# 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.
• Although your suggestion is often used for discrete variables, it is confusing for continuous variables, because--being the analog of writing "$f(X=x)$" for the conventional expression $f(X)$--it is pleonastic.
• "$P_{X, Y}(a, b)$ would be used for joint distribution, so the underscore is consistent with braces." I did not mention this notation. I mentioned $P_{X \mid Y}(a,b)$. The information regarding the distribution to follow is in the subscript and the parentheses contain the values that the RVs involved take Jun 26, 2022 at 20:23