I found in an article a fact that is commonly used in scientific papers which mentions that:
$\mathbb{P}(f(r)>T) = \mathbb{E}[\mathbb{P}(f(r)>T\,|\,r)]$
($\mathbb{E}$ is with respect to $r$)
That means the CCDF of $f(r)$ is equal to the expectation ,with respect to $r$, of the probability of $f(r)$ greater than $T$ given $r$.
I tried to prove this fact but without any success. Can someone help me to prove that? Is it true?
Thanks.
By relying on the tower property of conditional expectation, and the fact that the expectation $\mathbb{E}[\mathbb{1}_{\{A\}}]$ of the indicator function is just the probability $\mathbb{P}(A)$ of the event on which the indicator function is defined, you have the following chain of equalities: $$ \mathbb{P}[f(r)>T]= \mathbb{E}\left[\mathbb{1}_{\{f(r)>T\}}\right]=\mathbb{E}\left[\mathbb{E}[\mathbb{1}_{\{f(r)>T\}}|\,r]\right]= \mathbb{E} \left[ \mathbb{P}(f(r)>T\,|\,r) \right].$$