# Distribution of the sample mean of a binomial distribution with prior information

Let's say that there is a finite population of $N$ elements which are either $A$ or $B$ (equivalently this could be a binomial random variable with $N$ repetitions and unknown $P(A)=\theta$).

If I sample $N_0$ elements of the population ($N_0 < N$) with no repetition, the sample mean of the number of elements that are $A$ is denoted by $\hat{\theta}_{A,N_0}$. From basic statistics, $\hat{\theta}_{A,N_0}$ is normally distributed with mean $\theta$ and variance $\sigma_{N_0}=\sqrt{\frac{\theta(1-\theta)}{N_0}}$. However, since $\theta$ is unknown, the sample variance is used $\hat\sigma_{N_0}=\sqrt{\frac{\hat\theta_{A,N_0}(1-\hat\theta_{A,N_0})}{N_0}}$ instead of $\sigma_{N_0}$

Given $\hat{\theta}_{A,N_0}$, $\hat\sigma_{N_0}$, and $N_0$: I want to compute the sample mean distribution of the number elements which are $A$ in the rest of the population (those elements that have not been sampled). That is the distribution of $\hat\theta_{A,\bar{N}_0}$, where $\bar{N}_0=N-N_0$ elements.

As with $\hat{\theta}_{A,N_0}$, I know that $\hat{\theta}_{A,\bar{N}_0}$ is normally distributed with mean $\theta$ and variance $\sigma_{\hat{N}_0}$.

I can get a confidence interval for $\theta$, that is, $\theta \in \hat{\theta}_{A,N_0} \pm \delta$, using $\hat{\theta}_{A,N_0}$, $\hat\sigma_{N_0}$, and the fact that $\hat{\theta}_{A,N_0}$ is the realization of a normal random variable.

Finally, I have that $\hat{\theta}_{A,\bar{N}_0}$ is normally distributed, with mean in the interval $\hat{\theta}_{A,N_0} \pm \delta$ and variance $\sigma_{\bar{N}_0}$ (which since depends on the mean is an interval as well).

My questions are: 1) is my rationale ok?, and 2) can I get more information about the distribution of $\hat{\theta}_{A,\bar{N}_0}$ ?