Does every random variable have a density?

No; only random variables whose distribution is absolutely continuous have a density. Discrete variables place mass on individual points, and rarer singular continuous distributions have no density even though they have no atoms.

Why is the distribution function defined with less-than-or-equal rather than strictly less?

The less-than-or-equal convention makes the distribution function right-continuous, which is the natural choice that puts it in clean correspondence with the underlying probability measure and its atoms.

Random Variables and Distribution Functions

A random variable is a measurable map from a probability space to the real line, and its distribution function, the probability that the variable does not exceed a given level, is the universal way to describe how its values are spread.

Definition

A random variable is a measurable function from a probability space to the real numbers, and its distribution function maps each real number to the probability that the variable takes a value less than or equal to it.

Scope

The topic covers measurability of real and vector-valued random variables, the cumulative distribution function and its defining properties of monotonicity, right-continuity, and limits, the correspondence between distribution functions and probability measures on the line, densities and the Lebesgue decomposition into discrete, absolutely continuous, and singular parts, and joint distributions of random vectors with their marginals.

Core questions

What does it mean for a function on a sample space to be a random variable?
Which properties characterize a cumulative distribution function, and how does it determine the distribution?
When does a distribution have a density, and what are the alternatives?
How are the joint and marginal distributions of several random variables related?

Key concepts

measurable function
cumulative distribution function
probability density
Lebesgue decomposition
joint and marginal distributions

Key theories

Distribution function correspondence: Every probability measure on the real line corresponds to a unique non-decreasing, right-continuous distribution function with limits zero and one, and conversely, giving a complete and concrete description of one-dimensional distributions.
Lebesgue decomposition of a distribution: Any distribution on the line splits uniquely into a discrete part supported on atoms, an absolutely continuous part with a density, and a singular continuous part, clarifying when a probability density exists and when it does not.

Clinical relevance

Distribution functions are what empirical data estimate and what statistical models posit; the empirical distribution function underlies goodness-of-fit testing and the bootstrap, quantiles derived from the distribution function define value-at-risk and reference ranges, and densities are the objects fitted in most likelihood-based inference.

History

The recognition that a random variable is simply a measurable function and that its behavior is captured by a distribution function arose with the measure-theoretic reformulation of probability in the early twentieth century, replacing the earlier case-by-case treatment of particular distributions.

Key figures

Andrey Kolmogorov
William Feller
Henri Lebesgue

Seminal works

billingsley1995

Frequently asked questions

Does every random variable have a density?: No; only random variables whose distribution is absolutely continuous have a density. Discrete variables place mass on individual points, and rarer singular continuous distributions have no density even though they have no atoms.
Why is the distribution function defined with less-than-or-equal rather than strictly less?: The less-than-or-equal convention makes the distribution function right-continuous, which is the natural choice that puts it in clean correspondence with the underlying probability measure and its atoms.