Radioactive particles are emitted randomly over time from a source at an average rate of per second. In $n$ time periods of varying lengths $t_1,t_2,\dots,t_n$ (seconds), the numbers of particles emitted (as determined by an automatic counter) were $y_1,y_1,\dots,y_n$ respectively.
(b) Suppose that instead of knowing the $y_i$s, we know only whether or not there was one or more particles emitted in each time interval. Making a suitable assumption, give the likelihood function for $\theta$ based on these data, and describe how you can find the maximum likelihood estimate of $\theta$.
What would be a "suitable assumption"? The assumption I made was that each interval contains $1$ event if it has an event and no events if it does not have any event. This seems to be a trivializing and unsuitable assumption. What would the correct one be then?