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Given $N$ random iid variables, $X_1, \ldots, X_N$, with a uniform probability distribution on $[0, 1)$ what is the distribution of $\displaystyle \max_{i = 1 \ldots N}(X_i)$?

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Shai Covo's post below is an easier way to the right answer. But I wanted to mention that this is a special case of an order statistic. – Rahul Nov 25 '10 at 20:03
Order statistics is better than what I need. – Alexandru Nov 25 '10 at 20:32
up vote 12 down vote accepted

Hint: $\max \lbrace X_1,...,X_N \rbrace \leq x$ if and only if $X_i \leq x$ for all $i=1,...,N$.

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