(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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})

2. Relevant equations

Order Statistics

3. The attempt at a solution

As shown above.

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

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# Homework Help: Probability & Statistics: Order Statistics

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