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Let $M_{n} = \max(U_{1}... , U_{n})$ be the maximum of a sample size $n$ from $U(0 , 1)$ distribution.

In my statistics textbook it says that $M_{n}$ normalized is equal to $n(1 - M_{n})$ but I'm not sure how they got to this?

I know that any order statistic derived from a $U(0 , 1)$ distribution follows a Beta distribution, but I'm not sure if this will help me get to the answer.

kjetil b halvorsen
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Daniel De Wet
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  • What does 'Normalization of $M_n$' mean? Have you left out any details? – StubbornAtom Oct 10 '21 at 15:43
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    I think it means subtracting by the mean and dividing by the standard deviation. – Daniel De Wet Oct 10 '21 at 15:47
  • Do you know which Beta distribution corresponds to that of $M_n$? If so, what is its mean and standard deviation? – Henry Oct 10 '21 at 16:41
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    $n(1-M_n)$ is an appropriate norming and scaling of $M_n$ so that it converges to a standard (non-degenerate) distribution. Maybe that is the context which you do not mention. – StubbornAtom Oct 10 '21 at 16:48
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    "Normalized" likely means in the sense of the FTG theorem, as I explain at https://stats.stackexchange.com/a/153067/919. The techniques shown there apply to the present situation (and the analysis is much simpler). – whuber Oct 10 '21 at 17:44

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