Math 2.0

definition

Random variable with a density

Definition

Let XX be a real random variable on a probability space (Ω,F,P)(\Omega, \mathcal{F}, P). XX has a density, or is absolutely continuous, if there is a Borel measurable function f ⁣:R→[0,∞)f \colon \mathbb{R} \to [0, \infty) with the following property. For each Borel set B⊆RB \subseteq \mathbb{R}, P(X∈B)=∫Bf(x) dx,P(X \in B) = \int_B f(x) \, dx, where the integral is the Lebesgue integral.

Remarks

Such a function ff is a probability density function of XX. If XX has a density, then P(X=x)=0P(X = x) = 0 for each xx, and the cumulative distribution function FX(x)=∫−∞xf(t) dtF_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 Rn\mathbb{R}^n and the Lebesgue measure on Rn\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.

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Sources

  1. Web https://en.wikipedia.org/wiki/Probability_density_function, Absolutely continuous univariate distributions

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