Let $X_1, X_2$ be independent (not necessarily identically distributed) random variables. Assume we have estimates for the means $m_1, m_2$ and confidence internals $\left[a_1, b_1\right],\left\{a_2, b_2\right]$ i.e. $\mathbb{E}\left[X_1\right]=m_1, \mathbb{E}\left[X_2\right]=m_2, \mathbb{P}\left(X_1 \in\left[a_1, b_1\right]\right) \geq 95 \%$ $\mathbb{P}\left(X_2 \in\left[a_2, b_2\right]\right) \geq 95 \%$. We are interested in the sum $Y:=X_1+X_2$. We know that the mean behaves nicely i. e. $\mathbb{E}[Y]=m_1+m_2$. What would be the best way to derive confidence intervals for $Y$ by computational means? What would be the statistical assumptions for each approach be?

On a side note. In my setting, I have many $X_1, \ldots, X_n$. The confidence intervals are symmetric around the mean.

I have thought about assuming gaussian $X_1, \ldots, X_n$ and then reverse-engineering the standard deviation for bootstrap or simply sum of standard deviation or something like that, but did not make it work.

Let $X_1, X_2$ be independent (not necessarily identically distributed) random variables. Assume we have estimates for the means $m_1, m_2$ and confidence internals $\left[a_1, b_1\right],\left[a_2, b_2\right]$ i.e. $\mathbb{E}\left[X_1\right]=m_1, \mathbb{E}\left[X_2\right]=m_2, \mathbb{P}\left(X_1 \in\left[a_1, b_1\right]\right) \geq 95 \%$ $\mathbb{P}\left(X_2 \in\left[a_2, b_2\right]\right) \geq 95 \%$. We are interested in the sum $Y:=X_1+X_2$. We know that the mean behaves nicely i. e. $\mathbb{E}[Y]=m_1+m_2$. What would be the best way to derive confidence intervals for $Y$ by computational means? What would be the statistical assumptions for each approach be?

On a side note. In my setting, I have many $X_1, \ldots, X_n$. The confidence intervals are symmetric around the mean.

I have thought about assuming gaussian $X_1, \ldots, X_n$ and then reverse-engineering the standard deviation for bootstrap or simply sum of standard deviation or something like that, but did not make it work.

Let $X_1, X_2$ be independent (not necessarily identically distributed) random variables. Assume we have estimates for the means $m_1, m_2$ and confidence internals $\left[a_1, b_1\right],\left\{a_2, b_2\right]$ i.e. $\mathbb{E}\left[X_1\right]=m_1, \mathbb{E}\left[X_2\right]=m_2, \mathbb{P}\left(X_1 \in\left[a_1, b_1\right]\right)=95 \%$ $\mathbb{P}\left(X_2 \in\left[a_2, b_2\right]\right) \geq 95 \%$. We are interested in the sum $Y:=X_1+X_2$. We know that the mean behaves nicely i. e. $\mathbb{E}[Y]=m_1+m_2$. What would be the best way to derive confidence intervals for $Y$ by computational means? What would be the statistical assumptions for each approach be?

On a side note. In my setting, I have many $X_1, \ldots, X_n$. The confidence intervals are symmetric around the mean.

I have thought about assuming gaussian $X_1, \ldots, X_n$ and then reverse-engineering the standard deviation for bootstrap or simply sum of standard deviation or something like that, but did not make it work.

Let $X_1, X_2$ be independent (not necessarily identically distributed) random variables. Assume we have estimates for the means $m_1, m_2$ and confidence internals $\left[a_1, b_1\right],\left\{a_2, b_2\right]$ i.e. $\mathbb{E}\left[X_1\right]=m_1, \mathbb{E}\left[X_2\right]=m_2, \mathbb{P}\left(X_1 \in\left[a_1, b_1\right]\right) \geq 95 \%$ $\mathbb{P}\left(X_2 \in\left[a_2, b_2\right]\right) \geq 95 \%$. We are interested in the sum $Y:=X_1+X_2$. We know that the mean behaves nicely i. e. $\mathbb{E}[Y]=m_1+m_2$. What would be the best way to derive confidence intervals for $Y$ by computational means? What would be the statistical assumptions for each approach be?

On a side note. In my setting, I have many $X_1, \ldots, X_n$. The confidence intervals are symmetric around the mean.

I have thought about assuming gaussian $X_1, \ldots, X_n$ and then reverse-engineering the standard deviation for bootstrap or simply sum of standard deviation or something like that, but did not make it work.