Definition talk:Random Variable/Discrete

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There is a non-equivalent (though essentially equivalent) definition here which only requires that a countable $B \subseteq \Omega'$ has $\map \Pr {X \in B} = 1$. (it's the one used in Cohn) I'm not going to bother implementing it, since any variable satisfying said condition can be readily identified (in the sense that $\map \Pr {X = Y} = 1$) with another random variable $Y$ with countable range that preserves all the probabilistic properties of $X$ - in the sense of same mass function, same moments, etc. In particular:

$\ds \map Y \omega = \begin{cases}\map X \omega & \map X \omega \in B \\ b & \text{otherwise}\end{cases}$

for some fixed $B$ with $\map \Pr {X \in B} = 1$, and $b \in B$, works.

I remember LF saying at some point that the site isn't ready for non-equivalent treatments yet. I'll put this here as a note for someone else to maybe pick up in the future, but I don't see it as too important for now, since it makes no difference practically speaking. The thing about $\map \Pr {X = Y} = 1$ meaning that $X$ and $Y$ are "for all intents and purposes the same" is something to be implemented in due course. Caliburn (talk) 18:48, 28 December 2021 (UTC)