Assume that for random variables $X$ and $Y$, for all real values $a$, we have
$E\{\max(X-a, 0)\}\leq E\{\max(Y-a, 0)\}$. Also assume
$$\forall t\geq 0 \;\;\; P(|Y|\geq t) \leq k e^{-bt}$$ for some $k\geq 1, b\geq 0$.
Prove that
$$\forall t\geq 0 \;\;\; P(|X|\geq t) \leq k e^{1-bt}$$
Assume that for random variables $X$ and $Y$, for all real values $a$, we have
$E\{\max(X-a, 0)\}\leq E\{\max(Y-a, 0)\}$. Also assume
$$\forall t\geq 0 \;\;\; P(|Y|\geq t) \leq k e^{-bt}$$ for some $k\geq 1, b\geq 0$.
Prove that
$$\forall t\geq 0 \;\;\; P(|X|\geq t) \leq k e^{1-bt}$$
Probability and Statistics
Probability and Statistics