The Joint Probability Density Function for Two Continuous Random Variables

Definition

Let $X, Y$ be continuous rvs. Then $f (x, y)$ is a joint probability density function for $X, Y$ if for any two-dimensional set $A$ :

P ((X, Y) \in A) = \iint_{A} f (x, y) d x d y

In particular, if $A$ is a rectangle $A = [a, b] \times [c, d]$ then:

P ((X, Y) \in A) = P (a \leq X \leq b, c \leq Y \leq d) = \int_{a}^{b} \int_{c}^{d} f (x, y) d y d x

Definition

A joint pdf of continuous rvs $X_{1}, X_{2}, \dots, X_{n}$ is a function $f (x_{1}, x_{2}, \dots, x_{n})$ such that for any $n$ intervals $[a_{1}, b_{1}], \dots, [a_{n}, b_{n}]$ then:

P (a_{1} \leq X_{1} \leq b_{1}, \dots, a_{n} \leq X_{n} \leq b_{n}) = \int_{a_{1}}^{b_{1}} \dots \int_{a_{n}}^{b_{n}} f (x_{1}, \dots, x_{n}) d x_{n} \dots d x_{1}

More generally, for any $n$ -dimensional set $A$ consisting of values for $x_{1}, \dots, x_{n}$ then $P ((X_{1}, \dots, X_{n}) \in A)$ is the multiple integral of $f$ over the region $A$ .