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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

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