## Thursday, December 12, 2019

Are you strong in math stats or probability?  I have a conjecture.  It is about order statistics.  The result is a bit surprising and I haven't figured out why it is so beautiful.

First,  an easy problem.  Suppose
$X_i \stackrel{iid}{\sim }\mbox{G},\; i=1,\ldots,I$
is a sequence of independent and identically distributed continuous random variables with some distribution $$G$$.  For example, $$G$$ could be a standard normal distribution.  And suppose the same for $$Y_i$$:
$Y_i \stackrel{iid}{\sim }\mbox{G},\; i=1,\ldots,I.$
Further, define $$A=\{i: \mbox{ such that } X_i<Y_i\}$$.  Let $$n$$ be the size of $$A$$, that is, n is the the number of times $$X_i<Y_i$$.  It should come as no surprise that $$n$$ is distribution as binomial with a probability of .5 and a size of $$I$$, and this holds regardless of $$G$$.  So, for example, if $$I=5$$, then the probability distribution of $$n$$ is:

That is the easy problem.  Here is the hard problem.  Suppose we do the same, but instead  if comparing the samples, we sort  them first so  $$X_1<X_2<\dots<X_I$$ and $$Y_1<Y_2<\ldots<Y_I$$.  Then, my contention is that $$n$$ is distributed as a discrete uniform with mass $$1/(I+1)$$ on points $$0,\ldots,I$$.  The following is some R code:

I=5
M=100000
n=1:M
for (m in 1:M){
X=sort(rnorm(I))
Y=sort(rnorm(I))
n[m]=sum(X<Y)}
p=table(n)/M

Here is the graph of the probability mass function:

Cool result.  I just can't prove it.  Any help?  Be a math-stat hero.  Thanks!

Thanks for posting this interesting problem!

If you are still interested, I posted a proof here

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