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A parameter of an object is normally distributed with a mean m and a std. dev. s. If r such objects are randomly selected, what is the maximum expected value M of the parameter in the selection?

Cross-posted from: https://mathoverflow.net/questions/105903/what-is-the-maximum-expected-value-in-a-finite-selection-an-object-with-normal-di

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Shashank Sawant
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  • The maximum expected value or the expected value of the maximum? (This question is a bit rhetorical; answers can be given for both, but one is more interesting than the other.) Also, I recall a recent duplicate or near-duplicate; it would be worth doing a quick site search. – cardinal Aug 31 '12 at 18:22
  • I wanted the expected value of the maximum, but I don't see it to be different from the maximum expected value. Can you explain my error? – Shashank Sawant Aug 31 '12 at 18:39
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    Note that for large $n$, the asymptotic theory result was given here: http://stats.stackexchange.com/q/34418/5739 – StasK Aug 31 '12 at 18:50
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    The expected value is m, so there is no point in asking about its maximum. – StasK Aug 31 '12 at 21:34
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    @StasK: (+1) There is an interesting and somewhat subtle feature to note here. Since convergence in distribution does not imply that the means converge to that of the limit distribution, we may wonder whether that holds in this particular case, especially since we are dealing with maxima, where the prospect of mass "escaping to infinity" would seem to be more plausible. In fact, the means do converge, as proven by Pickands (1968). – cardinal Sep 01 '12 at 22:15
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    @caardinal, that's an interesting note. – StasK Sep 01 '12 at 23:14

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Let $\Phi(x) = P[X\leq x]$ when X is normal with mean m and standard deviation s. Then

$$ \Phi^n(x)= P[\max (X_1,X_2,..., X_n) \leq x]. $$

So $E[\max (X_1,X_2,..., X_n)] = \int_{-\infty}^{\infty} x n \Phi^{n-1}(x) \Phi^{\prime}(x)dx$.

Michael R. Chernick
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