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Let $a_1, a_2, a_3, \ldots, a_{10}$ be ten randomly chosen real numbers in the interval [0,1].

Let $m$ be the maximal value out of these 10 numbers.

What is the expected value of $m$?

(i.e. If i had $n$ sets of 10 numbers and took the maximum $m$ for each set and averaged the values of all the $m$'s, what would this value become as $n \rightarrow \infty$?

John Smith
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  • While true that doesn't really matter here @Masacroso. There exists a distribution of the maximal values of that sample. – Zaros May 20 '16 at 07:30
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    If $a_\imath$ are independent, uniformly distributed (are they?) then the answer is $10 \over 11$, see http://math.stackexchange.com/questions/150586/expected-value-of-max-of-iid-variables . – Abstraction May 20 '16 at 07:30
  • Note the usual confusion, addressed in the comment above: "randomly chosen" does not really mean anything mathematically, what you mean seems to be "chosen independently and uniformly at random." (A probability distribution that puts mass $1$ on ${1}$ is still a valid probability distribution over $[0,1]$, but choosing 10 numbers "randomly" from it will give you $1$ ten times, almost surely.) – Clement C. May 20 '16 at 07:37
  • @ClementC. All true, and it is good to encourage precision, but isn't the uniform distribution the obvious one to use if none is mentioned? – almagest May 20 '16 at 07:49
  • It is -- but between the independence and the uniformity, it's always good to be precise. – Clement C. May 20 '16 at 08:13

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