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If $Y$ is continuous and $X$ is discrete with a finite number of points in the support, how to write the joint density of $(Y,X)$? For example, to write the joint density function evaluated at $(Y,X)=(y,x)$, is it true that I can only write it as $f(y|x)P_x$, where $f(\cdot|x)$ is the conditional density of $Y$ given $X=x$ and $P_x$ is the marginal pmf of $X=x$.

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