I have random variable $Y=XU(X-t)$ (here $U(X)$ is the unit step function, $Y$ has non-negative support and depends on other random variable $X$) which has a CCDF $P(Y>t)$. I want to write the expected value of $Y$ in terms of CCDF of $Y$. I have seen in the literature that it can be written in following way but I do not know how to prove it. Can any body provide the proof if the following expression is right? $$E(Y)=\int_t^\infty P(Y>x)dx+tP(Y>t).$$
I can prove it for the case of $t=0$ but for general positive values of $t$ I do not know how to proceed. Any help in this regard will be much appreciated.