I know that for a nonnegative random variable X, $E[X]=\int_{0}^\infty P(X>x) dx$.
However, I want to calculate the expectation of $E[X\cdot\mathbb 1\{X>b\}]$ which is equal to $E[X]-E[X\cdot\mathbb 1\{X\leq b\}]$, with the complementary CDF of $X$. I expect it to be $E[X\cdot\mathbb 1\{X>b\}]=\int_{b}^{\infty}P(X>x)dx$. But I am not sure about this.
