Obtaining confidence intervals from composite defined functions

Assume you have a set of $n$ independent random variables $X_1, X_2, \dots, X_n$ with unknown distribution and mean (finite) values $\mu_1, \mu_2, \dots, \mu_n \in \mathbb{R}$ . Moreover, there are $n$ known probabilities $p_1, p_2, \dots, p_n$ with $p_i > 0$ and $p_1 + p_2 + \dots + p_n = 1$ . Upon it, we construct a random variable $X$ defined as: $$X = \begin{cases} X_1 & \text{with probability }p_1\\ X_2 & \text{with probability }p_2\\ \dots\\ X_n & \text{with probability }p_n \end{cases}$$ So, basically speaking, $X$ (which also has an unknown finite mean $\mu$ ) takes the value from one random variable $X_1, X_2, ..., X_n$ with a certain probability. We can create i.i.d. $k$ samples $x_1, x_2, \dots, x_k$ from $X$ and can calculate a confidence interval for the mean $\mu$ , e.g., by using the Student's $t$ distribution. The question is now the following: Assume that we do not sample directly from $X$ , but take $k_1$ samples from $X_1$ , $k_2$ samples from $X_2$ , and so on. We can calculate for each individual random variable $X_1, X_2, \dots, X_n$ the confidence intervals for the means $\mu_1, \mu_2, \mu_3$ , but can we also compute for $X$ a confidence interval for the mean $\mu$ from these $k_1 + k_2 + \dots + k_n$ samples, as these are not i.i.d. samples for $X$ anymore? Example: We know beforehand that $n = 3$ and $p_1 = 0.2, p_2 = 0.4, p_3 = 0.4$ . Furthermore, we assume that each $X_i$ is Bernoulli distributed with an unknown mean $\mu_i$ , so it is either $0$…