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December 13th, 2017, 01:59 PM   #1
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Exclamation 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 $W_n=n(1-X_{(n)})$.
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?
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December 13th, 2017, 04:56 PM   #2
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I get that

$f_{W_{(n)}}(w) = \left(1 - \dfrac w n \right)^{n-1},~w \in [0,n]$

$\lim \limits_{n\to\infty}~f_{W_{(n)}}(w) = e^{-w},~w \in [0,\infty)$
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December 13th, 2017, 05:18 PM   #3
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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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