1) 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{X1,X2,...,Xn}, 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?
2) (more about order statistics)
Here we have n random variables X1,X2...,Xn and we see Fx(x) here. Why can we label it just based on one single varaible "x" instead of x1,x2,...,xn? Don't we have to treat them separately as x1,x2,...xn 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 x1,x2,...xn. But consider the following case:
Let X1,X2,...,Xn 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(x1,x2,...,xn)=x1x2...xn, and is definitely NOT (x1)^n
So we've seen two different situations. In the first case, we can say x=x1=x2=...=xn, 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 X1,...Xn are iid with COMMON density fX(x). If this is the case, then the JOINT density would be only a function of a single variable "x" which doesn't make any sense to me (the joint density should be a function of n variables x1,x2,...,xn)
Thank you for clearing my doubts!