Joint Distribution of Discrete and Continuous Suppose i have two random variables, $X$ and $Y$. WLOG assume $X$ is discrete, and $Y$ is continuous.
How do we define the joint distribution between $X$ and $Y$? Furthermore, how can we extend this to a countably infinite set of random variables?
EDIT: i should note that, I'm wondering how can it be defined and calculated in terms of PDF and CDF.
 A: The joint distribution can be defined using the measure-theoretic definition: you have to know
$\mathbb{P}((X,Y)\in A) $ for all $A\in \mathbb{R}\times \mathbb{N}$ for instance,
and this works for any two random variables, you can extend this to any space of definition for $X$ and $Y$.
On the other hand, to define a pdf you would need a reference measure. For continuous distribution, the reference measure is the lebesgue measure, for discrete distribution the reference is the counting measure.
Take the following example : $X\sim Unif((0,1))$, $Y \sim Ber(X)$. You have that $X$ and $Y$ are dependent and you can show that for a function $\phi$,
$$E[\phi(X,Y)]= \int_{0,1}\sum_{y=0}^1 \phi(x,y) x^y(1-x)^{1-y} dx $$
hence, the density with respect to the tensor product lebesgue times counting measure is
$$f(x,y)= \begin{cases} x^y(1-x)^{1-y} &if \quad x \in (0,1), y \in \{0,1\}\\ 0 &else \end{cases}$$
Then, indeed you get a pdf, with respect to the tensor product lebesgue times counting measure, but as you saw it most of the time it is more natural to describe such random variables using, if possible, a hierarchical definition where one random variable depends of the other even though it may not always be possible.
