(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!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Probability & Statistics: Order Statistics

**Physics Forums | Science Articles, Homework Help, Discussion**