Joint Probability Density Function¶

Definition¶

A random pair $(X,Y)$ is jointly continuous with joint PDF $f_{X,Y}$ if, for every Borel region $B\subseteq\mathbb{R}^2$ $R^{2}$ , such as a rectangle, a disc or a half-plane,

\begin{aligned} \mathbb{P}((X,Y)\in B)=\iint_B f_{X,Y}(x,y)\,\mathrm{d}x\,\mathrm{d}y. \end{aligned}

Continuous marginals alone do not guarantee a joint density: with $Y=X$ both marginals can be continuous while all the probability lies on a line.

The double Definite Integral adds density over a rectangle or other region in the joint support, giving the probability that the pair falls there.

Interpretation¶

A joint density describes how probability is spread over the plane: $f_{X,Y}(x,y)$ is probability per unit area near $(x,y)$ , and the probability of a region is the volume under the surface above it.

The triangle above the diagonal of the unit square, where the joint density is 2, with the part below y = 1/2 shaded; its probability is 1/4.

The illustrated density has height $2$ over the triangle $0<x<y<1$ of area $1/2$ , so its total volume is $2(1/2)=1$ $1$ . The red part of the triangle below $y=1/2$ has area $1/8$ , so $\mathbb{P}(Y<1/2)=2\cdot1/8=1/4$ $1/4$ .

Choose $a$ and $b$ : $\mathbb{P}(X\le a,Y\le b)$ is the volume under $f_{X,Y}$ over the shaded rectangle. Try the course examples.

Any nonnegative function with total integral $1$ is a joint density. The value $f_{X,Y}(x,y)$ is not itself a probability and may exceed $1$ , as the height $2$ shows. For a small rectangle with sides $\Delta x$ and $\Delta y$ around $(x,y)$ , the probability is approximately $f_{X,Y}(x,y)\,\Delta x\,\Delta y$ . Points and curves have area zero and hence probability zero, which is why the pair $Y=X$ $X$ , whose probability lies on a line, has no joint density.

This is the one-variable density from Probability Theory with area in place of length, and the continuous counterpart of the joint PMF, with double integrals in place of double sums. From it follow marginal and conditional densities and expectations by two-dimensional LOTUS. For independent variables the joint density is the product $f_X(x)f_Y(y)$ .

The region where the density is positive fixes the integration limits. For $\mathbb{P}(Y<1/2)$ on the triangle, $x$ runs from $0$ to $y$ , not from $0$ to $1$ ; $0$ to $1$ would give probability $1$ .

Key Results¶

$f_{X,Y}\ge0$ and $\iint_{\mathbb{R}^2}f_{X,Y}(x,y)\,\mathrm{d}x\,\mathrm{d}y=1$ $1$ .
A region of planar area zero has probability zero under a joint density.

Examples¶

For density $1$ on $(0,1)^2$ , $\mathbb{P}(X<1/2,Y<1/3)=1/6$ $1/6$ .
For density $2$ on $0<x<y<1$ $1$ , $\mathbb{P}(Y<1/2)=\int_0^{1/2}\int_0^y2\,\mathrm{d}x\,\mathrm{d}y=1/4$ $1/4$ .

Prerequisites¶

Sources¶

Nicky van Foreest · Probability Distributions · Week 2

Aliases: joint PDF, joint density, joint density function, bivariate density Symbols: delta, element of, integral, real numbers, subset

Last updated: 20 October 2026 03:02