$h(\cdot)$ denotes a strict monotonic increasing transformation such as $\log$.
Another inequality I do not quite get is that
$$\mathsf{P}\left(h(X) \le h(x)\right) \ge \mathsf{P}\left(X \le h(x)\right)$$
Some help would be very much appreciated!
$h(\cdot)$ denotes a strict monotonic increasing transformation such as $\log$.
Another inequality I do not quite get is that
$$\mathsf{P}\left(h(X) \le h(x)\right) \ge \mathsf{P}\left(X \le h(x)\right)$$
Some help would be very much appreciated!
No name that I know for the property in the title, which is a simple consequence of the identity, valid for any strictly increasing function $h$, $$ \{\omega\in\Omega\mid h(X(\omega))\leqslant h(x)\}=\{\omega\in\Omega\mid X(\omega)\leqslant x\}. $$ Note: The inequality in the body cannot be true in general.