Common distribution function of continuous and discrete random variables

Question

Given two random variables, one discrete (X) and the other one continuous (Y), and given $P_X$ and $f_Y$, how would you refer to their common distribution function?

As $P_{X,Y}$ or $f_{X,Y}$?

Answer 1

I'd refer to the CDF, denoting it by $F_{X,Y}$ or similar. Such a 2D variable won't have a PMF or a PDF.

Answer 2

I think you may be confused between distribution (a measure) and density (an integrable function). The joint (not common) distribution always exists, but it will not have a density (otherwise the integral of that density over the second variable would be a density for the first variable).