This question is motivated by my previous post on binomial distribution.
Question: Given two independent random variables $X\sim B(n_1,p)$ and $Y\sim B(n_2,p)$ where $n_1<n_2,$ is it true that there exists a random variable $Z\sim (n_2-n_1,p)$ such that $Z$ is independent from $X$?
I know that one can simply take $$Z = Y-X$$ and $Z$ is binomially distributed with $n = n_2-n_1$ and probability of success $p.$ But I think $Z$ is not independent from $X.$