A dice is thrown. If you get $1$, $2$ or $3$ another dice is thrown. If you get any other number, the second dice is not thrown. Let $Y$ be the random variable which represents a number on the second dice and $X$ be the number on first dice. Find their joint distribution.
I understand that $P(X = 1,Y= 1) = \frac{1}{36}$, but will that define their joint distribution?
I understand their marginal distribution should be $P(X=x) = \frac{1}{6} x$ Belongs to $\{1,2,3,4,5,6\}$, similarly for $Y$. But I don’t understand how one will derive this from their joint distribution?