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Draw N samples from unif(a,b) and consider the largest realization $X_{(N)}$. What's the probability that this value falls within a certain subset of (a, b)? What expression has to be integrated (what's the p.d.f. of $X_{(N)}$)?

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For ease of typing, let $Y$ be the maximum. Then $Y\le y$ if and only if all the $X_i$ are $\le y$.

For $y$ between $a$ and $b$, this is $$\left(\frac{y-a}{b-a}\right)^N.$$ This gives us the cumulative distribution function of $Y$. If we want to be fussy, the cdf $F_Y(y)$ is $0$ for $y\lt a$, is given by the above expression for $a\le y\le b$, and is $1$ for $y\gt b$.

The cumulative distribution function is what is most useful for computations. If, however, you want the density function, just differentiate.

Remark: The cumulative distribution function of the minimum is also given by a simple expression. Indeed one can find without too much trouble the distribution of the $k$-th order statistic for any $k$ between $1$ and $N$.

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Thanks! This is much simpler than I thought and makes intuitive sense. Except I have a little trouble grasping that the Nth realization is also "given" the probability weight between a and y. If I would have had to guess, I would have conditioned on this maximum value. –  Wuschelbeutel Kartoffelhuhn Sep 28 '12 at 6:00
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To add to what Andre said that is not part of the question. This answer can be generalized to other distributions. The exact cdf can be derived or at least expressed as an integral. Also there is asymptotic theory that tells you that there is a way to normalize the maximum so that it converges to having one of the three extreme value distributions in Gnedenko's theorem and the tail behavior of the population distribution determines the type. This is just a short description and not all distributions have an extreme value type as a limit. –  Michael Chernick Sep 28 '12 at 14:19
    
This is for iid samples but extends to some stationary sequences. –  Michael Chernick Sep 28 '12 at 14:21
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