# an inverse permutation test

**R – Xi'an's Og**, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)

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**A** straightforward but probabilistic riddle this week in the Riddler, which is to find the expected order of integer i when the sequence {1,2,…,n} is partitioned at random into two sets, A and B, each of which is then sorted before both sets are merged. For instance, if {1,2,3,4} is divided in A={1,4} and B={2,3}, the order of 2 in {1,4,2,3} is 3. An R rendering of the experiment is

m=rbinom(1,n,.5)

if (m*(n-m)>0){

fist=sort(sample(1:n,m))

return(order(c(fist,sort((1:n)[-fist])))[i])

}else{

return(i)}

[\sourcecode]

It is rather easy to find that the probability that the order of i takes the value j is

if j*
*

if $j>i$ (i is in B), the case i=j being the addition of both cases, but the mean can be found (almost) immediately by considering that, when i is in A, its average value is (i+1)/2 and when it is in B, its average value is (n+i)/2 [by symmetry]. Hence a global mean of (2i+n+1)/4….

Filed under: Books, Kids, R, Statistics Tagged: order statistics, permutation tests, R, sorting, The Riddler

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**R – Xi'an's Og**.

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