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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?

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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.

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  • $\begingroup$ Your suggestion made me realise an improvement to my question. Making it. I guess you're suggestion doesn't make sense for continuous distributions. X = a is never happening. $\endgroup$ Jun 26, 2022 at 8:52
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    $\begingroup$ 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. $\endgroup$
    – whuber
    Jun 26, 2022 at 15:50
  • $\begingroup$ "$P_{X, Y}(a, b)$ 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 $\endgroup$ Jun 26, 2022 at 20:23
  • $\begingroup$ @MiloMinderbinder yes and I refer to exactly that, the convention is to use “,” or “|” consistently between underscore and braces. $\endgroup$
    – Tim
    Jun 26, 2022 at 20:33
  • $\begingroup$ Ohhh. Okay. My bad. I see your point now $\endgroup$ Jun 26, 2022 at 21:09

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