Marginal Probability Density Function¶

Definition¶

For a jointly continuous pair $(X,Y)$ with joint density $f_{X,Y}$ , the marginal PDF of $X$ is

\begin{aligned} f_X(x)=\int f_{X,Y}(x,y)\,\mathrm{d}y, \end{aligned}

where the integral runs over the values allowed for the other coordinate; the density is zero outside the joint support.

Interpretation¶

The marginal density of $X$ is the density of $X$ on its own, ignoring $Y$ . Integrating out the other coordinate of the joint density collapses the probability in the plane onto the $x$ -axis.

Two panels: the triangle with joint density 2 and a vertical and a horizontal slice, and the resulting marginal densities 2(1 - x) for X and 2y for Y.

For the illustrated triangular density, the red vertical slice at $x$ runs from $y=x$ to $y=1$ and has length $1-x$ $x$ , so $f_X(x)=\int_x^1 2\,\mathrm{d}y=2(1-x)$ for $0<x<1$ $1$ . The green horizontal slice at $y$ has length $y$ , so $f_Y(y)=\int_0^y2\,\mathrm{d}x=2y$ for $0<y<1$ $1$ .

The value $f_X(x)\,\Delta x$ approximates the probability of a thin vertical strip of width $\Delta x$ at $x$ , which is the volume under the joint density over that strip. When the support is not a rectangle, the limits of the inner integral depend on $x$ , as in the triangle. This is the continuous version of adding up a row of a joint PMF table to get a marginal PMF.

Marginal densities are the denominator of a conditional density, and the continuous law of total probability rewrites the same integral with conditional densities. The marginals do not determine the joint density: on the unit square, both $f(x,y)=1$ and $f(x,y)=1+(1-2x)(1-2y)$ have $\operatorname{Unif}(0,1)$ marginals. The joint density is the product of the marginals exactly in the independent case.

Ignoring the support is the usual error: integrating $2$ over $y\in(0,1)$ gives the constant $2$ , which integrates to $2$ in $x$ , so it cannot be a density.

Key Results¶

$f_Y(y)=\int_{\mathbb{R}}f_{X,Y}(x,y)\,\mathrm{d}x$ $\int_{R} f_{X, Y} (x, y) d x$ .
Each marginal integrates to one, since the joint density does.

Examples¶

If $f_{X,Y}(x,y)=2$ on $0<x<y<1$ $1$ , then $f_X(x)=2(1-x)$ for $0<x<1$ $1$ .
For the same triangle, $f_Y(y)=\int_0^y2\,\mathrm{d}x=2y$ for $0<y<1$ $1$ .

Prerequisites¶

Joint Probability Density Function

Sources¶

Nicky van Foreest · Probability Distributions · Week 2

Aliases: marginal PDF, marginal density, marginal density function Symbols: delta, element of, integral, real numbers

Last updated: 20 October 2026 03:02