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Let $X_1,...,X_n$ be a random sample from a $\mathcal{N}(\mu,1)$ distribution. Only the largest observation $Y = \max(X_1,...,X_n)$ is reported.

What is the density of $Y$? How do I get there?

Michael
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    If this is for some course, or for your own study, could you add the [self-study](http://stats.stackexchange.com/tags/self-study/info) tag please? – Glen_b Sep 30 '13 at 07:08
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    $P(Y\leq y) = P(X_1 \leq y).P(X_2 \leq y)\ldots P(X_n \leq y)$. Can you do it now? – Glen_b Sep 30 '13 at 07:17
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    This is a duplicate (at the very least, in spirit, if not the exact problem) of *several* questions on this site. There are bound to be duplicates on math.SE as well. – cardinal Sep 30 '13 at 11:59
  • Thanks for the comments so far! So the upper suggestion leads to $P(Y\leq y) = (\Phi(y-\mu))^{n-1}$, right (for normal distributions)? I also know the solution which is $f_Y(y) = n((\Phi(y-\mu))^{n-1} \phi(y-\mu))$. But the way to the solution is still not completely clear. To my background: I am a biologist taking an advanced stat class, so be nice! :-) – Michael Sep 30 '13 at 15:37
  • Michael, you may want to take a look at [this answer](http://stats.stackexchange.com/a/32353/21054) from Macro. The question is basically the same as yours, just for a Uniform distribution. More specifically: I think the exponent in your first expression should be $n$ instead of $n-1$. And the solution you give is the *density function* ($f_{Y}$), not the CDF ($F_{Y}$). But the PDF can - in this case - be obtained by differentiation: $f_{Y}(y)=\frac{d}{dy}F_{Y}(y)$. – COOLSerdash Sep 30 '13 at 16:48
  • Although the duplicate explicitly discusses the distribution of a *minimum,* its solution applies directly to the maximum (by replacing each $X_i$ by $-X_i$, which merely changes $\mu$ to $-\mu$). Although the duplicate is a low-voted question, I selected it among many possible duplicates because its answer responds to the issues raised above in comments: namely, how to obtain the PDF from the CDF. – whuber Sep 30 '13 at 20:49
  • OK, I understood how to get to the solution and why I works like this, thanks to all! How can I mark this thread as solved without an answer? – Michael Sep 30 '13 at 21:22
  • Michael, you do not need to mark it anymore, because it has been linked to another question as a duplicate--but thank you for your attention to that matter! I am glad you found our comments helpful. – whuber Sep 30 '13 at 21:45

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