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I was wondering how does the rules of probability apply if we have a discrete random variable $Y$ and a continuous random variable $X$.

What would $P(Y=y,X=x)$ be equal to? Is joint probability defined or is it joint density function? (note: I denote probability as $P$ and density functions with $f$)

Is it correct to write:

$$P(Y=y|X=x) = \frac{f_{X,Y}(y,x)}{f_X(x)}$$

or should I write:

$$f_{Y|X}(y|x) = \frac{f_{X,Y}(y,x)}{f_X(x)}$$

How does the continuous random variable affect the joint properties of the two random variables? Is it correct to say that:

If the random variables are both discrete, then the we can state their relationships with the probability $P$-function, but if either one or both of them are continuous then we have to describe the probabilistic relationship using density functions $f$.

For example, if I denote random variable $Z$ as $Z = (X,Y)$ (vector), then is the space where $Z$ is defined in continuous or discrete?

Hope my question is clear enough :) In summary: I'm just trying to figure out how does the mixing of discerete and continuous random variables affect the probability space :)

Thnx for any help!

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Related: math.stackexchange.com/a/608906/6179 –  Did Dec 18 '13 at 22:30
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