Two continuous random variables associated with the same experiment are jointly continuous and can be described in terms of a joint PDF $f_{X,Y}$ if $f_{X,Y}$ is a nonnegative function that satisfies
$$\b{P}((X, Y) \in B) = \iint_{(x, y) \in B } f_{X,Y}(x, y) \d x \d y$$
for every subset $B $ of the two-dimensional plane.
$\int_{ -\infty }^{ +\infty } \int_{ -\infty }^{ +\infty } f_{X, Y}(x, y) \d x \d y = 1$(Normalization)
The marginal PDF $f_X$ of $X $ is given by
$f_X(x) = \int_{ -\infty }^{ +\infty } f_X,Y(x, y) \d y$
Similarly,
$f_Y(y) = \int_{ -\infty }^{ +\infty } f_X,Y(x, y) \d x$
If $g: \R \times \R \to \R $, the expectation is:
$$E[g(X, Y)] = \int_{ -\infty}^{ +\infty } \int_{ -\infty }^{ +\infty } g(x, y) f_{X, Y}(x, y) \d x \d y $$
For any scalar $a$, $b$, and $c$, we have
$$E[aX + bY + c] = aE[X] + bE[Y] + c$$
If $X $ and $Y $ are two random variables associated with the same experiment, we define their joint CDF by
$F_{X, Y}(x, y) = \b{P}(X \leq x, Y \leq y) = \int_{ -\infty }^{ x } \int_{ -\infty }^{ y } f_{X, Y}(s, t) \d t \d s$
If $X, Y $ are jointly continuous random variables, they are independent if:
$$f_{X,Y}(x, y) = f_X(x)f_Y(y) $$
This implies,
for any subsets $A \subseteq \R, B \subseteq \R,$
$$\b{P}(x \in A, y \in B) = \b{P}(x \in A) \b{P}(y \in B)$$
Also:
$$E[XY] = E[X]E[Y] $$