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In a discrete space, everything is centered around the function "P". Every event has a probability associated with it, and random variables simply map outcomes to numbers. P(X = x) is basically just a weird way of saying P({e | X(e) = x}). From "P", people can define f(x) = P(X = x), g(x|y) = P(X = x, Y = y) / P(Y = y), and so on.

But in the continuous space it seems like everything is reversed! Whenever I look in a textbook, authors will just define a pdf (e.g. f(x,y)) and then derive "P" from it! (e.g. P((X,Y) $\in$ A) = $\int\int$f(x,y). How can this be?

A related question would be how random variables work in continuous spaces, too. Do they still map events to probabilities? Surprisingly, none of my textbooks talk about this...

Hi Michael Chernick, thank you for directing me to those other posts, but I just read the accepted answer in the first one, and it only talked about discrete variables. The second post supposes that I know what sigma algebras are, which I've never heard of before. The accepted answer talks about it, but also doesn't really answer why textbooks go from pdf -> P()

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  • $\begingroup$ You need to read more than just the accepted answer. Usually many, if not all, answers provide insight, often from different points of view, some of which you might understand better than the accepted answer. $\endgroup$ – whuber Aug 24 at 18:18