Real Probability Densities

Why

We want to talk about probability when the set of outcomes is the real numbers.

Definition

A probability density or probability density function is a real-valued function on the reals which is nonnegative and normalized. A real-valued function on the reals is normalized if it integrates to 1.

Notation

We call $f: \R \to \R $ a probability density if $f \geq 0$ and $\int f = 1$.