in probability and statistics, a probability density function is a function that characterizes any continuous probability distribution. for a random variable "x", the probability density function of x is sometimes written as formula_1. the integral of the probability density function in the interval formula_2 yields the probability that a given random variable with the given density is contained in the interval provided. by definition, the probability density function is non-negative throughout its domain, on which the integral sums up to 1.