Definition: Consider a set of random variables X(adsbygoogle = window.adsbygoogle || []).push({}); _{1},X_{2},...,X_{n}, the "order statistics" are the same random variables arranged in increasing (actually nondecreasing) order, i.e. X_{(1)}<X_{(2)}<...<X_{(n)}

After reading a few more paragraphs, I immediately ran into deep confusion because I cannot make any sense of it...

My confusions:

1. X_{1},X_{2},...,X_{n}are random VARIABLES (not constants), so you don't know exactly what the outcome will be. For example the set of possible values of each X_{i}may be the interval [0,1]. How can you possibly order something that you can't even tell the exact value? (this is like saying [0,1] is less than [0,1]) This makes no intuitive sense to me...

2. "Distributions of X_{i}'s and X_{(i)}'s are NOT the same."

How can this be true? They are just the SAME random variables ordered in a different way, how can they distributions possibly be different? I can't make sense of this either...

3. "If the random variables X_{1},X_{2},...,X_{n}are continuous, the equality signs in X_{(1)}<X_{(2)}<...<X_{(n)}can be ignored."

Now...I can't see WHY?

Could someone please explain? I have read the related material in 2 different textbooks, but they are saying pretty much the same thing and do not clear my doubts at all.

Any help would be appreciated!

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# Order Statistics

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