Naming a probability mapping of mixed continuous and discrete valued random variables

Question

Consider the random pair $(X, Y)$ such that $X$ is a continuous-valued random variable taking values in the set $\Theta$, while $Y$ is a discrete-values random variable taking values in the set $\{1, 2, \ldots, k\}$. Suppose I know the mapping $f: \Theta \times \{ 1, 2, \ldots, k \} \longrightarrow [0,1]$ where $f(x,y) \in [0,1]$ is the joint probability that $X$ and $Y$ take values $x \in \Theta$ and $y \in \{ 1, 2, \ldots, k \}$, respectively. What should this mapping $f$ be called? Given the mixture of continuous-valued and discrete-valued random variables in the pair, I cannot call it a probability density nor mass function I presume. I would appreciate any hints.

Answer 1

Given the mixture of continuous-valued and discrete-valued random variables in the pair, I cannot call it a probability density nor mass function I presume. I would appreciate any hints.

In the mixed case, it is called a mixed probability density function.

Answer 2

You can get answer for your equation in the link wiki I think, can calling mixture density function, or joint distribution function.