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I have an exam in the morning and there is still one question I cannot do.

$X_1, \ldots, X_n$ are iid random variables each having distribution with density $f_{X_i}(x;\theta)= 1/\theta$, for $x \in [0,\theta]$ where $\theta>0$ compute the CDF of the random variable $\max(X_1,\ldots X_n)$ and prove that $n(\theta-\max(X_1,\ldots,X_n)) \to W$ in distribution and state the CDF of $W$.

How can I do this? I have worked out that the CDF of $\max(X_1,\ldots,X_n)$ is $ (x/\theta)^n$ but that is all :(

Thanks.

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    @EmreA What if X_i > c for more than one $i$?2012-06-13

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You've already worked out the CDF of $\max(X_1,...,X_n)$. Now just find the CDF of $n(\theta-\max(X_1,\ldots,X_n))$ (use the definition of CDF, it's not hard) and take the limit as $n\to\infty$ (use $\lim_{n\to\infty} (1+x/n)^n = e^x$).

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    Looks about right to me. Note that the your CDF for $\max(X_1,...,X_n)$ is valid for $0 \le x \le \theta$, which makes the later CDF valid for $0 \le x \le n\theta$. As $n \to \infty$ this range of validity extends to all of $[0,\infty)$.2012-06-12