I have $2^m$ independent random variables. All have binomial distributions, each with $m$ samples. The probability of success for each binomial distribution is somewhere in the range $[0,p]$ (therefore, their expectation is in the range $[0,pm]$).
When I sample from the random variables, I only know the total number of variables for whom this sample is over $(p+\epsilon)m$.
Let $V_i$ be the number of binomial variables whose outcome in sample i is over $(p+\epsilon)m$
Let $W$ be the random variable indicating the number of binomial variables whose outcome is over $(p+2\cdot \epsilon)m$ in some sample.
I'm never going to be able to figure out exactly which are all the expectations of the binomial variables (even approximating them is impossible using only samples from $V_i$).
Luckily I only want to know the expectation and variance of $W$, up to an error of $m$. So my question is, is this possible using only samples from $V_i$, and if so, how many samples will I need?