# Expected time to wait for no events to occur within a sliding window assuming Poissson process

I wish to model the following:

I am maintaining a sliding window (history) of 10 samples of the output of a signal detector.

I model the probability of a detection failure (i.e absence of signal) as a Poisson process with a probability of (say) 0.2, so an average rate of 2 detection failures in any window of 10 samples, assuming the failures are independent.

I wish to model the expected time & pdf, from the point where there are 10 consecutive detection failures, to the point where there are no detection failures present within the sliding window.

• Welcome to CV. As I understand the problem, you have a Bernoulli i.i.d sequence $X_n$ for $n=1$, $2$, $\dots$ where $X_n=0$ is "no failure" and $X_n = 1$ is "failure". Then you are interested by the waiting time for the first run of ones with length $10$. Even if the time $n$ is to be replaced by a Poisson arrival $T_n$ (for a process independent of the $X_i$), solving the former problem will be needed. You can use the keyword "runs". – Yves Feb 4 at 15:40