I have a question about a homework exercise of my statistics class, concerning poisson processes:
Consider a poisson process with rate $\lambda$. An event is detected only with the prohability $p$. How is the number of detected/missed events per timeframe distributed?
I know that in a possion process the number of events per timeframe $n$ is distributed with the poisson distribution $P_{\lambda}(n) = \frac{\lambda^n}{n!}e^{-\lambda}$. Now the detection of the events itself is a bernoulli experiment and therefore follows a binomial distribution $Bi(k|n,p) = {n\choose k} p^k (1-p)^{n-k}$ with $n$ beeing the number of experiments and $k$ the number of successes. So in this case, $n$ is again the number of events per time frame and $k$ is the number if those that are detected. I could also approximate the binomial process by another possion process with $\tilde{\lambda} = np$. But now my question: How do I get the final distribution of detected events per time frame?