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## Homework Statement

Q1) About "order statistics", sometimes it's denoted x

_{(1)}and sometimes it's denoted X

_{(1)}. What is the difference between the two?

Also, for X

_{(1)}=min{X

_{1},X

_{2},...,X

_{n}}, it's a random variable. What does it mean to be the minimum of a bunch of random variables? If they are SPECIFIC observed values, then we can order them (e.g. if we have 6,3,8,7, then ordering them gives 3,6,7,8)...that I understand. But if they are random variables, HOW can we order them?

Q2) (more about order statistics)

http://www.geocities.com/asdfasdf23135/stat7.JPG

Here we have n random variables X

_{1},X

_{2}...,X

_{n}and we see F

_{X}(x) here. Why can we label it just based on one single varaible "x" instead of x

_{1},x

_{2},...,x

_{n}? Don't we have to treat them separately as x

_{1},x

_{2},...x

_{n}instead of just one "x"? Well, you may say it is because they're identically distributed, so we can just use a single "x" to represent each of x

_{1},x

_{2},...x

_{n}. But consider the following case:

Let X

_{1},X

_{2},...,X

_{n}be iid random variables with density f(x

_{i})=x

_{i}, 0<x

_{i}<sqrt2, then in this case the joint density must be f(x

_{1},x

_{2},...,x

_{n})=x

_{1}x

_{2}...x

_{n}, and is definitely NOT (x

_{1})

^{n}

So we've seen two different situations. In the first case, we can say x=x

_{1}=x

_{2}=...=x

_{n}, but not so in the second case. What is going on? Can someone please explain? I am always confused between these two cases. I am confused whenever they say X

_{1},...X

_{n}are iid with COMMON density f

_{X}(x). If this is the case, then the JOINT density [f

_{X}(x)]

^{n}would be a function of only a single variable "x" which doesn't make any sense to me (the joint density should be a function of n variables x

_{1},x

_{2},...,x

_{n})

## Homework Equations

Order Statistics

## The Attempt at a Solution

As shown above.

Thank you for clearing my doubts! I appreciate your great help!