Probability Densities

Why

What if the set of outcomes is the real line, $\R $?

Definition

The principal difficulty is assigning nonzero numbers to infinitely many elements of a set. The solution is instead to assign probabilities to the events of outcomes, not to the individual outcomes (elementary events, real numbers), themselves.

A probability density (or probability density function (pdf)) is a function $f: \R \to \R $ satisfying $f \geq 0$ and $\int f = 1$.