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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.

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