Im trying to understand some definitions of the setup in two-sided hypothesis-testing. The probability of observing extreme values when doing two-tailed hypothesis-testing can be evaluated (if I'm not wrong) in the following integral (where $f_t$ refers to the $t$-distribution)
$P(t > |t_0|) \cup P(t < -|t_0|) = \int_{-\infty}^{-|t_0|} f_t(x)dx + \int_{|t_0|}^{\infty} f_t(x)dx - \int_{-|t_0|}^{|t_0|} f_t(x)dx$
In order to simplify to
$P(t > |t_0|) + P(t < -|t_0|) = \int_{-\infty}^{-|t_0|} f_t(x)dx + \int_{|t_0|}^{\infty} f_t(x)dx = $
$P(|t| > |t_0|) = 2\int_{|t_0|}^{\infty} f_t(x)dx$
i think I'd need some kind of confirmation that the areas $P(t > |t_0|)$ and $P(t < -|t_0|)$ are independent, but I'm unsure how to. I couldn't find any claims that the fractiles can go to negative numbers when freedom degrees $df \to \infty$, which would then contradict to the fact that they are independent for all confidence-levels.
I hope I make sense, any answers appreciated.
The events $t > |t_0|$ and $t < -|t_0|$ are not independent, but they are mutually exclusive since $-|t_0| \le |t_0|$ for all real $t_0$
Luckily this is what you want if you are going to add their probabilities together, since their intersection has probability $0$