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Is it correct to say that each value of a random variable (at least in the case of discrete random variables) corresponds to an event?

These corresponding events are mutually exclusive and their probabilities add up to 1.

I think these follow from the definition of a random variable as a function assigning a value to every element of the sample space. However, I could not find them in the textbooks, so I just want to ask here.

kjetil b halvorsen
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Sanyo Mn
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    Check: http://stats.stackexchange.com/a/54894/35989 – Tim Aug 24 '16 at 13:06
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    Everything you write is correct about any random variable $X$ (discrete or not) and real number $x$ provided "correspond to" is understood as the set $X^{-1}(x)$ (defined as $\{\omega\,|\,X(\omega)=x\}$) and "add up to" is understood as "integrate to" in the sense that $$1=\int_{\mathbb{R}}dF_X(x)=\int_{\mathbb{R}}d\mathbb{P}\left(X^{-1}((-\infty,x])\right).$$ It's not clear, though, that these are what you mean. – whuber Aug 24 '16 at 13:22

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