# calculating confidence intervals for a weighted binomial distribution

We have n binomial distributions {$$b_i$$} - each with m trials, and a probability of success $$p_i$$ somewhere in the range [0,1]. Also, each binomial distribution $$b_i$$ is assigned some weight $$w_i$$, such that $$\sum (w_i) = 1$$. Let the distribution D be as follows - with probability $$w_i$$ we sample from distribution $$b_i$$.

In general, I will be interested to know how many samples I need to estimate all $$w_i$$ and $$p_i$$.

Also assuming $$\exists \bar{i}$$ s.t. $$w_\bar{i} > \frac{1}{2^n}$$ and $$\forall i \neq \bar{i}$$ $$p_\bar{i} > p_i +\frac{2}{n}$$, Is there a way, with fewer samples, to estimate $$p_\tilde{i}$$?