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December 13th, 2017, 12:59 PM  #1 
Senior Member Joined: Oct 2015 From: Antarctica Posts: 128 Thanks: 0  Interpretation of Limiting Distribution
Let $X_1, ..., X_n$ be random variables independent and identically distributed on $Uniform(0,1)$. Let $X_{(n)}=MAX{(X_1,...,X_2)}$. Define $W_n=n(1X_{(n)})$. Find the limiting distribution of $W_n$ as $n$ increases without bound. Can you identify this limiting distribution? Give an interpretation of the result obtained above. So, I managed to find that the limiting distribution follows an exponential distribution with mean 1. However, I'm not quite sure how to "interpret" this. What's so special about $W_n$ that makes this result significant when $n$ grows large? 
December 13th, 2017, 03:56 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,124 Thanks: 1103 
I get that $f_{W_{(n)}}(w) = \left(1  \dfrac w n \right)^{n1},~w \in [0,n]$ $\lim \limits_{n\to\infty}~f_{W_{(n)}}(w) = e^{w},~w \in [0,\infty)$ 
December 13th, 2017, 04:18 PM  #3 
Senior Member Joined: Oct 2015 From: Antarctica Posts: 128 Thanks: 0 
Right, that's exactly what I found; it's exponential with mean 1. But the real question here is: how do you "interpret" that? 

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distribution, interpretation, limiting 
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