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Do we interpret the support to be - the set of outcomes in Ω which have a non-zero probability, - the set of values that X can take with non-zero probability? Neither, actually. Consider a random variable that has a uniform density in $ [0,1]$, with $\Omega = \mathbb {R}$. Then the support is the full interval $ [0,1]$ - which is a subset of ...
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