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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?

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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 $t_1=\cdots=t_n$.

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Perhaps, the arrival of particles are independent of each other.

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