where is .5?

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A Riddler’s riddle on breaking the unit interval into 4 random bits (by which I understand picking 3 Uniform realisations and ordering them) and finding the length of the bit containing ½ (sparing you the chore of converting inches and feet into decimals). The result can be found by direct integration since the ordered Uniform variates are Beta’s, and so are their consecutive differences, leading to an average length of 15/32. Or by raw R simulation:

simz=t(apply(matrix(runif(3*1e5),ncol=3),1,sort))
mean((simz[,1]>.5)*simz[,1]+
  (simz[,1]<.5)*(simz[,2]>.5)*(simz[,2]-simz[,1])+
  (simz[,2]<.5)*(simz[,3]>.5)*(simz[,3]-simz[,2])+
  (simz[,3]<.5)*(1-simz[,3]))

Which can be reproduced for other values than ½, showing that ½ is the value leading to the largest expected length. I wonder if there is a faster way to reach this nice 15/32.

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This article is originally published at https://xianblog.wordpress.com
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