# Finite-sample confidence interval for the sample mean of N iid Beta-Binomial random variables

Let $$m,n,l,N$$ be 3 integers, and $$C_1,\dots,C_N$$ i.i.d. Beta-Binomial RV with the following distribution:

$$C_i\sim\frac{1}{m}\text{Binom}(m,M)$$ where $$M\sim\text{Beta}(n+1-l,l)$$

The sample mean is

$$\bar{C}=\frac1N\sum_{i=1}^N C_i$$

I would like to compute a valid $$(1-\alpha)-$$confidence interval for the expectation of $$\bar{C}$$, i.e., confidence interval that has coverage at least $$1-\alpha$$, for finite $$N$$ (sample size). Is it possible to do that? Note that $$0\leq C_i\leq1\ \forall i$$, and thus the same is true of $$\bar{C}$$, as well as of its expectation.