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I recently learned that to find the pdf of the median of say $X_1,X_2, X_3$, you first find the Cdf via $$ P(M \le x) =P(\text{at least 2 are}\, \le x) = P( \text{exactly 2 are}\, \le x) + P(\text{all 3 are} \le x)$$ where $M$ denotes the median. Finaly you differentiate to get the required pdf.

My questions:

  1. How does one find the cdf/pdf of an arbitrary number of order statistics?
  2. Is there a generalized formula?

Thanks

Edit:

Let's suppose the $X_i$'s are iid.

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    Michael Hardy's answer hints at the combinatoric types of things you want to be doing to answer this. For (say) iid $X_i$ you can get the joint distribution of any number of order statistics. [Here are some notes I found online that might be useful.](http://almaak.usc.edu/~larry/courses/Ma541a/s05/order-stat.pdf]) It derives the joint distribution of all the order statistics, and the marginal of any single order statistic. One can get formulas the joint distribution of any subvector of the order statistics in terms of the pdf $f$ and cdf $F$ in the iid case, but I don't think it has all that.2012-07-24
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    @guy: The link you gave me is broken ;)2012-07-24
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    The site is fine, I just accidentally put an extra bracket in at the end :). [I fixed it!](http://almaak.usc.edu/~larry/courses/Ma541a/s05/order-stat.pdf) It actually has more material than I thought; it addresses the general case, exchangeable case, and finally the iid case, and gets joint distributions.2012-07-24
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    @guy Thanks. ${}{}{}{}{}{}$2012-07-24

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