probability density function

A probability density function is a probability distribution which (indirectly) maps every interval of a continuous random variable to a probability.

Formally,
given a continuous random variable $X$ , the probability density function of $X$ is a function $f (x)$ such that for any two $a \leq b$ in the image of $X$ ,

P (a \leq X \leq b) = \int_{a}^{b} f (x) d x

where $f (x)$ is non-negative and $\int_{- \infty}^{\infty} f (x) d x = 1$ .

More formally,
let $(Ω, F, P)$ be a probability space, with $P : F \to [0, 1]$ .
Given a measurable space $(S, Σ)$ and random variable $X : Ω \to S$ ,
the density of $X$ is the Radon-Nikodym derivative of the pushforward measure $X_{*} P$ with respect to the reference measure $μ$ on $(S, Σ)$ :

f = \frac{d X_{*} P}{d μ}

then the probability that the value of $X$ lies in $A \subseteq S$ is

P (X \in A) = \int_{A} f d μ

where typically, the reference measure $μ$ is the Lebesgue measure.

The probability mass function of a discrete random variable can be defined as a probability density function against the counting measure.