I'm not sure how to solve the following: $X_1,\dotsc,X_n \sim U(0,2p)$
What is the confidence interval to the probability that $X>1$?
I've managed to find the CI for $p$ using the estimator of $p$, but I'm clueless regarding the above
a similar follow-up: if $X_1,\dotsc, X_n \sim\text{Pois}(\lambda)$, how to construct the CI of the probability of $X$ being $0$?
Details of Comment: From what you say, I assume you have lower and upper bounds for a 2-sided 95% CI for $p$. Say that these bounds are $L$ and $U$. Then $$P(L < p < U) = .95.$$ Manipulate the inequality $L < p < U$ to get $2L < 2p < 2U,$ then $-1/2L < - 1/2p < -1/2U$ and finally $$P(1-1/2L < 1 - 1/2p = \tau < 1-1/2U) = .95,$$ which would give the 95% CI $(1-1/2L, 1 - 2/U)$ for $\tau = P(X > 1) = 1 - 1/2p.$
All of this depends on what you have told me. I have not checked the correctness of that. I'm just filling in details of my comment as you requested.
This is not fundamentally different (or more difficult) than manipulating the probability statement $$P(-1.96 < Z = \frac{\bar X - \mu}{\sigma/\sqrt{n}} < 1.96) = .95$$ to get the CI $\bar X \pm 1.96 \sigma/\sqrt{n}$ for $\mu$ when sampling from normal data with $\sigma$ known.