Bound for expectation of non-negative random variable truncated to large values, with respect to, probability of truncated region.

Question

For $X$ a non-negative random variable, and can we get an upper bound to $E [X~ 1 (X\geq \alpha) ]$ in terms of $P(X\geq \alpha)$ as $\alpha \uparrow \infty$ ?

Answer 1

If you are asking if there exists a constant $C$ such that $EX I_{\{X \geq \alpha \}} \leq C P\{X\geq \alpha\}$ for all $\alpha$ (sufficiently large) the answer is n0. In fact $EX I_{\{X \geq \alpha \}} \geq \alpha P\{X\geq \alpha\}$ so such an inequality would imply $\alpha P\{X\geq \alpha\} \leq C P\{X\geq \alpha\}$ which implies $\alpha \leq C$ if we choose $X$ such that $P\{X\geq \alpha\} \neq 0$ for any $\alpha$. This is a contradiction.