Problem
This question arises from a paper I am reading now. The original reasoning could be translated into following
$$ \Pr[X>\alpha+\beta+t] \leq f(t) \Rightarrow \mathbb{E}[X] \leq \alpha+\beta + \int_{t}f(t)dt $$ where $x$ is a random variable and $\alpha, \beta$ are constants.
It seems that the author tries to do integration on both sides and somehow remove the probability sign and replace it with expectation. But to my understanding, I need pdf of $X$ to compute its expectation and $\Pr[X>\alpha+\beta+t] \leq f(t)$ is not enough.
So could someone help me with the reasoning the author uses here?