Estimating Poisson $\theta$ only from which percentage of intervals have events 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?

Suitable assumptions might be that $t_i$ are known and that this is a Poisson process with a uniform rate, and that the time intervals don't overlap.
Let
$$
X_i = \left.\begin{cases} 1 & \text{if at least one emission in the $i$th time period}, \\
0 & \text{otherwise}, \end{cases}\right\} = \begin{cases} 1 & \text{if }y_i\ge 1, \\
0 & \text{if }y_i=0. \end{cases}
$$
Then $\Pr(X_i=0)= e^{-\theta t_i}$ and $\Pr(X_i=1)=1-e^{-\theta t_i}$.  So
$$
L(\theta) = \prod_{i=1}^n (e^{-\theta t_i})^{1-X_i} (1-e^{-\theta t_i})^{X_i}
$$
This allows a lot of simplification in the case where $t_1=\cdots=t_n$.
