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What is $EY$, if $Y=max(X_{1},X_{2},...,X_{n})$ where $X_{i}$ are observations from uniform distribution over set $(0,a)$, $EY$ goes to $a$ as $n$ goes to infinity ?

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Sounds like homework. If it falls within the scope of the 'homework' tag (which is not limited to formally set homework!), could you tag it as such please?

Presumably you intend the $X$'s to be independent?

Start by taking the $X$'s on $(0,1)$ - for which results are even simpler to calculate from first principles and are also readily available for checking your answers against - and multiply the answers by $a$.

The cdf of $Y$, $P(Y\leq y) = P(X_1 \leq y, X_2 \leq y, ... , X_n \leq y)$

It should be obvious from there.

You can check your answer against this:

http://en.wikipedia.org/wiki/Order_statistic#The_order_statistics_of_the_uniform_distribution

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