Let $\xi=(\xi\_1,\ldots,\xi\_n)$ be a sequence of independent random variables. 
Let us pick an index $\nu\in \{1,\ldots,n\}$, and replace the entry $\xi\_\nu$ by a constant $c$. The rest of the $\xi\_i$ remain unchanged. 


**Question:** Is it true that the altered sequence remains independent, even if $\nu$ is random, possibly dependent on $\xi$?


*Note:* If $\nu$ is a fixed constant index, then the independence is clearly preserved. It is less clear, however, what happens if $\nu$ is random and dependent on $\xi$. For example, we may pick the first hit to some set, or apply any other data-dependent rule to pick the index $\nu$. 



Is independence preserved if a random entry in an independent sequence is replaced by a constant?
The answer is no. E.g., assume that $n=2$, $\xi\_i=X\_i$, and $X\_1,X\_2,\nu$ are any random variables (r.v's) each with values in the set $\{1,2\}$ and with $P(\nu=1)=p\in(0,1)$. 
With these conditions in place, the dependence between $X\_1,X\_2,\nu$ may be arbitrary. For instance, we may suppose that $\nu$ is independent of $X\_1,X\_2$, or we may suppose that $\nu=X\_1$, or ... . 


Take any $c\notin\{1,2\}$ and let $Y\_1,Y\_2$ be obtained from $X\_1,X\_2$ by replacing $X\_\nu$ by $c$, so that  

\begin{equation}
 (Y\_1,Y\_2)=\left\{
 \begin{aligned}
 (c,X\_2) &\text{ if }\nu=1,\\
 (X\_1,c) &\text{ if }\nu=2. 
 \end{aligned}
 \right.
\end{equation}
Then 
$$P(Y\_1=c,Y\_2=c)=0\ne p(1-p)=P(\nu=1)P(\nu=2)=P(Y\_1=c)P(Y\_2=c). 
$$
So, $Y\_1,Y\_2$ are not independent. 


