I heard that the number of meteorites of the given size that hit the Earth follows the Poisson distribution. I am wondering how to estimate the Poisson parameter $\lambda$ and its 95% c.i. if I have only one measurement for a given meteorite size.
Let's propose that we are considering 100 year period of time. For that period of time, we observed $k=5$ meteorites with a diameter greater than 1 meter.
Two ideas came into my mind about how we can estimate $\lambda$.
A. First is to use the formalism described here: Determining confidence interval with one observation (for Poisson distribution) So we will get that (for 95%): $$ \hat{\lambda}_\mathrm{MLE} = 4\\ 95\%\,\, \mathrm{c.i.}: (k+2-1.96\sqrt{k+1},k+2+1.96\sqrt{k+1}=\\ (1.6,10.4) $$ B. The second idea is to fix $k=4$ and calculate the probability that true $\lambda$ will appear in interval from $\lambda_1$ to $\lambda_2$: $$P(\lambda_1<\lambda<\lambda_2)= \int_{\lambda_1}^{\lambda_2}f(k=4,\lambda)d\lambda $$ where $f$ is Poisson PMF. From here, taking $\lambda$ in range from 0.1 to 10 with step 0.1 I can estimate the median value and 95% c.i., which will be: $$\lambda_\mathrm{median} = 4.6\\ 95\%\,\, \mathrm{c.i.}: (1.9,9) $$ My question is: are both estimations (A and B) okay?
An additional question is can I use the same estimations if I consider some big number of small meteorites ($k=50$) or $k=0$ if we consider super-big meteorites.
