Let $X$ be a discrete random variable and $Y$ be a continuous random variable. I can't seem to come up with an explanation as to why $X|Y$ is always discrete. How do I see this intuitively?
2 Answers
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Intuitively speaking, conditioning does not change the set of values that a random variable can take, it only changes the probabilities assigned to them. If the distribution of $X$ is concentrated on a discrete set $A$, then the distribution of $X$ conditioned on anything is still concentrated on (a subset of) the same set $A$.
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The total probability mass of the joint distribution of $X$ and $Y$ lies on a set of vertical lines in the $x$-$y$ plane, one line for each value that $X$ can take on. Along each line $x$, the probability mass (total value $P(X = x)$) is distributed continuously, that is, there is no mass at any given value of $(x,y)$, only a mass density. Thus, the conditional distribution of $X$ given a specific value $y$ of $Y$ is discrete; travel along the horizontal line $y$ and you will see that you encounter nonzero density values at the same set of values that $X$ is known to take on (or a subset thereof); that is, the conditional distribution of $X$ given any value of $Y$ is a _discrete distribution.
