Determinate the joint probability marginal distribution I have a random vector $(X,Y,Z,W)$ with the following Probability Mass Function: $ in a similar table with its own marginals?

From the table, you can determine the joint marginal distribution of $(X,Y)$ as follows:- $$p_{(X,Y)}(x,y)=\sum_{z}\sum_{w}p_{(X,Y,Z,W)}(x,y,z,w)$$ Practically, this means that for each of the $4$ possible values of $(x,y)$ i.e. $(0,0),(0,1),(1,0),(1,1)$ you need to add up the relevant entries of $p_{(X,Y,Z,W)}(x,y,z,w)$.