Probability density function

A probability density function of a probability measure $P$ on $\mathbb R^d$ is a nonnegative measurable function $f$ such that, for every Borel set $A$ ,

P(A)=\int_A f(x)\,dx,

where the integral is with respect to Lebesgue measure.

Properties

Necessarily $\int_{\mathbb R^d}f(x)\,dx=1$ . Densities are unique only up to changes on sets of Lebesgue measure zero. The density of a random variable means a density of its law.

A density can exceed one; probabilities are integrals over sets. Not every probability measure has a density with respect to Lebesgue measure.

General reference measures

Relative to another measure $\nu$ , a density is a Radon–Nikodym derivative $dP/d\nu$ . A probability mass function is a density relative to counting measure on a countable state space.

Examples

The normal distribution has a Gaussian density; the uniform distribution on an interval has constant density on that interval.