Probability density function

A continuous variable takes values in an interval, and each isolated value has probability zero: \(P(X=a)=0\). So probability is assigned to intervals, and the density \(f\) is the function whose area makes that possible. It is read as a concentration: where \(f\) is large, the variable falls more often in short intervals around that point.

Every density satisfies two conditions: \(f(x)\ge 0\) and \(\int_{-\infty}^{\infty}f(x)\,dx=1\). When a density comes with an unknown constant, as in \(f(x)=kx\) on \([0,2]\), the second condition determines it: \(2k=1\). Since single points do not count, \(P(a\le X\le b)\) and \(P(a<X<b)\) are equal, unlike the discrete case.

A density is not a probability and can exceed 1: \(f(x)=2\) on \([0,\tfrac12]\) has area 1. What is a probability is the area. If \(f(x)=2x\) on \([0,1]\), then \(P(X\le\tfrac12)=\int_0^{1/2}2x\,dx=\tfrac14\). A typical mistake is reading \(f(a)\) as \(P(X=a)\).

For continuous variables the density plays the role the mass function plays for discrete ones, with integrals in place of sums. Later the cumulative distribution function unifies both cases, and the integral of \(x\,f(x)\) defines the continuous expectation.