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Mar19-06, 07:57 PM
Sci Advisor
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Hurkyl's Avatar
P: 16,099
First case, we say that X and |X| are two different random variables. But I do not see any condition for |X| to be a random variable given that X is a random variable, so I think this statement is not true at all.
I repeat, the same method you use to construct random variables such as X, 1/X, and (X+3) also works to construct |X|.

Second, since we can not map back a real number to the sample space, say using |X|, then the fact that |X| is a random variable does not imply that X is a random variable too, so this statement is false.
What kind of thing is X? It makes absolutely no sense to speak about |X| unless you've given a meaning to X.