I would like to upper bound the expectation $$ \mathbb{E}[X \, \textbf{1}\{X > t\}], $$ where $\textbf{I}\{p\}$ evaluates to $1$ if $p$ is true, $0$ otherwise, and $X$ is some non-negative random variable which may be assumed to be sub-Gaussian.
The discrete case is easy, but I'm having trouble with the continuous case. Is anyone familiar with existing results or useful inequalities for this?
edit: assume also that $\mathbb{E}[X] = \mu < \infty$
The obvious estimate $\mathbb E[X \; 1\{X>t\}] \le \mathbb E[X]$ is best-possible in the sense that it is an equality if $\mathbb P(0 < X < t) = 0$.
Somewhat more generally, if $\mathbb E[X^p] < \infty$ with $p \ge 1$ you have $$\mathbb E[X \; 1\{X > t\}] \le t^{1-p} \mathbb E[X^p]$$
EDIT: And, if the moment generating function $M(s) = \mathbb E[e^{sX}] < \infty$ for some $s > 0$, this implies $$\mathbb E[X \; 1\{X > t\}] \le \dfrac{t}{e^{st}-1} (M(s) - 1)$$ so this gives you exponential decay.