---
tag: 0C21RZG5V2
kind: definition
name: Random variable with a density
msc: ["60E05"]
defines: 045D8P03E0
uses: [0DKRR22AEQ, 0BVHZXWKCD]
sources:
  - url: https://en.wikipedia.org/wiki/Probability_density_function
    locator: Absolutely continuous univariate distributions
---

## Definition

Let $X$ be a real [[random variable|0DKRR22AEQ]] on a probability space $(\Omega, \mathcal{F}, P)$.
$X$ has a density, or is absolutely continuous, if there is a Borel measurable function $f \colon \mathbb{R} \to [0, \infty)$ with the following property. For each Borel set $B \subseteq \mathbb{R}$,
$$P(X \in B) = \int_B f(x) \, dx,$$
where the integral is the Lebesgue integral.

## Remarks

Such a function $f$ is a [[probability density function|04YG690N5V]] of $X$.
If $X$ has a density, then $P(X = x) = 0$ for each $x$, and the cumulative distribution function $F_X(x) = \int_{-\infty}^x f(t) \, dt$ is continuous.
A continuous cumulative distribution function does not imply a density. The Cantor distribution has a continuous cumulative distribution function and no density.
The same definition with $\mathbb{R}^n$ and the Lebesgue measure on $\mathbb{R}^n$ gives random vectors with a density.

## Explanations

### L1 human

For a random variable with a density, the probability of an interval is an area under a curve.
Single points have probability zero, just as a single point has no length.