Bounty: 50
Say, I have a Poisson process which was measured $N$ times, and each measurement produced $k_i$ value.
Also, $k_i$ are events that I have to detect and my detection probability is $p$. In fact, I detect $widetilde{k}_i$ which correlates with $k_i$ via $p$.
Knowing, the number of the detected events $widetilde{k}_i$, the constant probability $p$, and the constant measurement time $T$, how I find the confidence interval for estimation of Poisson’s $lambda$?