# Joint PDF and Joint CDF of a Discrete and Continuous random variables

Suppose we have a discrete random variable $X$ and a continuous random variable $Y$. I am trying to understand how one defines/ find the joint PDF and joint CDF of $X$ and $Y$.

The joint CDF of $X$ and $Y$ is given by $$F_{X, Y}(x, y) = P(X \le x, Y \le y)$$

If both the random variables were discrete (continuous) then we could have found the joint PMF (joint PDF). But since $X$ and $Y$ are discrete and continuous, respectively, can we define "hybrid" joint PDF or "hybrid" joint PMF? For instance, can we find the "hybrid" joint PDF $f_{X, Y}(x, y)$ such that the marginal distributions of $X$ and $Y$ are given by

$$f_Y(y) = \sum_{x}f_{X, Y}(x, y)$$ and $$P(X = x) = \int_{- \infty} ^ {\infty}f_{X, Y}(x, y) \; \mathrm{d} y$$

Since I don't know measure theory yet, any answers which don't use references to measure theory will be really helpful.

• This is what measure theory is for... – zoli May 10 '18 at 6:26