When defining jointly continuous, we are saying: if there exists a function $f(x,y)$ satisfying that
$ P\{X\in A, Y\in B\}=\int_B\int_a f(x,y)dxdy $
then $X,Y$ are called jointly continuous.
So not every pair of random variable $X$ and $Y$ are jointly continuous, because such $f(x,y)$ may not exists, right?
Then in Ross's book Introduction to Probability Models, there is something I'm confused:
He proved the formula
$ E[X+Y]=E[X]+E[Y] $
for jointly continuous random variables, but then use it for arbitrary pair of random variables.
When $X,Y$ is jointly continuous, it is easy to prove, since $ E[g(X,Y)]=\int_{\mathbb{R}}\int_{\mathbb{R}}g(x,y)f(x,y)dxdy $
Then let $g(X,Y)=X+Y$.
To be clear, my question is :
(1) Isn't every pair of random variable $X$ and $Y$ are jointly continuous, right?
(2) The formula $E[X+Y]=E[X]+E[Y]$ is valid for any pair $X, Y$ even if they are not independent, right?
(3) Ross in his book proved $E[X+Y]=E[X]+E[Y]$ under the assumption $X$ and $Y$ is jointly continuous. It is quite easy. How to prove general case?