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Can anyone please explain the logical reason behind how the joint PDF of a random sample of order statistics of size $n$ is $n!$ times the joint PDF of the random sample? I have derived it mathematically, but I fail to see the logic behind this.

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  • $\begingroup$ I have edited question, but am unable to write the notation because I don't hAve an appropriate keyboard having the required mathematical notations :( $\endgroup$
    – Sweta95
    Commented May 8, 2016 at 15:54
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    $\begingroup$ none of us do, that's why we use Mathjax for math typesetting. meta.math.stackexchange.com/questions/5020/… $\endgroup$
    – Sycorax
    Commented May 8, 2016 at 20:49

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Here is first round answer:

The order statistic orders the elements in the sample. Think of this as n slots that are filled by n units. Initially, any one of the n units may be placed in the first slot. Next, n-1 units may be placed in the second slot.

There are n*(n-1) possible combinations for these two slots to be filled.

The third slot may be filled by n-2 observational units, and so on.

Thus, there are n! possible orderings for the n units.

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