Random Variables and Distribution Functions

What is a Random Variable?

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A random variable is a bridge between the abstract probability space and concrete numerical values we can measure.

Definition: Random Variable

A random variable $X$ on a probability space $(\Omega ,\mathcal{F},\mathbb{P})$ is a function $X:\Omega \to \mathbb{R}$ (or more generally $X:\Omega \to {\mathbb{R}}^n$) such that for every Borel set $B\subseteq \mathbb{R}$, the preimage $X^{-1}(B)\in \mathcal{F}$. In other words, $X$ is an $\mathcal{F}$-measurable function.

The measurability condition ensures that we can assign probabilities to events like "$X\le x$" because the sets $\{\omega \in \Omega :X(\omega )\le x\}$ belong to the σ-algebra $\mathcal{F}$.

Example: Rolling a fair die defines a probability space $(\Omega ,\mathcal{F},\mathbb{P})$ where $\Omega =\{1,2,3,4,5,6\}$. A random variable $X$ might map each outcome to itself: $X(\omega )=\omega$. Another could be $Y(\omega )={\omega }^2$, assigning numerical values to each outcome.

The key insight is that random variables allow us to work with the abstract probability space indirectly—we don’t need to understand $\Omega$ itself, only the distribution of $X$.

What is a Distribution Induced by a Random Variable?

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A random variable $X$ on probability space $(\Omega ,\mathcal{F},\mathbb{P})$ induces a probability distribution on $(\mathbb{R},\mathcal{B})$ where $\mathcal{B}$ is the Borel σ-algebra.

Definition: Distribution (Push-forward Measure)

The distribution of a random variable $X$ is the probability measure ${\mathbb{P}}_X$ on $(\mathbb{R},\mathcal{B})$ defined by ${\mathbb{P}}_X(B)=\mathbb{P}(X^{-1}(B))=\mathbb{P}(\{\omega \in \Omega :X(\omega )\in B\})$ for every Borel set $B\in \mathcal{B}$.

This is also called the push-forward measure of $\mathbb{P}$ under $X$, often denoted ${\mathbb{P}}_X=X_{\ast }\mathbb{P}$.

Example: If we roll a fair die with $\mathbb{P}(\{i\})=1/6$ for each outcome, the distribution of $X(\omega )=\omega$ assigns probability ${\mathbb{P}}_X(\{i\})=1/6$ to each value $i\in \{1,2,3,4,5,6\}$. For a set like $B=\{1,3,5\}$ (odd numbers), ${\mathbb{P}}_X(B)=1/2$.

The distribution is the complete probabilistic characterization of the random variable. Once we know ${\mathbb{P}}_X$, we can compute probabilities of events involving $X$ without reference to the original probability space $(\Omega ,\mathcal{F},\mathbb{P})$.

What is a Cumulative Distribution Function?

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The cumulative distribution function (CDF) is the most common way to describe a distribution.

Definition: Cumulative Distribution Function

The cumulative distribution function of a random variable $X$ is the function $F_X:\mathbb{R}\to [0,1]$ defined by $F_X(x)=\mathbb{P}(X\le x)=\mathbb{P}(\{\omega :X(\omega )\le x\})$ for all $x\in \mathbb{R}$.

Key Properties of CDFs:

1. Non-decreasing: If $x<y$ then $F_X(x)\le F_X(y)$.
2. Right-continuous: ${\lim }_{h\downarrow 0}{F}_X(x+h)=F_X(x)$ for all $x$.
3. Boundary conditions: ${\lim }_{x\to -\infty }{F}_X(x)=0$ and ${\lim }_{x\to \infty }{F}_X(x)=1$.
4. Jump interpretation: $\mathbb{P}(X=x)=F_X(x)-{\lim }_{h\downarrow 0}{F}_X(x-h)=F_X(x)-F_X(x-)$.
5. Interval probabilities: For $a<b$, we have $\mathbb{P}(a<X\le b)=F_X(b)-F_X(a)$.
   Proof:

The CDF completely characterizes the distribution—two random variables with the same CDF have the same distribution. Conversely, any non-decreasing right-continuous function $F:\mathbb{R}\to [0,1]$ with ${\lim }_{x\to -\infty }F(x)=0$ and ${\lim }_{x\to \infty }F(x)=1$ is the CDF of some probability distribution (by the Lebesgue-Stieltjes theorem).

Example: For the fair die, $F_X(x)=0$ for $x<1$, $F_X(x)=k/6$ for $k\le x<k+1$ where $k\in \{1,2,3,4,5\}$, and $F_X(x)=1$ for $x\ge 6$.

What is a Density Function?

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A density function provides another way to describe distributions, useful when the distribution is “smooth” enough.

Definition: Probability Density Function

A random variable $X$ has a probability density function (pdf) $f_X:\mathbb{R}\to [0,\infty )$ if its CDF can be written as $F_X(x)={\int }_{-\infty }^x{f}_X(t)dt$ for all $x\in \mathbb{R}$.

When $f_X$ exists, it satisfies:

• ${\int }_{-\infty }^{\infty }{f}_X(x)dx=1$ (normalization)
• $\mathbb{P}(a<X\le b)={\int }_a^b{f}_X(x)dx$ (probability via integration)
• $F_X^{\prime }(x)=f_X(x)$ (derivative of CDF where the derivative exists)

Key distinction: For continuous distributions, $\mathbb{P}(X=x)=0$ for individual points, but $f_X(x)$ describes the relative likelihood in a small neighborhood of $x$. The density is a relative measure, not an absolute probability.

Not all distributions have a density. Discrete distributions (like the fair die) have no pdf because their CDFs have jumps rather than being continuously differentiable. Some distributions are mixtures of continuous and discrete parts (singular distributions) and also lack pdfs.

Example: The uniform distribution on $[0,1]$ has pdf $f_X(x)=1$ for $x\in [0,1]$ and $f_X(x)=0$ otherwise. The standard normal distribution has pdf $f_X(x)=\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2}$.

What are Measurable Maps?

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Measurability is the fundamental concept that allows us to work with probability on abstract spaces.

Definition: Measurable Function

Let $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ be measurable spaces. A function $f:X\to Y$ is $\mathcal{A}$-$\mathcal{B}$-measurable (or simply measurable) if for every set $B\in \mathcal{B}$, the preimage $f^{-1}(B)\in \mathcal{A}$.

Intuition: A function is measurable if preimages of measurable sets are measurable. This ensures we can “pull back” σ-algebras through the function—whatever we can measure in the codomain corresponds to something we can measure in the domain.

Why this matters for probability: If we want to assign a probability to the event "$X\in B$" for some set $B$, we need $X^{-1}(B)$ to be an event (i.e., in $\mathcal{F}$). Measurability guarantees this.

Theorem: Sufficient Conditions for Measurability

A function $f:(X,\mathcal{A})\to (\mathbb{R},\mathcal{B})$ is measurable if and only if for every $x\in \mathbb{R}$, $f^{-1}((-\infty ,x])\in \mathcal{A}.$

This is useful because we don’t need to check all Borel sets—just checking measurability on intervals is enough.

Intuition for measurability: Think of the σ-algebra as describing what we can “observe” or “measure” in that space. A measurable function is one that respects this structure—observations we can make in the codomain correspond to observations we can make in the domain.

Examples of measurable functions:

• Continuous functions are always measurable (with respect to Borel σ-algebras)
• Monotone functions are measurable
• Sums, products, and compositions of measurable functions are measurable

What are σ-Algebras Generated by Random Variables?

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A random variable naturally generates a σ-algebra, representing the information contained in that variable.

Definition: σ-Algebra Generated by a Random Variable

The σ-algebra generated by a random variable $X$ on probability space $(\Omega ,\mathcal{F},\mathbb{P})$ is $\sigma (X)=\sigma (\{X^{-1}(B):B\in \mathcal{B}\}),$ the smallest σ-algebra containing all sets of the form $X^{-1}(B)$ for Borel sets $B$.

This is equivalent to:

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

Interpretation: $\sigma (X)$ is the collection of all events we can describe using knowledge of $X$‘s value. If we know whether "$X\in B$" for all Borel sets $B$, we know whether every event in $\sigma (X)$ has occurred.

Theorem: Generated σ-Algebra Structure

For a random variable $X:\Omega \to \mathbb{R}$, $\sigma (X)=\{\omega \in \Omega :X(\omega )\in B\}=X^{-1}(B_1),X^{-1}(B_2),\dots$ where $B_i$ are Borel sets, and the smallest σ-algebra containing these is generated by countable unions, complements, and intersections.

Example: For a fair die with $X(\omega )=\omega$:

• The event “X is even” = $\{2,4,6\}=X^{-1}(\{2,4,6\})$ is in $\sigma (X)$
• The event “X ≤ 3” = $\{1,2,3\}=X^{-1}((-\infty ,3])$ is in $\sigma (X)$
• In fact, $\sigma (X)=2^{\Omega }$ (all subsets) because every point is in the range of $X$

Information interpretation: $\sigma (X)$ represents the information revealed by observing $X$. A larger σ-algebra means more information; $\sigma (X)\subseteq \mathcal{F}$ means observing $X$ reveals only partial information about the outcome $\omega$.

Multiple random variables generate larger σ-algebras: $\sigma (X,Y)=\sigma (\sigma (X)\cup \sigma (Y))$ represents the information from observing both variables.

What Operations on Random Variables Preserve Measurability?

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Since random variables must be measurable, it’s important to know which operations maintain measurability.

Theorem: Operations Preserving Measurability

Let $X,Y$ be random variables on $(\Omega ,\mathcal{F},\mathbb{P})$ taking values in $\mathbb{R}$. Then the following are measurable (i.e., random variables):

1. Arithmetic operations: $X+Y$, $X-Y$, $X\cdot Y$, and $X/Y$ (where $Y\ne 0$)
2. Monotone operations: $X\vee Y:=\max (X,Y)$ and $X\wedge Y:=\min (X,Y)$
3. Absolute value: $|X|$
4. Powers: $X^p$ for $p>0$
5. Limits: If $X_n$ is a sequence of random variables, then ${\sup }_n{X}_n$, ${\inf }_n{X}_n$, ${\mathrm{limsup}}_n{X}_n$, and ${\mathrm{liminf}}_n{X}_n$ are measurable
6. Continuous functions: If $g:\mathbb{R}\to \mathbb{R}$ is continuous, then $g(X)$ is measurable
7. Measurable functions: If $g:\mathbb{R}\to \mathbb{R}$ is measurable, then $g(X)$ is measurable
8. Indicators: For any $A\in \mathcal{F}$, the indicator function $1_A(\omega )=\{\begin{array}{ll}1& \text{if }\omega \in A\\ 0& \text{otherwise}\end{array}$ is a random variable

Key Principle: Composition with measurable functions preserves measurability. If $X$ is measurable and $g$ is measurable, then $g\circ X$ is measurable.

Proof sketch for arithmetic: For example, to show $X+Y$ is measurable, note that $\{X+Y\le x\}=\{(X,Y)\in \{(u,v):u+v\le x\}\}$. Since this is a preimage of a Borel set under the measurable function $(X,Y)$, it’s measurable.

Important limitation: Not all operations preserve measurability when dealing with uncountable sequences. For instance, the pointwise supremum of uncountably many measurable functions need not be measurable. However, countable suprema are always measurable.

Practical implication: This closure under operations means we can build up complex random variables from simple ones and be confident they remain properly defined (measurable).