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$h(\cdot)$ 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!

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    Which property do you want? The one in the title of your question or the one in the text of your question?2012-08-23

1 Answers 1

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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.

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    hm well ok than the statement just does not hold. thx alot!2012-08-23