This is a true story.
In 1898, the Prussian army wanted to verify if the deaths of soldiers by a horse kick was random or not.
During 20 years, 10 units where observed and 122 deaths occurred. This provided 200 samples of number of years (per army unit) that had 0 death, 1 death, 2 death etc.
The expectation was that 𝛌=0.61 of soldiers will be kicked to death during one year.
The idea on how to verify if the deaths are random or not is to divide the time to n small time segments, and conduct a Bernoulli test on each segment with a coin with probability p which will have the observed expectation 𝛌 (which is constant, not matter what the value of n). This means that for each small period – we ask whether or not a soldier will be killed in that period of time.
The Binomininal probability is then,
The expectation for Binomial distribution is np and so np = 𝛌 i.e, p = 𝛌/n .
if n goes to infinity we get :
i.e., the Poisson distribution.
Now, we can evaluate if the Poisson distribution correctly predict the observer Prussian’s soldiers death in each year by calculating the histogram of number of deaths by horse kick i.e, we will count how many years 0 soldiers died, 1 soldiers died etc.
Here are the Purssian’s data set compared to Poisson distribution (taken from the very good introduction to Poisson Distirbution from the University of Massachusetts)
Where p is the Poisson probability, E is the number of years-army-unit that had K number of deaths for K=0,..,6 (E was caluclated as E = p * 200) and A is the actual number of years-army-unit that had K deaths.
It can be seen that the results match and so this confirm that the soldier’s death was random and there is no much one can do to prevent it.
Of course, this idea can be used for calculating “success” not just on expectation over time but also for known expectation for “success” in geographical area e.g., the chance to find oil in Pacific ocean etc.
Here are some graphs of Poisson’s probability mass function (taken from Wikipedia)
Attributes of the Poisson distributions:
- It has single parameter 𝛌 for the expectation over a period of time or space
- It is not symmetric: it is not defined for negative integers only for all the positive integers
- it’s maximum is reached on the expectation value 𝛌 (same as the Binominal and Normal distributions).
To summarize, the Poisson distribution is the limit of Binominal distribution with a fix expectation.