Multivariant marginal distribution

Question

The marginal distribution of a individual RV of two discrete random variables $X$ and $Y$ is

$$pX(x) = \sum_y pXY(x,y) $$

And if you have 3 disrete RVs $X$, $Y$, $Z$

is this correct? $$pXY(x,y) = \sum_z pXYZ(x,y,z) $$

And why so?

Answer 1

True. $P(X=x,Y=y)=\sum_{z} P(X=x,Y=y,Z=z)$ because the events on the right side are mutually exlcusive and their union is the event on the left.