I know how to find the expectation of a function of a Random Variables, I was just wondering that does expectation of a joint distribution exists?
I think, since expectation is an average which by definition means a statistic: a single value that describes a distribution so can we capture the behaviour of the entire distribution in both x and y in a single number, shouldn't we need 2 values for it(one for x and the other for y)?
We can do so for a function of in x and y because that function outputs a single value hence the Expectation is a single number.
You're right. It makes sense to talk about the expectation of any PDF $f(x)$ on any $\mathbb R^n$. The definition is the same as for real-valued distributions. $$ \mathbb E X = \int _{\mathbb R^n} xf(x) dx$$
Here $x$ is a vector and $f(x)$ is a scalar. So we can integrate the vector $x f(x)$ with respect to the normal uniform measure on $\mathbb R^n$ to get a vector in $\mathbb R^n$.
Given two real-valued variables $X$ and $Y$ we define the joint CDF
$$F_{X,Y}(x,y) = P(X < x, Y < y)$$
and the joint PDF
$$f_{X,Y}(x,y) = \frac{\partial^2 F_{X,Y}(x,y)}{\partial x \partial y}$$
The expectation of $(X,Y)$ is defined as the expectation of that PDF:
$$ \mathbb E (X,Y) = \int _{\mathbb R^2} (x,y)f_{X,Y}(x,y) dx dy$$