I have some questions about computing the difference between two order statistics.
Given that for $i,j \in \{1,\cdots,\lambda\}$, denote Bin(s,p) to be a binomial random variable with success probability $p$ and $s$ independent experiments. Define $W_j=Bin(s_1,p)-Bin(n-s_1,p)$ for some $p \in (0,\frac{1}{2})$ and $Z_i=Bin(s_2,p)-Bin(n-s_2,p)$ for some $p \in (0,\frac{1}{2})$, where $n \in \mathbb{N}$.
All $W_j,Z_i$ are i.i.d random variables with $s_1<s_2$. Can we show that for arbitrary $\lambda \in \mathbb{N}$, there exists some $\epsilon >0$
$E(\min_{i\in \{1,\cdots,\lambda \}}Z_i)-E(\min_{j\in \{1,\cdots,\lambda \}}W_j)\geq \epsilon >0$ for $s_2>s_1$?
Or is my claim false? How do we see it?
Thank you for your time!