Creating a bivariate distribution from two independent variables If you have two random variables that are independent say $X\sim f_X (vars)$ and $Y \sim f_Y (vars)$.
Is this a way to produce a bivariate distribution $f_{(X,Y)}$? 
$f_{(X,Y)} = p(X=x \cap Y=y) = p(x) \times p(y) = f_X(x)\times f_Y(y)$.
Does this hold for the case of integrals as well in the continuous case?

Yes, the joint cumulative distribution function $F_{X,Y}(x,y)$, that is, the probability that $X\le x$ and $Y\le y$, is the product of the individual cdf $F_X(x)$ and $F_Y(y)$. 
If $X$ and $Y$ have continuous distributions with density functions $f_X(x)$ and $f_Y(y)$, then the joint density function $f_{X,Y}(x,y)$ is the product of the individual density functions.
The discrete case is as you described it. 
