Random variable

Definition

A random variable is a measurable function:

or

where $(\mathrm{\Omega },\mathrm{\Sigma })$ is a measurable space and $\mathcal{B}(-)$ is the corresponding Borel $\sigma$-algebra.
If $(\mathrm{\Omega },\mathrm{\Sigma },\mathbb{P})$ is a probabilistic space, we call $X$ a random variable on $\mathbb{P}$.
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Remarks:

1. Boundedness: The condition $\underset{\omega \in \mathrm{\Omega }}{sup}|X(\omega )|<\mathrm{\infty }$ is often added (e.g., for uniform integrability).
2. A random variable which only takes the values 0 or 1 encodes the same data as an event (more precisely, it is the indicator function ${\chi }_E$ of a unique event, which takes the value $1$ on $E$ and $0$ on its complement). More generally, we can construct events from random variables: for any Borel subset $E\subseteq \mathbb{R}$ the preimage $X^{-1}(E)$ is an event (the event that $X$ lies in $E$), often written $X\in E$, and so we can consider its probability $\mathbb{P}(X\in E)$.
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Probability distribution

Every random variable on a probabilistic space,

induces a probability measure ${\mu }_X$ on $({\mathbb{R}}^n,\mathcal{B}({\mathbb{R}}^n))$ by means of the pushforward measure

It is called the distribution of $X$. There is a relationship between famous probability distributions on $\mathbb{R}$ and the Lebesgue measure, fundamental to modern probability theory. Most common continuous distributions, like the Normal, Exponential, and Uniform distributions, are defined by "weighting" the Lebesgue measure with a function. These are called absolutely continuous distributions. The key notion is that of probability density function.

Category theory viewpoint

From a categorical perspective, a random variable

is simply a morphism in the category of measurable spaces $\mathbf{Meas}$. Now observe that:

• In Set, an element $a\in A$ corresponds to a map $1\to A$.
• In a general category $\mathcal{C}$, morphisms $X\to A$ are called generalized elements of $A$ of shape $X$.
• This expresses the idea that objects are understood in terms of the ways other objects “probe” them.

Thus:

• A random variable $X:\mathrm{\Omega }\to \mathbb{R}$ is a generalized element of $\mathbb{R}$ of shape $\mathrm{\Omega }$.
• Intuitively, it is a structured element of $\mathbb{R}$, where the structure comes from the underlying probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$.

In this sense, random variables are not just “values” in $\mathbb{R}$, but morphisms encoding how $\mathbb{R}$ is accessed through probabilistic structure. This viewpoint is directly inspired by Yoneda’s lemma, which tells us that an object is completely determined by all its generalized elements (i.e. by all morphisms into it). See category of sets#Generalized elements in other categories.