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Consider $n$ independent uniform random variables $X_i \sim U(-\theta,\theta)$, and let $Y_1 = \min(X_1, \ldots, X_n)$ and $Y_n = \max(X_1, \ldots, X_n)$ .

What is distribution of $Z = \max (-Y_1,Y_n)$, given that the joint PDF of $Y_1$ and $Y_n$ is $$ f(y_1, y_n) = \frac{n(n-1)}{(2θ)^n} (y_n-y_1)^{n-2} , \quad -\theta < y_1 < y_n < \theta ? $$

mpiktas
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user22094
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    Please double-check your equation for the joint PDF: what are you doing subtracting $\theta$ from it? – whuber Mar 16 '13 at 22:39
  • This looks like standard bookwork. If this is homework, or other work for some subject or just for the purpose of self study, you should mark it with the [`self-study`](http://stats.stackexchange.com/tags/self-study/info) tag. (Click the tag in this comment for details, and check the information relating to homework in the [faq](http://stats.stackexchange.com/faq).) – Glen_b Mar 17 '13 at 00:41
  • possible duplicate of [Likelihood Ratio of two-sample Uniform Distribution](http://stats.stackexchange.com/questions/81225/likelihood-ratio-of-two-sample-uniform-distribution) – Alecos Papadopoulos Jan 30 '14 at 00:35
  • The easy way to do this would be to look at $|Y|$ so you're back to a simple max problem $Z=max |Y_i|$. – Glen_b Jan 30 '14 at 01:34
  • Nice problem. _Prima facie_, it _seems_ attractive to treat it, by symmetry, as the max of two separate univariate problems, or equivalently, as the pdf of the sample maximum given the parent $Uniform(-\theta, \theta)$ with sample size $2n$... but that will yield an incorrect solution, because it misses the crucial interdependency. And the interdependency is that the domain of support for $Z = max(-Y_1,Y_n)$ is not $(-\theta, \theta)$, but rather $(0,\theta)$, because (i) if $y_n > 0$, then $Z > 0$, and (ii) if $y_n < 0$, then $y_1 < 0$ ---> $Z>0$. The pdf of $Z$ is a Power Function. – wolfies Jan 30 '14 at 16:07
  • First find the CDF of $Z$ as is done in [this](https://stats.stackexchange.com/a/164656/119261) answer. – StubbornAtom Nov 19 '19 at 19:03

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