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!

## 6 comments:

Thanks for posting this interesting problem!

If you are still interested, I posted a proof here

https://mossadnik.github.io/2020/01/10/sorted-random-variables.html

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