# Infer (supposed) Poissonian probability from data

Suppose to count the drops of rain in a square meter in 15 seconds, producing 16 observations: 40, 20, 24, 15, 23, 12, 39, 26, 29, 33, 16, 36, 17, 32, 40, 15.

What is the probability of counting 28 drops in a following measure?

By simple computations we can find the mean and sigma of this dataset, namely $$\mu = 26.06$$ and $$\sigma = 9.43$$.

My issue is the following: from the definition of the problem, I would expect the data to be distributed according to a Poisson distribution, being it a count process. By plotting the dataset in a histogram, however, it is clear that the distribution is far from a Poisson distribution or even a Gaussian.

I could only come up with two possible solutions to this problem:

1. I nonetheless suppose the dataset is Poissonian (and we would see a Poissonian appear with more measurements) and we can estimate $$\lambda$$ by taking the average value between $$\mu$$ and $$\sigma$$, since they should both converge to $$\lambda$$ for infinite observations. If this approach is true, the dataset is generated from

$$P(x) = \frac{\lambda^x}{x!}e^{-\lambda}$$

with $$\lambda = \frac{\mu+\sigma}{2} = 16.24$$. This approach would give a negligible probability and I would say is clearly wrong.

1. I forget all my assumptions about the process and only believe the data. In this case I would just find the probability from my histogram. by taking e.g. 6 bins with extremes (10-15|15-20|20-25|25-30|30-35|35-40) (right extreme included) I find the histogram (3, 3, 2, 2, 2, 4). The subset (26, 27, 28, 29, 30) appears with probability $$\frac{obs_{(25, 30]}}{obs_{tot}} = \frac{2}{16} = 12.5\%$$ and assuming all the numbers uniformly distributed in this subset, since there are 5 numbers, I get $$P(x=28) = \frac{12.5\%}{5} = 2.5\%$$.

The second result seems much more reasonable but is totally heuristic and obviously depends on the histogram chosen, although I am pretty sure we wouldn't get completely different results and this feels somewhat right.

Question

Is my second approach reasonable? What would be the "smart" way to infer this probability?

• Perhaps you could set up a hierarchical Bayesian model with a wide prior on the Poisson, but whose mean would be 26.06 and generate umpteen posterior predictive observations and use that as the basis of your estimate. Mar 13, 2022 at 19:10
• Could you please edit the question to include the source of this example? If it's from homework or a textbook there probably is some associated point that the author is trying to make; knowing details of the source might help point you in the right direction. If this is a homework or self-study question, please replace one of your tags with self-study and read its tag info.
– EdM
Mar 13, 2022 at 19:56
• The source is an old PhD admission test: phys.uniroma1.it/fisica/sites/default/files/allegati/29.pdf but it's in Italian. Mar 13, 2022 at 19:58
• Conventionally $\mu$ and $\sigma$ are population mean and standard deviation. The corresponding observed sample quantities are more commonly written as $\bar{x}$ and $s$ (or sometimes $s_n$ if the Bessel correction is not applied) Mar 13, 2022 at 23:08