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You have a lottery ticket with 10 slots. Behind each slot there's a equally distributed number between 0 and 1. Your payout is the maximum number between any of these slots. How much are you prepared to pay for the lottery ticket?

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What do you mean with maximum number between? $\max\{X_i\}$? Or $\max\{X_i-X_j\}? Can we make the additional assumption that the slot entries are independent? – Hagen von Eitzen Jul 29 '13 at 19:05

I assume you mean that the contents of the slots are $10$ independent random variables $X_1, X_2,\dots,X_{10}$, each of which has (continuous) uniform distribution on $[0,1]$.

Let $Y$ be the maximum of the $X_i$. Then $Y\le y$ if and only if all the $X_i$ are $\le y$. This has probability $y^{10}$. So (for $0\le y\le 1$) $Y$ has cumulative distribution function $y^{10}$.

Thus $Y$ has density function $10y^{9}$ on $[0,1]$ and $0$ elsewhere. Now find the expectation in the usual way.

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