Probability Density Function¶

Definition¶

A probability density function of an absolutely continuous $X$ is a nonnegative integrable function satisfying

\begin{gathered} \mathbb{P}(X\in A)=\int_Af_X(x)\,\mathrm{d}x,\\ \int_{\mathbb{R}}f_X(x)\,\mathrm{d}x=1. \end{gathered}

Densities that agree except possibly on a set of zero length, for example at finitely many points, describe the same distribution. This qualification is called almost everywhere here: values on the exceptional set do not affect the integrals. It does not require two densities to agree at every point.

Interpretation¶

A density describes how probability is spread over the real line: probabilities are areas under its graph, not heights.

A density curve with the area between a and b shaded, equal to the probability that X lies between a and b.

The shaded region between $a$ and $b$ has area $\int_a^bf_X(x)\,\mathrm{d}x=\mathbb{P}(a\le X\le b)$ $b)$ , the first formula of the Definition with $A=[a,b]$ $[a, b]$ , and the whole area under the curve is $1$ . Moving $b$ to the right adds the area of a thin strip. A strip of zero width has zero area, so $\mathbb{P}(X=a)=0$ and it does not matter whether the endpoints are included.

Probabilities are areas under the density. The narrow uniform density reaches height $2$ , which is fine: only areas, never heights, are probabilities.

The height $f_X(x)$ is probability per unit length. For a short interval of length $h$ starting at a point $x$ where $f_X$ is continuous, $\mathbb{P}(x<X\le x+h)\approx f_X(x)\,h$ $f_{X} (x) h$ . This is why a height above $1$ is allowed, as for the uniform density of height $2$ on $(0,\tfrac{1}{2})$ , and why changing $f_X$ at finitely many points changes no probability: no area changes. The definite integral measures these areas; for the piecewise continuous densities in the examples, Riemann integral calculations evaluate them.

For a continuous random variable the density plays the role that the PMF plays for a discrete one, with sums replaced by integrals. Integrating the density up to $x$ gives the CDF, and by the fundamental theorem of calculus differentiating the CDF gives back $f_X(x)$ at every point where $f_X$ is continuous. Density height itself is never a point probability.

Key Results¶

$\mathbb{P}(a<X\le b)=\int_a^bf_X(x)\,\mathrm{d}x$ $\int_{a}^{b} f_{X} (x) d x$ .
$F_X(x)=\int_{-\infty}^xf_X(t)\,\mathrm{d}t$ $\int_{- \infty}^{x} f_{X} (t) d t$ , so $F_X'=f_X$ except possibly on a set of zero length.

Examples¶

For $f_X(x)=2x$ on $(0,1)$ and zero elsewhere, $\mathbb{P}(X\le1/2)=1/4$ $1/4$ .
The uniform density on $(0,1/2)$ has height $2$ , total area $1$ , and $\mathbb{P}(X=1/4)=0$ $0$ .

Prerequisites¶

Sources¶

Probability Theory for EOR · Week 5
Nicky van Foreest · Probability Distributions · Refresher

Aliases: PDF, density function, probability density, p.d.f. Keywords: density Symbols: element of, infinity, integral, real numbers

Last updated: 20 October 2026 03:02