Absolute value of a random variable

Question

I have never encountered this concept before. Is this equation valid for $y>0$?

$$\mathbb{P}(|X|>y) = \mathbb{P}(-|X|<y<|X|)$$

What about this?

$$\mathbb{P}(|X|>y) = \mathbb{P}(X>y) + \mathbb{P}(X<-y)$$

Answer 1

I have never encountered this concept before. Is this equation valid for $y>0$?

$$\mathbb{P}(|X|>y) = \mathbb{P}(-|X|<y<|X|)$$

Yes, for all strictly positive $y$, then $-\lvert X\rvert < y <\lvert X\rvert$ is an equivalent event to $y<\lvert X\rvert $.

However, whether this is helpful is another matter.   Does it simplify anything?

What about this?

$$\mathbb{P}(|X|>y) = \mathbb{P}(X>y) + \mathbb{P}(X<-y)$$

Yes the event that $\lvert X\rvert > y$ is equivalent to the union of the events that $X>y$ or $-X>y$, and because for strinctly positive $y$, these are disjoint events, then $$\mathbb{P}(|X|>y) ~=~ \mathbb{P}(X>y) + \mathbb{P}(-X>y)$$

This may be much more helpful.

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Note: the requirement that $y>0$ is necessary.   Neither holds otherwise.