “What is the probability that $X$ is greater than 5?”. In set notation, we are asking for $\Pr(\{\omega : X(\omega) > 5\})$.

For this probability to be defined, the subset $\{\omega : X(\omega) > 5\}$ must be an event (i.e., it must belong to $\cF$), because the probability measure $\Pr$ is only defined on $\cF$. The condition $X^{-1}(B) \in \cF$ ensures precisely this: for any “reasonable” question we ask about the value of $X$ (represented by a Borel set $B$), the set of outcomes satisfying it is an event we can measure.

Sigma-field generated by X

Every random variable $X$ naturally comes with a $\sigma$-field that describes the information contained in $X$.

Definition: σ(X)

The $\sigma$-field generated by a random variable $X$, denoted by $\sigma(X)$, is the collection of all inverse images of Borel sets:

$$\sigma(X) = \{ X^{-1}(B) : B \in \cB \} = \{ \{\omega \in \Omega : X(\omega) \in B\} : B \in \cB \}.$$

Intuition: $\sigma(X)$ is the smallest $\sigma$-field on $\Omega$ that makes $X$ a random variable. It contains exactly the events whose occurrence (or non-occurrence) can be determined just by knowing the value of $X$. If $\sigma(X)$ is smaller than $\cF$, it means $X$ “forgets” some information about the outcome $\omega$.

Distribution of X

A random variable $X$ transports the probability measure from $(\Omega, \cF)$ to $(\R, \cB)$. This “transported” measure is what we usually call the distribution of $X$.

Definition: Distribution

The distribution (or law) of $X$ is a probability measure $\mu_X$ on $(\R, \cB)$ defined by:

$$\mu_X(B) = \Pr(X \in B) = \Pr(X^{-1}(B)) \quad \text{for all } B \in \cB.$$

Everything we check about the distribution of $X$ (like its PDF, CDF, mean, variance) is essentially a property of this measure $\mu_X$.

Examples

Examples

1. Discrete Space: Let $\Omega$ be a discrete space (at most countable elements) equipped with the power set $\sigma$-field $2^\Omega$. Then, any mapping $X: \Omega \to \R$ is a random variable.

   $$\Pr(X \in B) = \sum_{b \in B} \Pr(X=b)$$

2. Indicator Random Variable: Let $A$ be an event ($A \in \cF$). The indicator random variable of $A$, denoted $I_A$ (or $1_A$), is defined as:

   $$I_A(\omega) = \begin{cases} 1 & \text{if } \omega \in A \quad (A \text{ happens}) \\ 0 & \text{if } \omega \notin A \quad (A \text{ doesn't happen}) \end{cases}$$

Properties of Measurable Functions

Useful Facts

1. Compositions: If $X$ is a random variable and $f: \R \to \R$ is a measurable function (e.g., continuous functions), then $f(X)$ is also a random variable.
2. Limits: If $X_1, X_2, \dots$ is a sequence of random variables, then their pointwise limits are also random variables. That is, functions defined by $$(\limsup_{n \to \infty} X_n)(\omega) = \limsup_{n \to \infty} (X_n(\omega))$$ (and similarly for $\inf, \sup, \liminf$) are measurable. This relies on the fact that limits of measurable functions are measurable.

Independence of Random Variables

We say random variables are independent if the information they generate is independent.

Definition: Independent Random Variables

Random variables $X$ and $Y$ are independent if their generated $\sigma$-fields $\sigma(X)$ and $\sigma(Y)$ are independent.

This is equivalent to the condition that for any Borel sets $C, D \in \cB$:

$$\Pr(X \in C, Y \in D) = \Pr(X \in C)\Pr(Y \in D)$$

or in terms of CDFs: $F_{X,Y}(x,y) = F_X(x)F_Y(y)$.