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I'm having trouble with the theoretical definition of the p-value. Let $T$ be a test statistic with CDF $F(T)$ and supose the null hypothesis is true. If we were to reject the null hypothesis for small values of $T$ then the p-value is defined by $P:=F(T)$. But if we were to reject the null hypothesis for large values of $T$ then the p-value is defined as $P := 1-F(T)$.

What is a proof for this assertion and why is it defined as a random variable? I already know that $F(T)$ follow a Unif(0,1) distribution.

Thank you.

hugdelcur96
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    No theorem; no proof. Just a definition. But there is plenty of intuition behind the definition, and I think that's what you are looking for. Am I correct? – Peter Leopold May 29 '19 at 01:16
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    It's not *always* the case that a p-value is associated with one direction or another of the test statistic (though many common tests work this way). It's possible to construct tests where your rejection region is two or more non-contiguous regions and then p-values won't be of either of those forms. – Glen_b May 29 '19 at 03:57
  • See the post at https://stats.stackexchange.com/a/130772/919 for an example of what glen_b is writing about. – whuber May 29 '19 at 12:08
  • Thank you guys. I already understood the concept. – hugdelcur96 May 30 '19 at 18:20

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