Understanding conditional notation Which is the correct way to describe the conditional probability distribution of X conditioned upon Y where X = a, Y = b
$$P_{X \mid Y}(a \mid b) \tag1$$ or $$P_{X \mid Y}(a,b) \tag2$$
What is the difference between the two? I have seen (1) written at places. But, $P_{X \mid Y}()$ is a function in x and y. So I don't see why (2) is not the notation to describe it and what ambiguities does (1) resolve?

Why not write it as $P(X=a|Y=b)$? If you want to use underscore notation, $P_{X|Y}(a|b)$ is the more common one, e.g. you can find it used in Wikipedia. $P_{X, Y}(a, b)$ would be used for joint distribution, so the underscore is consistent with braces. The notation is described in greater detail here.
