A **probability density function** serves to represent a probability distribution in terms of integrals. If a probability distribution has density *f*(*x*), then intuitively the infinitesimal interval [*x*, *x* + d*x*] has probability *f*(*x*) d*x*. A probability density function can be seen as a "smoothed out" version of a histogram: if one empirically measures values of a random variable repeatedly and produces a histogram depicting relative frequencies of output ranges, then this histogram will resemble the random variable's probability density (assuming that the variable is sampled sufficiently often and the output ranges are sufficiently narrow).

Formally, a probability distribution has density *f*(*x*) if *f*(*x*) is a non-negative Lebesgue-integrable function **R** → **R** such that the probability of the interval [*a*, *b*] is given by

For example, the uniform distribution on the interval [0,1] has probability density *f*(*x*) = 1 for 0 ≤ *x* ≤ 1 and zero elsewhere. The standard normal distribution has probability density

- .

Two densities *f* and *g* for the same distribution can only differ on a set of Lebesgue measure zero.

See also: