User:Caliburn/s/prob/Definition:Probability Density Function

Definition
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X: \Omega \to \R$ be an absolutely continuous random variable on $\struct {\Omega, \Sigma, \Pr}$.

Let $P_X$ be the probability distribution of $X$.

Let $\map \BB \R$ be the Borel $\sigma$-algebra on $\R$.

Let $\lambda$ be the Lebesgue measure on $\struct {\R, \map \BB \R}$.

Then a probability density function of $X$, written $f_X$, is a Radon-Nikodym derivative of $P_X$ with respect to $\lambda$.

Also known as
Probability density function is often conveniently abbreviated as p.d.f. or pdf.

Sometimes it is also referred to as the density function.

Also see

 * Definition:Probability Mass Function