Joint Distributions for Continuous Random Variables

For two discrete random variables, the joint distribution is described by a joint pmf $p(x,y)$ that assigns a probability to each pair $(x,y)$ and sums to $1.$ For two continuous random variables, probabilities are spread over a region, so we use a joint probability density function (joint pdf).

A function $f(x,y)$ is a joint pdf for the continuous random variables $X$ and $Y$ if:

1. $f(x,y) \ge 0$ for all $(x,y),$ and
2. The total integral over the plane equals $1{:}$

$$\int_{-\infty}^{\infty}\!\!\int_{-\infty}^{\infty} f(x,y)\, dy\, dx = 1.$$

In practice, $f(x,y)$ is nonzero only on some region of the plane, and we only integrate over that region. For example, suppose $f(x,y) = c$ on the unit square $0 \le x \le 1,\ 0 \le y \le 1$ and $0$ elsewhere. To find $c,$ we require

$$\int_0^1 \!\!\int_0^1 c\, dy\, dx = c \cdot 1 \cdot 1 = 1,$$

so $c = 1.$ This is the uniform joint density on the unit square.