Let $X=(X\_1,\ldots,X\_n)$ be an iid sequence of random variables, and let $\nu$ be a *uniformly random* integer in the range $1,\ldots,n$. Then $\xi\_\nu$ is a random entry of $X$. Is it always true that after deleting such a random entry from $X$, the remaining sequence is still iid?


Due to the uniform randomness of the index $\nu$, it is tempting to answer that the leftover sequence always remains iid. This is correct, if we also assume that $\nu$ is chosen *independently* of $X$.


What happens, however, if no such independence is assumed? For example, consider a 0-1 valued iid sequence. Remove a random 1, chosen uniformly at random among all 1s. If there is no 1, then remove a random 0. In this situation, the position of the removed entry is still uniformly random, but not independent of the original sequence. Can such a removal destroy the iid property in the remaining sequence?



Can deleting a random entry from an iid sequence destroy the iid property?
The independence will be then in general lost. E.g., let $X\_1,\dots,X\_n$ be independent random variables each uniformly distributed on $[0,1]$. Let $M:=\max(X\_1,\dots,X\_n)=X\_\nu$, so that $\nu$ is uniformly distributed on $[n]:=\{1,\dots,n\}$. Let $(Y\_1,\dots,Y\_{n-1})$ be the leftover sequence, after the removal of $X\_\nu$. Then, conditionally on $M$, the $Y\_i$'s are iid uniformly distributed on $[0,M]$.


So, for $n\ge2$ and $i\in[2]$ we have
$E(Y\_i|M)=M/2$ and $E(Y\_1Y\_2|M)=E(Y\_1|M)E(Y\_2|M)=(M/2)^2$. So,
$$EY\_1Y\_2=E(M/2)^2>(EM/2)^2=EY\_1\,EY\_2,$$
with the inequality taking place because $Var\,M>0$.
Thus, $Y\_1$ and $Y\_2$ are not independent.




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Intuitively, it can be expected that the $Y\_i$'s are positively dependent. Indeed, if $M$ is small, then all $Y\_i$'s will be small. So, if $Y\_1$ turns out to be small, a reason for that may be that $M$ is small, and then $Y\_2$ will be small. Thus, the smallness of $Y\_1$ seems to make $Y\_2$ tend to be small.


